# Optimal Real-Time Bidding Frameworks Discussion

Weinan Zhang†, Kan Ren‡, Jun Wang†
†University College London, ‡Shanghai Jiao Tong University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This note is a complementary material for the solution of optimal real-time
bidding function in paper [1], where the estimated cost is taken as the bid
price, i.e., the upper bound of the true cost. Here we discuss a more general
bid optimisation framework with various utility and cost functions.

##  1 Preliminaries

Define the bidding function as $b(r)$ which returns the bid price given the
estimated click-through rate (CTR) $r$ of an ad impression.

Winning probability. Given the market price distribution $p_{z}(z)$ and the
bid price $b$, the probability of winning the auction is

$\displaystyle w(b)=\int_{0}^{b}p_{z}(z)dz.$ (1)

If the bid wins the auction, the ad is displayed and the utility and cost of
this ad impression are then observed.

Utility. The utility function given the CTR is denoted as $u(r)$. The specific
form of $u(r)$ depends on the campaign KPI. For example, if the KPI is click
number, then

$\displaystyle u_{\text{clk}}(r)=r.$ (2)

If the KPI is campaign’s net profit, with the advertiser’s true value on each
click is $v$, then

$\displaystyle u_{\text{rev}}(r)=vr-z.$ (3)

Cost. The expected cost if win given a bid $b$ is denoted as $c(b)$. In RTB ad
market we have first price auctions

$\displaystyle c_{1}(b)=b,$ (4)

and second price auctions

$\displaystyle
c_{2}(b)=\frac{\int_{0}^{b}zp_{z}(z)dz}{\int_{0}^{b}p_{z}(z)dz}.$ (5)

In [1], we used $c_{1}(b)$ to model the upper bound of the second price
auctions with possible soft floor prices. Here we shall first use the abstract
cost function $c(b)$ in the framework and then specify the implementation of
the cost function in specific tasks.

##  2 Bid Optimisation without Budget Constraint

For non-budget bid optimisation, only $u_{\text{rev}}(r)$ utility function is
meaningful111If there is no cost-related factors in the utility, one will bid
as high as possible to win every impression as there is no budget constraint.

$\displaystyle U_{\text{rev}}(b(\cdot))$
$\displaystyle=T\int_{r}\int_{z=0}^{b(r)}(vr-z)p_{z}(z)dz\cdot p_{r}(r)dr.$
(6)

Take the gradient of net profit $U_{\text{rev}}(b(\cdot))$ w.r.t. the bidding
function $b(r)$ and set it to 0,

$\displaystyle\frac{\partial U_{\text{rev}}(b(\cdot))}{\partial
b(\cdot)}=(vr-b(r))\cdot p_{z}(b(r))\cdot p_{r}(r)=0,$ (7)

which derives

$\displaystyle b(r)=vr$ (8)

i.e., the optimal bid price is set as the impression value. Thus the truth-
telling bidding is the optimal strategy when there is no budget considered.

##  3 Bid Optimisation with Budget Constraint

Assuming we want to find the optimal bidding function $b()$ to maximise the
campaign-level KPI $r$ across its auctions over the lifetime with total bid
requests volume $T$ and budget $B$.

$\displaystyle\max_{b()}~{}~{}$ $\displaystyle T\int_{r}u(r)w(b(r))p_{r}(r)dr$
(9) subject to $\displaystyle T\int_{r}c(b(r))w(b(r))p_{r}(r)dr=B$

The Lagrangian of the optimisation problem Eq. (9) is

$\displaystyle\mathcal{L}(b(r),\lambda)=$
$\displaystyle\int_{r}u(r)w(b(r))p_{r}(r)dr-\lambda\int_{r}c(b(r))w(b(r))p_{r}(r)dr+\frac{\lambda
B}{T},$ (10)

where $\lambda$ is the Lagrangian multiplier. Based on calculus of variations,
the Euler-Lagrange condition of $b(r)$ is

$\displaystyle\frac{\mathcal{L}(b(r),\lambda)}{\partial b(r)}=0,$ (11)

which is

$\displaystyle u(r)p_{r}(r)\frac{\partial w(b(r))}{\partial b(r)}-\lambda
p_{r}$ $\displaystyle(r)\Big{[}\frac{\partial c(b(r))}{\partial
b(r)}w(b(r))+c(b(r))\frac{\partial w(b(r))}{\partial b(r)}\Big{]}=0$ (12)
$\displaystyle\Rightarrow~{}~{}~{}\lambda\frac{\partial c(b(r))}{\partial
b(r)}w(b(r))=$ $\displaystyle\Big{[}u(r)-\lambda c(b(r))\Big{]}\frac{\partial
w(b(r))}{\partial b(r)}.$ (13)

Eq. (13) is a general condition of the optimal bidding function, where the
specific implementations of winning function $w(b)$, utility function $u(r)$
and cost function $c(b)$ are needed to derive the corresponding form of
optimal bidding function.

###  3.1 First-Price Auction

With cost function $c_{1}(b)$ in Eq. (4), Eq. (13) is written as

$\displaystyle\lambda\int_{0}^{b(r)}p_{z}(z)dz=(u(r)-\lambda b(r))\cdot
p_{z}(b(r))$ (14)

which still depends on the detailed form of market price distribution
$p_{z}(z)$ to solve the $b()$. In [1], the authors tried an implementation:

$\displaystyle p_{z}(z)$ $\displaystyle=\frac{l}{(l+z)^{2}},$ (15)
$\displaystyle\Rightarrow~{}~{}w(b)$ $\displaystyle=\frac{b}{b+l}.$ (16)

Taking Eq. (16) and click utility Eq. (2) into Eq. (14) we have

$\displaystyle\lambda\frac{b(r)}{b(r)+l}$ $\displaystyle=(u(r)-\lambda
b(r))\cdot\frac{b(r)}{(b(r)+l)^{2}}$ (17) $\displaystyle\Rightarrow~{}~{}b(r)$
$\displaystyle=\sqrt{\frac{u(r)l}{\lambda}+l^{2}}-l,$ (18)

which is the result in [1].

###  3.2 Second-Price Auction

Taking the definition of winning function Eq. (1) and the second-price cost
function Eq. (5) into Eq. (13), we have

$\displaystyle\lambda\frac{b(r)p_{z}(b(r))w(b(r))-p_{z}(b(r))\int_{0}^{b}zp_{z}(z)dz}{w(b(r))^{2}}w(b(r))=(u(r)-\lambda
c(b(r)))p_{z}(b(r))$ (19) $\displaystyle\Rightarrow~{}~{}~{}$
$\displaystyle\lambda(b(r)-c(b(r)))=u(r)-\lambda c(b(r))$ (20)
$\displaystyle\Rightarrow~{}~{}~{}$ $\displaystyle b(r)=\frac{u(r)}{\lambda},$
(21)

where we can see the optimal bidding function is linear w.r.t. CTR $r$.

The solution of $\lambda$ is obtained by taking Eq. (21) into the constraint

$\displaystyle T\int_{r}c(b(r))w(b(r))p_{r}(r)dr=B,$ (22)

which is rewritten as

$\displaystyle\int_{r}c\Big{(}\frac{u(r)}{\lambda}\Big{)}w\Big{(}\frac{u(r)}{\lambda}\Big{)}p_{r}(r)dr=\frac{B}{T}.$
(23)

We can see that essentially the solution of $\lambda$ makes the equation
between the expected cost and the budget. Furthermore, as both $w(r/\lambda)$
and $c(r/\lambda)$ monotonously decrease as $\lambda$ increases, it is quite
easy to find a numeric solution of $\lambda$.

##  4 Discussions

The implementation of the cost constraint in Eq. (9) needs careful modelling
based on the data. In [1] the authors used the cost upper bound, i.e., the bid
price, to control the cost and let the total value of cost upper bound be the
campaign budget. Here if we implement the expected cost in second price
auction Eq. (5), the cost constraint in Eq. (9) might not be controlled by the
budget. With about 50% probability, the budget will be exhausted in advance,
which is not expected in practice.

We can easily find in the first-price bidding function Eq. (18), the bid price
is jointly determined by utility function $u(r)$, $\lambda$ and market
parameter $l$. Specifically, the bid price is monotonic increasing w.r.t.
utility while decreasing w.r.t. $\lambda$. Moreover, different value settings
for parameter $l$ also influence the final bid decision as is shown in [1]. As
is defined in Eq. (16), we can tune the parameter $l$ to alter the winning
probability so as to fit different market environments. In fact, we may
conclude that: the market consideration influences bid price by tuned $l$, the
budget constraint controls bid function by $\lambda$, while advertiser’s
utility expectation $u(r)$ could finally determine the final decision.

Let’s take a look at the bid strategy under second-price auctions. In Eq.
(21), the bid price is mainly determined by utility $u(r)$. However, the bid
strategy constructs a bridge between utility and budget consideration by
$\lambda$. And let us move our attention onto Eq. (23) and we can find that
$\lambda$ is also controlled by this equation which takes both estimated CTR
$r$ and winning probability $w(\cdot)$ into consideration. But the latter two
factors have low effects on bid strategy through $\lambda$.

From the discussion above, we may find that both bid strategies under either
auction setting are influenced by three factors: advertiser’s expected
utility, budget constraints and market information. While under first-price
auction setting, the bid strategy takes utility function and market price
together, but the strategy upon second-price auctions reflects utility more
straightly.

For the bidding function under first-price auctions, we could take more
attention on the winning probability estimation problem and consequently to
optimize the bid strategy. We also think that it could be more crucial to take
the market information into consideration in the utility function for second-
price auctions. Finally, we may take a step forward that, to coordinates the
optimization for both CTR estimation and bidding strategy, considering market
information and budget capping, to dynamically bid in real-time and real-world
environments.

## References

  * [1] W. Zhang, S. Yuan, and J. Wang.  Optimal real-time bidding for display advertising.  In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 1077–1086. ACM, 2014.


11institutetext: L3S Research Center, Leibniz University Hannover, Hanover,
Germany
11email: {tai<EMAIL_ADDRESS>
22institutetext: Institute for Didactics of Mathematics and Physics, Leibniz
University Hannover, Hanover, Germany
22email<EMAIL_ADDRESS>33institutetext: Institute of Computer
Science, Free University Berlin, Berlin, Germany
33email<EMAIL_ADDRESS>
# Evaluation of group fairness measures in student performance prediction
problems

Tai Le Quy 11 0000-0001-8512-5854 Thi Huyen Nguyen 11 0000-0001-8195-716X
Gunnar Friege 22 0000-0003-3878-9230 Eirini Ntoutsi 33 0000-0001-5729-1003

###### Abstract

Predicting students’ academic performance is one of the key tasks of
educational data mining (EDM). Traditionally, the high forecasting quality of
such models was deemed critical. More recently, the issues of fairness and
discrimination w.r.t. protected attributes, such as gender or race, have
gained attention. Although there are several fairness-aware learning
approaches in EDM, a comparative evaluation of these measures is still
missing. In this paper, we evaluate different group fairness measures for
student performance prediction problems on various educational datasets and
fairness-aware learning models. Our study shows that the choice of the
fairness measure is important, likewise for the choice of the grade threshold.

machine learning educational data mining

###### Keywords:

fairness fairness measures student performance prediction

##  1 Introduction

Educational data mining (EDM) applies data mining, artificial intelligence
(AI), and machine learning (ML) to improve academic experiences. In recent
years, AI-infused technologies have been widely studied and deployed by many
educational institutions [3, 19]. One of the most important tasks in EDM that
attract great attention is student performance prediction. The early
estimation of student learning outcomes can help detect and notify students at
risk of academic failure. Besides, it supports institutional administrators in
identifying key factors affecting students’ grades and providing suitable
interventions for outcome improvement. The performance prediction process
relies on historical academic records and trains ML algorithms on labeled data
to predict students’ performance. Various datasets [37, 11, 26] and approaches
[16, 24, 41] have been proposed for the purpose. With the widespread use and
benefits of AI systems, fairness has become a crucial criterion in designing
such systems.

Non-discriminative ML models have been a topic of increasing importance and
growing momentum in education. Despite advances and superior accuracy of
recent ML models, some studies have shown that ML-based decisions can be
biased to protected attributes such as gender or race due to historical
discrimination embedded in the data [28, 32]. Endeavoring to reduce biases is
important and decisive in the applicability of an ML model in education. As an
example, a recent study has proposed approaches that aim at predicting
calculated grades of students in England as a replacement for actual grades
due to the cancellation of exams during COVID-19 [5]. However, the proposal
could not be applied as a consequence of some exposed historical biases.

A large variety of fairness measures have been introduced in ML area. However,
choosing proper measures can be cumbersome due to the dependence of fairness
on context. There are more than 20 different fairness measures introduced in
the computer science research area [36, 28]. In fact, no metric is universal
and fits all circumstances [15, 36, 28]. Model developers should explore
various fairness measures to decide the most appropriate notions for the
context. Fairness is a fundamental concept of education, whereby all students
must have an equal opportunity in study or be treated fairly regardless of
their household income, assets, gender, or race [29]. Fairness definitions in
education, hiring, and ML in the 50-year history have been discussed in the
research of [20]. However, no previous work exists on the efficiency of
different fairness metrics and how to choose them in educational settings.

In this paper, we provide a comprehensive study to evaluate the sufficiency of
various fairness metrics in student performance prediction. We consider a
group of the most prevalent fairness notions in ML. Various experiments are
conducted on diverse educational datasets and evaluated using different
fairness metrics. Our experiments provides users a broad view of unfairness
from diverse aspects in an educational context. Besides, the results also
guide the selection of suitable fairness measures to evaluate students’ grade
predictive models. We believe our contributions are crucial to alleviate the
burden of choosing fairness measures for consideration and motivate further
studies to improve the accuracy and fairness of student performance prediction
models.

The rest of the paper is organized as follows. In Section 2, we present some
closely related work on fairness-aware ML and student performance prediction.
Section 3 describes the most popular group fairness measures in ML. Next, we
conduct quantitative evaluations of predictive models on educational datasets
and discuss the choice of suitable fairness metrics in Section 4. Finally, we
conclude the paper in Section 5.

##  2 Related work

Extensive research efforts have been conducted to provide useful insights into
students’ performance analysis and prediction [38]. Various ML models were
tested on different problem settings. Cortez et al. [11] presented an early
study to predict the grades of secondary students in Portuguese and
Mathematics classes. Their results showed that good predictive accuracy could
be achieved when previous school period grades are available. Similarly,
Berhanu et al. [7] employed Decision Tree to predict students’ performance
using the agriculture college dataset. Some studies [41, 25] proposed diverse
approaches to forecast students’ grades in higher education. Besides, many
other studies were reviewed in multiple surveys [33, 34, 31, 33, 1]. They
pointed out the most common techniques such as Decision Tree, Naive Bayes,
Support Vector Machines, and neural networks and dominant factors impacting
predictive outcomes (i.e., Cumulative Grade Point Average, previous grades,
classroom attendance, etc.).

There are more than 20 fairness notions introduced for classification [36,
28]. One of the most well-known fairness measure is demographic parity, so-
called statistical parity. It requires an equal probability of positive
predictions in protected and non-protected groups. However, Dwork et al. [13]
argued that the metric fails to ensure individual fairness. To avoid this,
Hardt et al. [18] proposed equalized odds metric. It measures whether a
classifier predicts labels equally well for all values of attributes. Besides,
many other popular metrics were introduced and used in fairness ML studies
such as predictive parity, predictive equality [9], treatment equality [8],
etc. Despite a substantial number of fairness measures, there is no metric
that fits all circumstances [28, 36].

Following the evolution of fairness measures, recent studies have attempted to
evaluate fairness in an educational context [17, 39, 22]. Anderson et al. [6]
conducted two post-hoc fairness assessments for existing student graduation
prediction models. Renzhe et al. [39] studied different combinations of
student data sources for building highly predictive and fair models for
predictions of college success. Jiang et al. [23] proposed several strategies
to mitigate bias in the LSTM grade prediction model. They report experimental
results on the true positive rate (TPR), true negative rate (TNR), and
accuracy.

##  3 Fairness measures

Table 1: An overview of group fairness measures Measures | Proposed by | Published year | #Citations
---|---|---|---
Statistical parity | [13] | 2012 | 2,367
Equal opportunity | [18] | 2016 | 2,575
Equalized odds | [18] | 2016 | 2,575
Predictive parity | [9] | 2017 | 1,430
Predictive equality | [10] | 2017 | 878
Treatment equality | [8] | 2018 | 626
Absolute Between-ROC Area | [17] | 2019 | 84

This section presents the most prevalent group fairness notions used in ML.
The list of notions111The number of citations is reported by Google Scholar on
$1^{st}$ August 2022. is summarized in Table 1. To simplify, we consider the
student performance prediction problem as a binary classification task, which
is formalized as below:

Let $\mathcal{D}$ be a binary classification dataset with class attribute
$Y=\\{+,-\\}$, e.g., $Y=\\{pass,fail\\}$. $S$ is a binary protected attribute,
$S\in\\{s,\overline{s}\\}$, e.g., $S$ = “gender”, $S\in\\{female,male\\}$. In
which, s is the discriminated group (_protected group_), e.g., “female”, and
$\overline{s}$ is the non-discriminated group (_non-protected group_), e.g.,
“male”. The predicted outcome is denoted as $\hat{Y}=\\{+,-\\}$. The notions
$s_{+}$ ($s_{-}$), $\overline{s}_{+}$ ($\overline{s}_{-}$) are used to denote
the protected and non-protected groups for the positive (negative,
respectively) class.

We use a confusion matrix (Fig. 1) to demonstrate the group fairness measures
with an example of a dataset with 100 instances, class $Y=\\{pass,fail\\}$.
The protected attribute is “gender”, and the protected group is “female”; the
distribution of “female”:“male” is 46:54. Examples of fairness measures in the
following sub-sections are computed based on this confusion matrix.

Actual class
|  | Predicted class
---|---|---
| Positive + | Negative -
Positive + |  True Positive (TP) $TP_{prot}+TP_{non-prot}$ 70 (32:38) |  False Negative (FN) $FN_{prot}+FN_{non-prot}$ 10 (4:6)
Negative - |  False Positive (FP) $FP_{prot}+FP_{non-prot}$ 9 (4:5) |  True Negative (TN) $TN_{prot}+TN_{non-prot}$ 11 (6:5)
Figure 1: The confusion matrix with an example

###  3.1 Statistical parity

_Statistical parity_ (denoted as _SP_) is a well-known group fairness measure
[13], whereby the output of any classifier satisfies statistical parity if the
difference (bias) in the predicted outcome ($\hat{Y}$) between any two groups
under study (i.e., $s$ and $\overline{s}$) is up to a predefined tolerance
threshold $\epsilon$:

$P(\hat{Y}|S=s)-P(\hat{Y}|S=\overline{s})\leq\epsilon.$ (1)

We use the violation of statistical parity [35, 40, 27] to measure the bias of
a classifier:

$SP=P(\hat{Y}=+|S=\overline{s})-P(\hat{Y}=+|S=s).$ (2)

The value range: $SP\in[-1,1]$, with $SP=0$ indicating no discrimination,
$SP\in(0,1]$ designating that the protected group is discriminated, and
$SP\in[-1,0)$ standing for _reverse discrimination_ (the non-protected group
is discriminated). In our example (Fig. 1), this measure shows the proportion
of “pass” students between the two demographic subgroups.
$SP=\dfrac{38+6}{54}-\dfrac{32+4}{46}\approx 0.0322$.

###  3.2 Equal opportunity

_Equal opportunity_ (denoted as _EO_) is proposed by Hardt et al. [18],
whereby a binary predicted outcome $\hat{Y}$ satisfies equal opportunity
w.r.t. the protected attribute $S$ and the class attribute $Y$ if:

$P(\hat{Y}=+|S=s,Y=+)=P(\hat{Y}=+|S=\overline{s},Y=+).$ (3)

In other words, the protected and non-protected groups should have equal true
positive rates (TPR) [28, 36], $TPR=\dfrac{TP}{TP+FN}$ (i.e., the classifier
should give similar results for students of both genders with actual “pass”
class). A classifier with equal false negative rates (FNR),
$FNR=\dfrac{FN}{TP+FN}$, will also have equal TPR [36]. The equal opportunity
can be measured by:

$EO=|P(\hat{Y}=-|Y=+,S=\overline{s})-P(\hat{Y}=-|Y=+,S=s)|.$ (4)

The value range: $EO\in[0,1]$; with 0 standing no discrimination and 1
indicating maximum discrimination. In our example,
$EO=\mid\dfrac{38}{38+6}-\dfrac{32}{32+4}\mid\approx 0.0253$.

###  3.3 Equalized odds

A predictor $\hat{Y}$ is satisfied _equalized odds_ (denoted as EOd) w.r.t.
the protected attribute $S$ and class label $Y$, if “$\hat{Y}$ and $S$ are
independent conditional on $Y$” [18]. Specifically, predicted true positive
and false positive probabilities should be the same between male and female
student groups.

$P(\hat{Y}=+|S=s,Y=y)=P(\hat{Y}=+|S=\overline{s},Y=y),\phantom{aaa}y\in\\{+,-\\}.$
(5)

Therefrom, we can measure the equalized odds as the following [27, 21]:

$EOd=\sum_{y\in\\{+,-\\}}|P(\hat{Y}=+|S=s,Y=y)-P(\hat{Y}=+|S=\overline{s},Y=y)|.$
(6)

The value range: $EOd\in[0,2]$; with 0 standing for no discrimination and 2
indicating the maximum discrimination. In our example,
$EOd=|\dfrac{32}{32+4}-\dfrac{38}{38+6}|+|\dfrac{4}{4+6}-\dfrac{5}{5+5}|\approx
0.1253$.

###  3.4 Predictive parity

_Predictive parity_ [9] (denoted as _PP_) is satisfied if both protected and
non-protected groups have an equal positive predictive value (PPV) or
_Precision_ , $PPV=\dfrac{TP}{TP+FP}$, i.e., the probability of a student
predicted to “pass” actually having “pass” class should be the same, for both
male and female students.

$P(Y=+|\hat{Y}=+,S=s)=P(Y=+|\hat{Y}=+,S=\overline{s}).$ (7)

Therefore, we report the predictive parity measure as:

$PP=|P(Y=+|\hat{Y}=+,S=s)-P(Y=+|\hat{Y}=+,S=\overline{s})|.$ (8)

where $PP\in[0,1]$, with 0 standing for no discrimination and 1 indicating the
maximum discrimination. $PP=\dfrac{32}{32+4}-\dfrac{38}{38+5}\approx 0.0052$,
in our example.

###  3.5 Predictive equality

Predictive equality [10] (denoted as _PE_), also referred as false positive
error (FPR) rate balance [9] ($FPR=\dfrac{FP}{TN+FP}$), aims to the equality
of decision’s accuracy across the protected and non-protected groups. In
detail, the probability of students with an actual “fail” class being
incorrectly assigned to the “pass” class should be the same for both male and
female students.

$P(\hat{Y}=+|Y=-,S=s)=P(\hat{Y}=+|Y=-,S=\overline{s}).$ (9)

In practice, researchers report predictive equality measure by the difference
of $FPRs$ [21]:

$PE=|P(\hat{Y}=+|Y=-,S=s)-P(\hat{Y}=+|Y=-,S=\overline{s})|.$ (10)

The value range: $PE\in[0,1]$, 0 and 1 indicate no discrimination and maximum
discrimination, respectively. $PE=|\dfrac{4}{6+4}-\dfrac{5}{5+5}|=0.1$, in our
example.

###  3.6 Treatment equality

_Treatment equality_ [8] (denoted as _TE_) is satisfied if the ratio of false
negatives and false positives is the same for both protected and non-protected
groups.

$\frac{FN_{prot.}}{FP_{prot.}}=\frac{FN_{non-prot.}}{FP_{non-prot.}}.$ (11)

In our paper, we report the treatment equality by the difference between two
ratios described in Eq.11. The metric becomes unbounded if $FP_{prot.}$ or
$FP_{non-prot.}$ is
zero222https://docs.aws.amazon.com/sagemaker/latest/dg/clarify-post-training-
bias-metric-te.html. In our example, $TE=-0.2$, because the ratios of FN and
FP are 1 and 1.2 for female and male groups, respectively.

###  3.7 Absolute Between-ROC Area

_Absolute Between-ROC Area (ABROCA)_ [17] is based on the Receiver Operating
Characteristics (ROC) curve. It measures the divergence between the protected
($ROC_{s}$) and non-protected group ($ROC_{\overline{s}}$) curves across all
possible thresholds $t\in[0,1]$ of FPR and TPR. The absolute difference
between the two curves is measured to capture the case that the curves may
cross each other.

$\int_{0}^{1}\mid ROC_{s}(t)-ROC_{\overline{s}}(t)\mid\,dt.$ (12)

The value range: $ABROCA\in[0,1]$. The lower value indicates a lower
difference in the predictions between the two groups and, therefore, a fairer
model.

##  4 Evaluation

In this section, we evaluate the performances of predictive models w.r.t.
accuracy and fairness measures on five datasets and investigate the effect of
choosing grade threshold on fairness measures.

###  4.1 Datasets

We evaluate the fairness measures on popular educational datasets [27, 30,
38], which are summarized in Table 2. All datasets are imbalanced, as shown in
the imbalance ratio (IR) column.

Table 2: An overview of educational datasets

Datasets #Instances #Instances (cleaned) #Attributes Protected attribute Class
label IR (+:-) Law school 20,798 20,798 12 Race Pass the bar exam 8.07:1 PISA
5,233 3,404 24 Gender Reading score 1.35:1 Studden academics 131 131 22 Gender
ESP 3.70:1 Student performance 649 649 33 Gender Final grade 5.49:1 xAPI-Edu-
Data 480 480 17 Gender Grade level 2.78:1

Law school. The Law school
dataset333https://github.com/tailequy/fairness_dataset/tree/main/Law_school
contains the law school admission records from 163 law schools in the US in
1991. The target is to predict whether a candidate would pass the bar exam or
not. The protected attribute is “race” = $\\{white,non-white\\}$, where _“non-
white”_ is the protected group.

PISA dataset. The PISA dataset444https://www.kaggle.com/econdata/pisa-test-
scores contains information on the performance of American students [14]
taking the exam in 2009 from the Program for International Student Assessment
(PISA). The grade threshold (“readingScore” attribute) is chosen at 500 to
compute the class label = $\\{low,high\\}$ since the mean reading score is
497.6. The experiments are performed on the cleaned version of this dataset
with 3,404 instances after removing missing values.

Student academics performance dataset. The student academics performance
dataset555https://archive.ics.uci.edu/ml/datasets/Student+Academics+Performance
[19] consists of socio-economic, demographic, and academic information of
students from three different colleges in India with 22 attributes. The class
label is ESP (end semester percentage). In this paper, we encode class label
as a binary attribute with values {“pass”,“good-and-higher”}, where _“good-
and-higher”_ is a positive class.

Student performance dataset. The student performance
dataset666https://archive.ics.uci.edu/ml/datasets/student+performance [11] was
collected in two Portuguese schools in 2005 - 2006. It contains 33 features
describing demographics, grades, social and school-related information of
students. “gender” is considered the protected attribute. The target is to
predict the final outcome. The class label = $\\{pass,fail\\}$ is computed
based on the final grade (attribute “G3”) as {$<$10, $\geq$10}[11, 27].

Students’ academic performance dataset (xAPI-Edu-Data). xAPI-Edu-
Data777https://www.kaggle.com/datasets/aljarah/xAPI-Edu-Data [4] contains 480
student records described by 17 attributes collected from Kalboard 360
learning management system. We encode the class label as a binary attribute as
$\\{Low,Medium-High\\}$ corresponding to {L, M or H} in the original dataset.
The positive class is _“Medium-High”_.

###  4.2 Predictive models

We select four prevalent classifiers used for student performance prediction
problems based on the survey of Xiao et al. [38], and two well-known fairness-
aware classifiers, namely Agarwal’s [2] and AdaFair [21]. In which, Agarwal’s
method reduces the fair classification to a sequence of cost-sensitive
classification problems with the lowest (empirical) error subject to the
desired constraints, and AdaFair is based on AdaBoost that further updates the
weights of the instances in each boosting round. In brief, the predictive
models are: 1) Decision Tree (DT); 2) Naive Bayes (NB); 3) Multi-layer
Perceptron (MLP); 4) Support Vector Machines (SVM); 5) Agarwal’s; 6) AdaFair.
In our experiments, we use 70% of data for training and 30% for testing
(single split). Predicted models are implemented and executed with default
parameters provided by Scikit-learn and Iosifidis et al. [21]. Agarwal’s
method is implemented in the AI Fairness 360
toolkit888https://github.com/Trusted-AI/AIF360.

###  4.3 Experimental results

Law school dataset. The results are presented in Table 3. AdaFair is the best
predictive model w.r.t. fairness measures, although its balanced accuracy is
significantly lower than that of other models. Besides, the fairness measures
show a quite large variation across the classification methods, as
demonstrated in Fig. 7-a. Furthermore, the shape and position of the ROC
curves, as visualized in Fig. 2, have been changed across the predictive
models, which indicates the change in the performance of models w.r.t. each
value in the protected attribute. Because the datasets are imbalanced, we
report the performance of predictive models on both accuracy and balanced
accuracy measures.

Table 3: Law school: performance of predictive models Measures |  aaDTaa |  aaNBaa |  aaMLPaa |  aSVMa |  aAgarwal’s | AdaFair
---|---|---|---|---|---|---
Accuracy | 0.8458 | 0.8191 | 0.9042 | 0.8926 | 0.7952 | 0.8921
Balanced accuracy | 0.6301 | 0.7784 | 0.6596 | 0.5029 | 0.5848 | 0.5
Statistical parity | 0.1999 | 0.5250 | 0.2367 | 0.0052 | 0.0326 | 0.0
Equal opportunity | 0.1557 | 0.4665 | 0.1237 | 0.0014 | 0.0202 | 0.0
Equalized odds | 0.3253 | 0.8105 | 0.5501 | 0.0169 | 0.0953 | 0.0
Predictive parity | 0.1424 | 0.0130 | 0.0754 | 0.1857 | 0.1802 | 0.1885
Predictive equality | 0.1696 | 0.3440 | 0.4265 | 0.0154 | 0.0751 | 0.0
Treatment equality | -0.0667 | 22.440 | 0.7770 | 0.0039 | -1.9676 | 0.0
ABROCA | 0.0336 | 0.0316 | 0.0336 | 0.0833 | 0.0365 | 0.0822

(a) DT

(b) NB

(c) MLP

(d) SVM

(e) Agarwal’s

(f) AdaFair

Figure 2: Law school: ABROCA slice plots

PISA dataset. The interesting point is SVM and DT show their superiority in
terms of fairness measures, although AdaFair still has very good results on
fairness metrics and accuracy (Fig. 3 and Table 4). Furthermore, fairness
measures have the least variability in this dataset, as shown in Fig. 7-b.

Table 4: PISA: performance of predictive models Measures |  aaDTaa |  aaNBaa |  aaMLPaa |  aSVMa |  aAgarwal’s | AdaFair
---|---|---|---|---|---|---
Accuracy | 0.6360 | 0.6624 | 0.6526 | 0.6096 | 0.6614 | 0.6810
Balanced accuracy | 0.6224 | 0.6379 | 0.5732 | 0.5026 | 0.6340 | 0.6130
Statistical parity | -0.0200 | -0.0316 | -0.0771 | -0.0022 | -0.0096 | -0.0573
Equal opportunity | 0.0019 | 0.0262 | 0.0330 | 0.0043 | 0.0414 | 0.0164
Equalized odds | 0.0165 | 0.0709 | 0.1398 | 0.0068 | 0.0548 | 0.0752
Predictive parity | 0.1012 | 0.0683 | 0.0826 | 0.1108 | 0.0785 | 0.0868
Predictive equality | 0.0146 | 0.0446 | 0.1067 | 0.0024 | 0.0134 | 0.0588
Treatment equality | 0.5642 | 0.3855 | -0.0251 | -0.0033 | 0.4609 | 0.0260
ABROCA | 0.0070 | 0.0330 | 0.0223 | 0.0844 | 0.0326 | 0.0216

(a) DT

(b) NB

(c) MLP

(d) SVM

(e) Agarwal’s

(f) AdaFair

Figure 3: PISA: ABROCA slice plots

Student academics performance dataset. The AdaFair outperforms other models
w.r.t. fairness measures, however, the balanced accuracy is decreased
considerably (Table 5). Besides, all fairness measures have significant
variation across predictive models (Fig. 4 and 7-c).

Table 5: Student academics: performance of predictive models Measures |  aaDTaa |  aaNBaa |  aaMLPaa |  aSVMa |  aAgarwal’s | AdaFair
---|---|---|---|---|---|---
Accuracy | 0.7750 | 0.8750 | 0.8750 | 0.9250 | 0.8750 | 0.9
Balanced accuracy | 0.6528 | 0.8194 | 0.8194 | 0.6250 | 0.8194 | 0.5
Statistical parity | -0.1278 | -0.1328 | -0.1328 | 0.0526 | 0.0677 | 0.0
Equal opportunity | 0.1455 | 0.0991 | 0.2105 | 0.0 | 0.0123 | 0.0
Equalized odds | 0.1455 | 0.5991 | 0.7105 | 0.5 | 0.5124 | 0.0
Predictive parity | 0.0042 | 0.0588 | 0.0552 | 0.0397 | 0.0556 | 0.01
Predictive equality | 0.0 | 0.5 | 0.5 | 0.5 | 0.5 | 0.0
Treatment equality | -3.0 | $N/A$ | $N/A$ | 0.0 | $N/A$ | 0.0
ABROCA | 0.0728 | 0.2059 | 0.1316 | 0.1285 | 0.0317 | 0.0372

(a) DT

(b) NB

(c) MLP

(d) SVM

(e) Agarwal’s

(f) AdaFair

Figure 4: Student academics: ABROCA slice plots

Student performance dataset. In general, all models show good accuracy
(balanced accuracy) on predicting students’ performance (Table 6). MLP and
AdaFair models fairly guarantee the fairness of results on most measures.
Besides, the values of fairness measures also do not vary significantly across
predictive models (Fig. 7-d), although the ABROCA slices are quite different
in shape (Fig. 5).

Table 6: Student performance: performance of predictive models Measures |  aaDTaa |  aaNBaa |  aaMLPaa |  aSVMa |  aAgarwal’s | AdaFair
---|---|---|---|---|---|---
Accuracy | 0.9333 | 0.8974 | 0.9077 | 0.9231 | 0.8923 | 0.9487
Balanced accuracy | 0.8639 | 0.8595 | 0.7840 | 0.7441 | 0.8565 | 0.8240
Statistical parity | -0.0382 | -0.0509 | -0.0630 | 0.0151 | -0.0209 | -0.0255
Equal opportunity | 0.0125 | 0.0174 | 0.03 | 0.0183 | 0.0176 | 0.0092
Equalized odds | 0.1316 | 0.2198 | 0.1252 | 0.3279 | 0.2200 | 0.1877
Predictive parity | 0.0456 | 0.0591 | 0.0601 | 0.0944 | 0.0577 | 0.0639
Predictive equality | 0.1190 | 0.2024 | 0.0952 | 0.3095 | 0.2024 | 0.1786
Treatment equality | 2.0 | 7.5 | 0.3333 | 0.5 | 9.75 | 0.3333
ABROCA | 0.0575 | 0.0686 | 0.0683 | 0.0231 | 0.0762 | 0.0887

(a) DT

(b) NB

(c) MLP

(d) SVM

(e) Agarwal’s

(f) AdaFair

Figure 5: Student performance: ABROCA slice plots

xAPI-Edu-Data dataset This is a surprising dataset because the traditional
classification methods show a better performance not only in terms of
accuracy/balanced accuracy measures but also w.r.t. fairness measures (Table
7). In addition, variation in the values of fairness measures across the
predictive models is not significant, as shown in Fig. 7-e, except for the
ABROCA measure with a noticeable change in the shape (Fig. 6).

Table 7: xAPI-Edu-Data: performance of predictive models Measures |  aaDTaa |  aaNBaa |  aaMLPaa |  aSVMa |  aAgarwal’s | AdaFair
---|---|---|---|---|---|---
Accuracy | 0.8333 | 0.8750 | 0.8750 | 0.8611 | 0.8681 | 0.8056
Balanced accuracy | 0.8 | 0.8970 | 0.8545 | 0.8505 | 0.8859 | 0.8162
Statistical parity | -0.1274 | -0.2608 | -0.2112 | -0.2209 | -0.2505 | -0.2292
Equal opportunity | 0.0282 | 0.0974 | 0.0654 | 0.0308 | 0.0974 | 0.0538
Equalized odds | 0.1329 | 0.1954 | 0.1262 | 0.2706 | 0.1684 | 0.3207
Predictive parity | 0.0752 | 0.0074 | 0.0654 | 0.0088 | 0.0122 | 0.0057
Predictive equality | 0.1047 | 0.0980 | 0.0608 | 0.2399 | 0.0709 | 0.2669
Treatment equality | 1.0667 | -8.0 | 0.0 | -0.2667 | -2.0 | -1.1667
ABROCA | 0.0665 | 0.0216 | 0.0263 | 0.0796 | 0.0293 | 0.1065

(a) DT

(b) NB

(c) MLP

(d) SVM

(e) Agarwal’s

(f) AdaFair

Figure 6: xAPI-Edu-Data: ABROCA slice plots.

(a) Law school

(b) PISA

(c) Student academic

(d) Student performance

(e) xAPI-Edu-Data

(f) TE measure

Figure 7: Variation of fairness measures

Regarding the _treatment equality_ measure, this measure is entirely different
from all other measures with an extensive range of values, which is visualized
in Fig. 7-f999We use the abbreviations of the fairness measures and datasets
in Fig. 7. On the _PISA_ datasets, this TE measure shows the best values
across predicted models, followed by _Law school_ and _Student Academics_
datasets.

Summary of results: In general, _ABOCA_ is the measure with the lowest
variability across predictive methods and datasets. It also clearly presents
the ML model’s accuracy variation over each value of the protected attribute.
_Equal opportunity_ and _predictive parity_ also have a slight variation
across methods and datasets. _Equalized odds_ , to some extent, can represent
two measures _equal opportunity_ and _predictive equality_ as it is the sum of
the other two metrics. Furthermore, _treatment equality_ has a very wide range
of values (sometimes the value may not be bounded), making it difficult to
compare and evaluate.

###  4.4 Effect of varying grade threshold on fairness

Grade thresholds are often chosen as a basis for determining whether a
candidate passes or fails an exam. In the student performance dataset, 10 (out
of 20) is selected as the grade threshold [11, 27]. However, the selection of
a threshold can affect the fairness of the predictive models, as shown in the
IPUMS Adult dataset [12]. Hence, we investigate the effect of grade threshold
on fairness by varying the threshold in a range of [4, 16], corresponding to
25% to 75% of the maximum grade (20). The results in Fig. 8 show that all
fairness measures are affected by the grade threshold. When the grade
threshold is gradually increased, the predictive models tend to be fairer
(shown on the measures: equalized odds, predictive equality, and ABROCA). The
opposite trend is observed in the remaining measures (except the treatment
equality measure). Regarding the balanced accuracy, two models (DT and
AdaFair) tend to predict more accurately. The NB model has a decreasing
accuracy after the threshold is increased.

(a) Accuracy

(b) Balanced accuracy

(c) Statistical parity

(d) Equal opportunity

(e) Equalized odds

(f) Predictive parity

(g) Predictive equality

(h) Treatment equality

(i) ABROCA

Figure 8: Accuracy and fairness interventions with varying grade threshold on
Student performance dataset (Decision Tree method).

##  5 Conclusion and outlooks

In this work, we evaluate seven popular group fairness measures for student
performance prediction problems. We conduct experiments using four traditional
ML models and two fairness-aware ML methods on five educational datasets. Our
experiments reflect variations and correlations of fairness measures across
datasets and predictive models. The results provide a overview picture for the
selection of fairness measure in a specific case. Besides, we investigate the
effect of varying grade thresholds on the accuracy and fairness of ML models.
The preliminary results suggest that choosing the threshold is an important
factor contributing to ensuring fairness in the output of the ML models. In
the future, we plan to extend our evaluation of fairness w.r.t. more than one
protected attribute, such as gender and race, and further explore the
correlation between groups of fairness notions.

## Acknowledgments

The work of the first author is supported by the Ministry of Science and
Culture of Lower Saxony, Germany, within the Ph.D. program “LernMINT: Data-
assisted teaching in the MINT subjects”. The work of the second author is
funded by the German Research Foundation (DFG Grant NI-1760/1-1), project
“Managed Forgetting”.

## References

  * [1] Abu Saa, A., Al-Emran, M., Shaalan, K.: Factors affecting students’ performance in higher education: a systematic review of predictive data mining techniques. Technology, Knowledge and Learning 24(4), 567–598 (2019)
  * [2] Agarwal, A., Beygelzimer, A., Dudík, M., Langford, J., Wallach, H.: A reductions approach to fair classification. In: ICML. pp. 60–69. PMLR (2018)
  * [3] Alvero, A., Arthurs, N., Antonio, A.L., Domingue, B.W., Gebre-Medhin, B., Giebel, S., Stevens, M.L.: AI and holistic review: Informing human reading in college admissions. In: AIES. p. 200–206. ACM (2020). https://doi.org/10.1145/3375627.3375871
  * [4] Amrieh, E.A., Hamtini, T., Aljarah, I.: Preprocessing and analyzing educational data set using x-api for improving student’s performance. In: AEECT. pp. 1–5. IEEE (2015). https://doi.org/10.1109/AEECT.2015.7360581
  * [5] Anders, J., Dilnot, C., Macmillan, L., Wyness, G.: Grade expectations: How well can we predict future grades based on past performance? CEPEO Working Paper No. 20-14 (2020)
  * [6] Anderson, H., Boodhwani, A., Baker, R.S.: Assessing the fairness of graduation predictions. In: EDM (2019)
  * [7] Berhanu, F., Abera, A.: Students’ performance prediction based on their academic record. International Journal of Computer Applications 131(5), 0975–8887 (2015)
  * [8] Berk, R., Heidari, H., Jabbari, S., Kearns, M., Roth, A.: Fairness in criminal justice risk assessments: The state of the art. Sociological Methods & Research 50(1), 3–44 (2021). https://doi.org/10.1177/0049124118782533
  * [9] Chouldechova, A.: Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big data 5(2), 153–163 (2017)
  * [10] Corbett-Davies, S., Pierson, E., Feller, A., Goel, S., Huq, A.: Algorithmic decision making and the cost of fairness. In: KDD. pp. 797–806 (2017)
  * [11] Cortez, P., Silva, A.M.G.: Using data mining to predict secondary school student performance (2008), https://hdl.handle.net/1822/8024
  * [12] Ding, F., Hardt, M., Miller, J., Schmidt, L.: Retiring adult: New datasets for fair machine learning. NeurIPS 34, 6478–6490 (2021)
  * [13] Dwork, C., Hardt, M., Pitassi, T., Reingold, O., Zemel, R.: Fairness through awareness. In: ITCS. pp. 214–226 (2012). https://doi.org/10.1145/2090236.2090255
  * [14] Fleischman, H.L., Hopstock, P.J., Pelczar, M.P., Shelley, B.E.: Highlights from pisa 2009: Performance of us 15-year-old students in reading, mathematics, and science literacy in an international context. nces 2011-004. National Center for Education Statistics (2010)
  * [15] Foster, I., Ghani, R., Jarmin, R.S., Kreuter, F., Lane, J.: Big data and social science: A practical guide to methods and tools. crc Press (2016)
  * [16] Francis, B.K., Babu, S.S.: Predicting academic performance of students using a hybrid data mining approach. Journal of medical systems 43(6), 1–15 (2019). https://doi.org/10.1007/s10916-019-1295-4
  * [17] Gardner, J., Brooks, C., Baker, R.: Evaluating the fairness of predictive student models through slicing analysis. In: LAK19. pp. 225–234 (2019). https://doi.org/10.1145/3303772.3303791
  * [18] Hardt, M., Price, E., Srebro, N.: Equality of opportunity in supervised learning. Advances in neural information processing systems 29 (2016)
  * [19] Hussain, S., Dahan, N.A., Ba-Alwib, F.M., Ribata, N.: Educational data mining and analysis of students’ academic performance using weka. Indonesian Journal of Electrical Engineering and Computer Science 9(2), 447–459 (2018)
  * [20] Hutchinson, B., Mitchell, M.: 50 years of test (un) fairness: Lessons for machine learning. In: FAT. pp. 49–58 (2019). https://doi.org/10.1145/3287560.3287600
  * [21] Iosifidis, V., Ntoutsi, E.: AdaFair: Cumulative fairness adaptive boosting. In: CIKM. pp. 781–790 (2019). https://doi.org/10.1145/3357384.3357974
  * [22] Jiang, W., Pardos, Z.A.: Towards equity and algorithmic fairness in student grade prediction. In: AIES. pp. 608–617. ACM (2021). https://doi.org/10.1145/3461702.3462623
  * [23] Jiang, W., Pardos, Z.A.: Towards equity and algorithmic fairness in student grade prediction. In: AIES. pp. 608–617 (2021). https://doi.org/10.1145/3461702.3462623
  * [24] Khan, A., Ghosh, S.K.: Student performance analysis and prediction in classroom learning: A review of educational data mining studies. Education and information technologies 26(1), 205–240 (2021). https://doi.org/10.1007/s10639-020-10230-3
  * [25] Khan, N.A.U., Khan, I.U., Alamri, L.H., Almuslim, R.S.: An improved early student’s academic performance prediction using deep learning. International Journal of Emerging Technologies in Learning (iJET) (2021)
  * [26] Kuzilek, J., Hlosta, M., Zdrahal, Z.: Open university learning analytics dataset. Scientific data 4(1), 1–8 (2017). https://doi.org/10.1038/sdata.2017.171
  * [27] Le Quy, T., Roy, A., Vasileios, I., Wenbin, Z., Ntoutsi, E.: A survey on datasets for fairness-aware machine learning. WIREs Data Mining and Knowledge Discovery 12(3) (2022). https://doi.org/10.1002/widm.1452
  * [28] Mehrabi, N., Morstatter, F., Saxena, N., Lerman, K., Galstyan, A.: A survey on bias and fairness in machine learning. ACM Computing Surveys (CSUR) 54(6), 1–35 (2021). https://doi.org/10.1145/3457607
  * [29] Meyer, K.: Education, justice and the human good: Fairness and equality in the education system. Routledge (2014)
  * [30] Mihaescu, M.C., Popescu, P.S.: Review on publicly available datasets for educational data mining. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 11(3), e1403 (2021). https://doi.org/10.1002/widm.1403
  * [31] Namoun, A., Alshanqiti, A.: Predicting student performance using data mining and learning analytics techniques: A systematic literature review. Applied Sciences 11(1), 237 (2020). https://doi.org/10.3390/app11010237
  * [32] Ntoutsi, E., Fafalios, P., Gadiraju, U., Iosifidis, V., Nejdl, W., Vidal, M.E., Ruggieri, S., Turini, F., Papadopoulos, S., Krasanakis, E., et al.: Bias in data-driven artificial intelligence systems—an introductory survey. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 10(3), e1356 (2020). https://doi.org/10.1002/widm.1356
  * [33] Saleem, F., Ullah, Z., Fakieh, B., Kateb, F.: Intelligent decision support system for predicting student’s e-learning performance using ensemble machine learning. Mathematics 9(17), 2078 (2021). https://doi.org/10.3390/math9172078
  * [34] Shahiri, A.M., Husain, W., et al.: A review on predicting student’s performance using data mining techniques. Procedia Computer Science 72, 414–422 (2015)
  * [35] Simoiu, C., Corbett-Davies, S., Goel, S.: The problem of infra-marginality in outcome tests for discrimination. The Annals of Applied Statistics 11(3), 1193–1216 (2017). https://doi.org/10.1214/17-AOAS1058
  * [36] Verma, S., Rubin, J.: Fairness definitions explained. In: 2018 IEEE/ACM International Workshop on Software Fairness (FairWare). pp. 1–7 (2018). https://doi.org/10.23919/FAIRWARE.2018.8452913
  * [37] Wightman, L.F.: LSAC national longitudinal bar passage study. LSAC research report series. (1998)
  * [38] Xiao, W., Ji, P., Hu, J.: A survey on educational data mining methods used for predicting students’ performance. Engineering Reports 4(5), e12482 (2022). https://doi.org/10.1002/eng2.12482
  * [39] Yu, R., Li, Q., Fischer, C., Doroudi, S., Xu, D.: Towards accurate and fair prediction of college success: Evaluating different sources of student data. In: EDM (2020)
  * [40] Žliobaitė, I.: On the relation between accuracy and fairness in binary classification. In: FAT/ML 2015 workshop at ICML. vol. 15 (2015)
  * [41] Zohair, A., Mahmoud, L.: Prediction of student’s performance by modelling small dataset size. International Journal of Educational Technology in Higher Education 16(1), 1–18 (2019). https://doi.org/10.1186/s41239-019-0160-3


# Power-series summability methods in
de Branges–Rovnyak spaces

Javad Mashreghi  Département de mathématiques et de statistique, Université
Laval, Québec City (Québec), Canada G1V 0A6<EMAIL_ADDRESS>Pierre-Olivier Parisé  Département de mathématiques et de statistique,
Université Laval, Québec City (Québec), Canada G1V 0A6. Department of
Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822, U.S.A.
<EMAIL_ADDRESS>Thomas Ransford  Département de mathématiques et de
statistique, Université Laval, Québec City (Québec), Canada G1V 0A6.
<EMAIL_ADDRESS>
###### Abstract.

We show that there exists a de Branges–Rovnyak space $\mathcal{H}(b)$ on the
unit disk containing a function $f$ with the following property: even though
$f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor
partial sums of $f$ nor their Cesàro, Abel, Borel or logarithmic means
converge to $f$ in $\mathcal{H}(b)$.

A key tool is a new abstract result showing that, if one regular summability
method includes another for scalar sequences, then it automatically does so
for certain Banach-space-valued sequences too.

###### Key words and phrases:

De Branges–Rovnyak spaces, polynomial approximations, summability methods,
logarithmic means, Abel means, Borel means

###### 1991 Mathematics Subject Classification:

Primary 40C15; Secondary 40G10, 41A10, 46E20, 40J05

JM supported by an NSERC Discovery Grant. POP supported by an NSERC Alexander-
Graham-Bell Scholarship and a scholarship from FRQNT. TR supported by grants
from NSERC and the Canada Research Chairs program.

##  1\. Introduction

We denote by $\mathbb{D}$ the open unit disk and
$\operatorname{Hol}(\mathbb{D})$ the space of holomorphic functions on
$\mathbb{D}$. Polynomial approximation in Banach spaces of holomorphic
functions on the unit disk has attracted attention recently. In [12], the
authors proved that, for a Banach space of holomorphic functions $X$ on the
open unit disk having the approximation property and containing a dense set of
polynomials, there exist linear bounded operators $T_{n}:X\rightarrow X$ such
that, for each $f\in X$, the functions $T_{n}(f)$ are polynomials and
$T_{n}(f)\rightarrow f$ in the norm of $X$. The authors called the
aforementioned sequence $(T_{n})$ a linear polynomial approximation scheme.

For some spaces, the operators $T_{n}$ are explicitly known. For example, if
$X=H^{2}$, the Hardy space, then the bounded linear operators can be chosen to
be the $n$-th partial sums $s_{n}[f]$ of the Taylor expansion of a function
$f(z)=\sum_{n\geq 0}a_{n}z^{n}$ belonging to $H^{2}$. This is an easy
consequence of the definition of the norm in $H^{2}$. When $X=H^{p}$, the
Hardy spaces with $1<p<\infty$, we can still choose the partial sums
$s_{n}[f]$ of a function $f\in H^{p}$ as a linear polynomial approximation
scheme. The proof is more elaborate and is based on a result of Riesz on the
boundedness of the Hilbert transform. The reader is referred to [10, p. 108].
When $X=A(\mathbb{D})$, the disk algebra, due to a slight variant of du Bois-
Reymond’s theorem that establishes the existence of a continuous function on
the unit circle whose Fourier series diverges at one point, the partial sums
$s_{n}[f]$ do not converge to $f$ in the norm of $A(\mathbb{D})$. Instead, we
can use the Cesàro means of order 1 of the Taylor expansion of a function
$f\in A(\mathbb{D})$, defined as

$\sigma_{n}[f]:=\frac{1}{n+1}\sum_{k=0}^{n}s_{k}[f].$

This is essentially Fejér’s theorem. This procedure also works for certain
other spaces, for example the Hardy space $H^{1}$ and the weighted Dirichlet
spaces $\mathcal{D}_{\omega}$ for superharmonic weights
$\omega:\mathbb{D}\rightarrow(0,\infty)$.

However there are some spaces for which we do not explicitly know the linear
polynomial approximation scheme. One such family of spaces are the de
Branges–Rovnyak spaces $\mathcal{H}(b)$, where $b\in H^{\infty}$ is a non-
extreme point of the unit ball of $H^{\infty}$. Despite the fact that the set
of polynomials is dense in $\mathcal{H}(b)$, the authors of [7] showed that,
for certain choices of $b$, the partial sums $s_{n}[f]$ and the Cesàro means
$\sigma_{n}[f]$ may fail to converge in $\mathcal{H}(b)$ to the initial
function $f$. Therefore the attention turned to other linear summability
methods that do not give a polynomial approximation, but have a better chance
to approximate the function $f$ in the norm of $\mathcal{H}(b)$, namely the
dilates of $f$, which are in fact the Abel means of the partial sums
$s_{n}[f]$ :

$f_{r}(z):=\sum_{n\geq 0}a_{n}r^{n}z^{n}=(1-r)\sum_{n\geq
0}s_{n}[f](z)r^{n}\quad(r\in[0,1),\,z\in\mathbb{D})$

for $f(z)=\sum_{n\geq 0}a_{n}z^{n}$. Nevertheless, they showed that even this
summability method, which includes all the Cesàro summability methods of order
$\alpha>-1$, may fail for de Branges–Rovnyak spaces with non-extreme symbols
$b$. Furthermore, the same authors showed in the same article that there is a
constructive way to obtain a polynomial approximation of a given function
$f\in\mathcal{H}(b)$ for any non-extreme point $b$ of the unit ball of
$H^{\infty}$. However, the procedure is highly non-linear and it does not
correspond to a polynomial approximation scheme.

In this article, we study another summability method which includes the Abel
method : the logarithmic method. Its means are defined as

$L_{r}[f](z):=\frac{r}{\log\left(\frac{1}{1-r}\right)}\sum_{n\geq
0}\frac{1}{n+1}s_{n}[f](z)r^{n}$ (1.1)

for $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ and for $r\in[0,1)$, $z\in\mathbb{D}$. It
was introduced by Borwein in [2] through a power-series method.

If $f\in\mathcal{H}(b)$, it is well known that $f_{r}\in\mathcal{H}(b)$. We
will show that also $L_{r}[f]\in\mathcal{H}(b)$. However, our main result
reveals that $L_{r}[f]$ may diverge.

######  Theorem 1.1.

There exist a non-extreme point $b$ of the unit ball of $H^{\infty}$ and a
function $f\in\mathcal{H}(b)$ such that

$\lim_{r\rightarrow 1^{-}}\|L_{r}[f]\|_{b}=\infty.$

Theorem 1.1 is proved in §4. It is a consequence of the example already
constructed in [7] to show that $s_{n}[f]$ and $\sigma_{n}[f]$ may diverge in
$\mathcal{H}(b)$, together with an integral formula that links the dilates of
a function to its logarithmic means.

As a corollary, we obtain the following result concerning generalized Abel
methods $A_{r}^{\alpha}$, where $\alpha>-1$ and where the generalized Abel
means $A_{r}^{\alpha}[f]$ are defined by the following expression:

$A_{r}^{\alpha}[f](z):=(1-r)^{1+\alpha}\sum_{n\geq
0}\binom{n+\alpha}{\alpha}s_{n}[f](z)r^{n}\quad(r\in[0,1)).$

######  Corollary 1.2.

For every $\alpha>-1$, there exist a non-extreme point $b$ of the unit ball of
$H^{\infty}$ and a function $f\in\mathcal{H}(b)$ such that
$A_{r}^{\alpha}[f]\not\rightarrow f$ in $\mathcal{H}(b)$ as $r\rightarrow
1^{-}$.

This corollary is a consequence of an abstract result in functional analysis.
It enables us to compare the summability of a sequence of elements in a Banach
space with respect to two summability methods, based on the inclusion of one
summability method in the other for scalar sequences. This theorem is proved
in §5, and Corollary 1.2 is deduced in §6.

##  2\. Sequence-to-function summability methods

We start by defining some terminology in summability theory. Our main
references are Hardy [8] and Boos [1]. Throughout this section, we let $X$
denote a Banach space over the complex numbers $\mathbb{C}$ and
$\|\cdot\|_{X}$ be its norm. We denote by $c(X)$ the space of convergent
sequences in $X$.

A sequence-to-function summability method $K$ is given by a sequence
$(k_{n})_{n\geq 0}$ of functions $k_{n}:[0,R)\rightarrow\mathbb{C}$ where
$R\in(0,\infty]$. The $K$-means of a sequence of vectors $x:=(x_{n})_{n\geq
0}\subset X$ are defined by the following formal series

$\displaystyle K_{r}[x]:=\sum_{n\geq 0}k_{n}(r)x_{n}\quad(r\in[0,R)).$

We say that a sequence $x:=(x_{n})_{n\geq 0}\subset X$ is $K$-summable or
summable by the method $K$ if the series defining $K_{r}[x]$ converges for
every $r\in[0,R)$ and, moreover, $K_{r}[x]$ converges in norm to some $y\in X$
as $r\rightarrow R^{-}$. We say that a sequence-to-function summability method
$K$ is regular if, whenever $X$ is a Banach space and $(x_{n})_{n\geq 0}\in
c(X)$, then $(x_{n})_{n\geq 0}$ is $K$-summable and $\lim_{r\rightarrow
R^{-}}K_{r}[x]=\lim_{n\rightarrow\infty}x_{n}$. Necessary and sufficient
conditions for a sequence-to-function summability method to be regular are
given in the following theorem, which is a slight modification of Theorem 5 in
[8, p. 49].

######  Theorem 2.1.

Let $X$ be a Banach space. A sequence-to-function summability method $K$ is
regular if and only if the following conditions hold :

  * •

there exists an $R_{0}\in[0,R)$ such that the function $r\mapsto\sum_{n\geq
0}|k_{n}(r)|$ is uniformly bounded on $[R_{0},R)$;

  * •

for each $n\geq 0$, we have $k_{n}(r)\rightarrow 0$ as $r\rightarrow R^{-}$;

  * •

the function $k(r):=\sum_{n\geq 0}k_{n}(r)$ converges to $1$ as $r\rightarrow
R^{-}$.

###### Proof.

By considering the homeomorphism $r\mapsto\log\left(\frac{R}{R-r}\right)$ from
$[0,R)$ onto $[0,\infty)$, we may restrict our attention to the situation to
the interval $[0,\infty)$.

Suppose that the method $K$ is regular. Since $\mathbb{C}$ embeds
isometrically into $X$ and the functions $k_{n}(r)$ are complex-valued, $K$ is
regular for the space of convergent complex-valued sequences. Then the result
follows from the classical case $X=\mathbb{C}$ (see Theorem 5 in [8, p. 49]).

If the conditions are satisfied, then elementary estimates using the
conditions show that $K_{r}[x]\rightarrow\lim_{n\rightarrow\infty}x_{n}$ as
$r\rightarrow R^{-}$ for any sequence $(x_{n})_{n\geq 0}\in c(X)$. ∎

Let $K$ and $H$ be two sequence-to-function summability methods. We say that
$K$ is included in $H$, denoted by $K\subseteq H$, if, whenever
$(x_{n})_{n\geq 0}$ is a $K$-summable sequence in a Banach space $X$, then
$(x_{n})_{n\geq 0}$ is also $H$-summable and

$\lim_{r\rightarrow R^{-}}H_{r}[x]=\lim_{r\rightarrow R^{-}}K_{r}[x].$

If $K\subseteq H$ and $H\subseteq K$, we say that the two summability methods
are equivalent. If $X=\mathbb{C}$, all of the aforementioned definitions will
be preceded by the word “scalar”. For example, when we write “the summability
method $K$ is scalar-equivalent to the summability method $H$”, this means
that they are equivalent when applied to scalar-valued sequences, that is
$(x_{n})_{n\geq 0}\subset\mathbb{C}$.

We now give some examples of sequence-to-function summability methods.

###  2.1. Matrix summability methods

Let $R=\infty$ and, for each $n\geq 0$, let the function $k_{n}$ be constant
on each interval $[m,m+1)$, where $m\geq 0$ is an integer. Then, for each
$r\in[m,m+1)$, the expression of the $K$-mean becomes

$\displaystyle K_{r}[x]=\sum_{n\geq 0}k_{n}(r)x_{n}=\sum_{n\geq
0}k_{n}(m)x_{n}.$

Hence the method given by the sequence of functions $k_{n}$ can be viewed as
an infinite matrix $(k_{n}(m))_{m,n\geq 0}$. In these circumstances, we call
the summability method a matrix summability method. The necessary and
sufficient conditions for a matrix summability method to be regular are
attributed to Silverman and Toeplitz (see [8, p. 43]). The conditions in
Theorem 2.1 now become:

  * •

there is a number $M>0$ such that $\sum_{n\geq 0}|k_{n}(m)|\leq M$ for every
$m\geq 0$;

  * •

for each $n\geq 0$, $\lim_{m\rightarrow\infty}k_{n}(m)=0$;

  * •

we have $\sum_{n\geq 0}k_{n}(m)\rightarrow 1$, as $m\rightarrow\infty$.

There is a generalization of the Silverman-Toeplitz Theorem, due to Robinson,
to matrix summability methods that are given by a matrix $(C_{m,n})_{m,n\geq
0}$, where $C_{m,n}$ are bounded linear operators on $X$ (see [13, Theorem
IV]).

###  2.2. Power-series summability methods

Let $p(r):=\sum_{n\geq 0}p_{n}r^{n}$ be a power series with a radius of
convergence $R_{p}>0$, where $p_{0}>0$ and $p_{n}\geq 0$ for $n\geq 1$.
Following §3.6 of [1], we say that a sequence $x:=(x_{n})_{n\geq 0}\subset X$
is _summable by the power-series method_ $(p)$, or is _$P$ -summable_, if the
series

$\displaystyle P_{r}[x]:=\frac{1}{p(r)}\sum_{n\geq 0}p_{n}x_{n}r^{n}$

converges for each $r\in[0,R)$ and there exists a $y\in X$ such that

$\displaystyle P_{r}[x]\rightarrow y\quad(r\rightarrow R^{-}).$

According to Theorem 2.1, a power-series method $(p)$ is regular if and only
if $p(r)\rightarrow\infty$ as $r\rightarrow R_{p}^{-}$. The Abel summability
method is a special case of the power-series method $(p)$ with
$p(r)=(1-r)^{-1}$ and $r\in[0,1)$. The $A_{r}$-means are

$\displaystyle A_{r}[x]=(1-r)\sum_{n\geq 0}x_{n}r^{n}\quad(0\leq r<1).$

In this paper, we apply a power-series method to the sequence of partial sums
$(s_{n}[f])_{n\geq 0}$ of the Taylor expansion of a function
$f\in\operatorname{Hol}(\mathbb{D})$. The expression of the means defined by a
power-series method $(p)$ are

$\displaystyle P_{r}[f](z):=\frac{1}{p(r)}\sum_{n\geq
0}p_{n}s_{n}[f](z)r^{n},$

where $p$ has a radius of convergence $R_{p}\geq 1$. Since $|s_{n}[f](z)|\leq
C(R)R^{n}$ for any $R>1$ and some constant $C(R)>0$, the function $P_{r}[f]$
is well-defined for each $z\in\mathbb{D}$. Also, the series defining
$P_{r}[f]$ converges uniformly on compact subsets of $\mathbb{D}$, and thus it
defines a function holomorphic on all of $\mathbb{D}$.

A useful power-series method is the logarithmic method. As we shall see later
in the paper, this power-series method is convenient because it contains many
other summability methods. The logarithmic method is the power-series method
associated with the power series

$\displaystyle l(r):=\sum_{n\geq
0}\frac{r^{n}}{n+1}=\frac{1}{r}\log\frac{1}{1-r}\quad(0\leq r<1).$

Thus, the expression of the logarithmic mean of the partial sums $s_{n}[f]$ is

$\displaystyle L_{r}[f](z):=\frac{r}{\log\frac{1}{1-r}}\sum_{n\geq
0}\frac{s_{n}[f](z)}{n+1}r^{n}\quad(0\leq r<1).$

We will also study the generalized Abel methods. Given a number $\alpha>-1$,
the generalized Abel means of order $\alpha$ are associated with the power
series

$\displaystyle a_{r}^{\alpha}(r):=\sum_{n\geq
0}\binom{n+\alpha}{\alpha}r^{n}=\frac{1}{(1-r)^{1+\alpha}}\quad(r\in[0,1)).$

Applied to the partial sums $(s_{n}[f])_{n\geq 0}$ of the Taylor series of
$f\in\operatorname{Hol}(\mathbb{D})$, the expression of the mean
$A_{r}^{\alpha}[f]$ is

$\displaystyle A_{r}^{\alpha}[f](z):=(1-r)^{\alpha+1}\sum_{n\geq
0}\binom{n+\alpha}{\alpha}s_{n}[f](z)r^{n}.$

If $\alpha<\beta$, then the generalized Abel method of order $\beta$ is
scalar-included in the generalized Abel method of order $\alpha$ (see [2,
Theorem 2]). When $\alpha=1$, we obtain the classical Abel means. As we
mention earlier in the paper, the expression of $A_{r}^{1}[f]$ can be
rearranged to give

$\displaystyle A_{r}^{1}[f](z)=\sum_{n\geq 0}a_{n}r^{n}z^{n},$

which is the dilate $f_{r}$ of $f\in\operatorname{Hol}(\mathbb{D})$. We
present a pointwise relation between the logarithmic means and the dilates of
a function. It will be used later in §4 and its proof is straightforward.

######  Lemma 2.2.

For any $f\in\operatorname{Hol}(\mathbb{D})$, $r\in[0,1)$ and
$z\in\mathbb{D}$, we have

$L_{r}[f](z)=\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r}\frac{f_{t}(z)}{1-t}\,dt.$
(2.1)

##  3\. Background on de Branges–Rovnyak spaces

Let $\phi\in L^{\infty}(\mathbb{T})$. The Toeplitz operator
$T_{\phi}:H^{2}\rightarrow H^{2}$ with symbol $\phi$ is defined as

$\displaystyle T_{\phi}f:=P_{+}(\phi f)\quad(f\in H^{2}),$

where $P_{+}:L^{2}(\mathbb{T})\rightarrow H^{2}$ is the orthogonal projection
of $L^{2}(\mathbb{T})$ onto $H^{2}$. This is clearly a bounded operator with
$\|T_{\phi}\|\leq\|\phi\|_{L^{\infty}(\mathbb{T})}$. (In fact,
$\|T_{\phi}\|=\|\phi\|_{\infty}$ by a theorem of Brown and Halmos, but we do
not need this here.) The adjoint of $T_{\phi}$ is $T_{\overline{\phi}}$. If
$\phi\in H^{\infty}$, then $T_{\phi}$ is simply the operator of multiplication
by $\phi$. We now introduce the de Branges–Rovnyak space associated to a
function $b\in H^{\infty}$, where $\|b\|_{\infty}\leq 1$. This is Sarason’s
definition taken from [14].

######  Definition 3.1.

Let $b\in H^{\infty}$ with $\|b\|_{\infty}\leq 1$. The associated de
Branges–Rovnyak space, denoted by $\mathcal{H}(b)$, is the range space
$(I-T_{b}T_{\overline{b}})^{1/2}H^{2}$ equipped with the following inner
product

$\displaystyle\left\langle(I-T_{b}T_{\overline{b}})^{1/2}f,(I-T_{b}T_{\overline{b}})^{1/2}g\right\rangle_{b}:=\left\langle
f,g\right\rangle_{2},$

where $f,g\in H^{2}\ominus\ker(I-T_{b}T_{\overline{b}})^{1/2}$.

The definition of the inner product $\left\langle\cdot,\cdot\right\rangle_{b}$
makes the operator $(I-T_{b}T_{\overline{b}})^{1/2}:H^{2}\rightarrow H^{2}$ a
partial isometry from $H^{2}$ onto $\mathcal{H}(b)$. Its cousin, the space
$\mathcal{H}(\overline{b})$, is defined similarly by interchanging the roles
of $b$ with $\overline{b}$ in the above definition.

There is a close relationship between $\mathcal{H}(b)$ and
$\mathcal{H}(\overline{b})$. This is the content of the next theorem.

######  Theorem 3.2.

[14, §II-2]. A function $f\in H^{2}$ belongs to $\mathcal{H}(b)$ if and only
if $T_{\overline{b}}f$ belongs to $\mathcal{H}(\overline{b})$. In this case,
we have that

$\displaystyle\|f\|_{b}^{2}=\|f\|_{2}^{2}+\|T_{\overline{b}}f\|_{\overline{b}}^{2}\text{.}$

The structure of a de Branges–Rovnyak space depends strongly on whether $b$ is
an extreme or a non-extreme point of the unit ball of $H^{\infty}$.

######  Theorem 3.3.

[7, Theorem 2.2]. Let $b\in H^{\infty}$ with $\|b\|_{\infty}\leq 1$. The
following statements are equivalent:

  1. i)

$b$ is a non-extreme point of the unit ball of $H^{\infty}$;

  2. ii)

$\mathcal{H}(b)$ contains all functions holomorphic in a neighbourhood of
$\overline{\mathbb{D}}$;

  3. iii)

$\mathcal{H}(b)$ contains all polynomials;

  4. iv)

polynomials are dense in $\mathcal{H}(b)$.

From now on, we shall simply say that $b$ is “extreme” or “non-extreme” to
indicate that $b$ is correspondingly an extreme or non-extreme point of the
unit ball of $H^{\infty}$. The function $b$ is non-extreme if and only if
$\log(1-|b|^{2})\in L^{1}(\mathbb{T})$ (see [6, Theorem 7.9]). In this case,
there exists a unique outer function $a\in H^{\infty}$, normalized so that
$a(0)>0$, such that $|a|^{2}+|b|^{2}=1$ a.e. on $\mathbb{T}$. We call $(b,a)$
the Pythagorean pair associated to $b$.

There is a useful characterization of $\mathcal{H}(b)$ when $b$ is non-
extreme.

######  Theorem 3.4.

[14, §IV-1]. Let $b$ be non-extreme, let $(b,a)$ be the corresponding
Pythagorean pair, and let $f\in H^{2}$. Then $f\in\mathcal{H}(b)$ if and only
if $T_{\overline{b}}f\in T_{\overline{a}}H^{2}$. In this case, there exists a
unique function $f^{+}\in H^{2}$ such that
$T_{\overline{b}}f=T_{\overline{a}}f^{+}$, and

$\displaystyle\|f\|_{b}^{2}=\|f\|_{2}^{2}+\|f^{+}\|_{2}^{2}\text{.}$

The authors of [5] obtained an explicit formula for the $\mathcal{H}(b)$-norm
of functions $f$ holomorphic in a neighbourhood of $\overline{\mathbb{D}}$.
Notice that, if $(b,a)$ is a pair, then $\phi:=b/a\in N^{+}$, the Smirnov
class.

######  Theorem 3.5.

[5, Theorem 4.1] Let $(b,a)$ be a pair, and let $\phi:=b/a$, say
$\phi(z)=\sum_{j\geq 0}c_{n}z^{n}$. Let $f$ be holomorphic in a neighbourhood
of $\overline{\mathbb{D}}$ with expansion $f(z)=\sum_{j\geq 0}a_{n}z^{n}$.
Then the series $\sum_{j\geq 0}a_{j+n}\overline{c}_{j}$ converges absolutely
for each $n$ and

$\displaystyle\|f\|_{b}^{2}=\sum_{n\geq 0}|a_{n}|^{2}+\sum_{n\geq
0}\Big{\lvert}\sum_{j\geq 0}a_{j+n}\overline{c}_{j}\Big{\rvert}^{2}.$ (3.1)

We remark two consequences of this result that will be useful in what follows.

######  Corollary 3.6.

Let $b$ be non-extreme and let $f\in\mathcal{H}(b)$. Then, for each $R>1$, the
partial sums of the Taylor series of $f$ satisfy $\|s_{n}[f]\|_{b}=O(R^{n})$
as $n\to\infty$.

###### Proof.

Fix $S$ with $1<S^{2}<R$. Let $(b,a)$ be the Pythagorean pair corresponding to
$b$, and let $\phi:=b/a$. Since both $f$ and $\phi$ are holomorphic on
$\mathbb{D}$, their respective Taylor coefficients satisfy $a_{j}=O(S^{j})$
and $c_{j}=O(S^{j})$ as $j\to\infty$. Feeding this information into the
formula (3.1), applied to $s_{N}[f]$ in place of $f$, we obtain

$\|s_{N}[f]\|_{b}^{2}=O((N+1)S^{2N})+O((N+1)^{3}S^{4N})\quad(N\to\infty).$

This implies that $\|s_{N}[f]\|_{b}=O(R^{N})$. ∎

######  Corollary 3.7.

Let $(b,a)$ be a pair, and let $\phi:=b/a$, say $\phi(z)=\sum_{j\geq
0}c_{n}z^{n}$. Let $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ be a function in $H^{2}$
and let $r\in[0,1)$. Then there is a constant $C(\phi,r)$, which depends only
on $\phi$ and $r$, such that

$\displaystyle\|f_{r}\|_{b}^{2}\leq C(\phi,r)\|f\|_{H^{2}}^{2}.$

The dependence $r\mapsto C(\phi,r)$ can be chosen to be increasing.

###### Proof.

Since $f_{r}(z):=\sum_{n\geq 0}a_{n}r^{n}z^{n}$, using formula (3.1), we get

$\displaystyle\|f_{r}\|_{b}^{2}$ $\displaystyle=\sum_{n\geq
0}r^{2}|a_{n}|^{2}+\sum_{n\geq 0}\Big{\lvert}\sum_{j\geq
0}r^{j+n}a_{j+n}\overline{c}_{j}\Big{\rvert}^{2}$
$\displaystyle\leq\|f\|_{H^{2}}^{2}+\sum_{n\geq 0}\Big{(}\sum_{j\geq
0}r^{j+n}|a_{j+n}||c_{j}|\Big{)}^{2}.$

By the Cauchy-Schwarz inequality, we obtain

$\displaystyle\|f_{r}\|_{b}^{2}$
$\displaystyle\leq\|f\|_{H^{2}}^{2}+\sum_{n\geq 0}\Big{(}\sum_{j\geq
0}r^{j+n}|a_{j+n}|^{2}\Big{)}\Big{(}\sum_{j\geq 0}r^{j+n}|c_{j}|^{2}\Big{)}$
$\displaystyle=\|f\|_{H^{2}}^{2}+\sum_{n\geq
0}r^{n}\|(S^{*})^{n}(f_{\sqrt{r}})\|_{H^{2}}^{2}\Big{(}\sum_{j\geq
0}r^{j}|c_{j}|^{2}\Big{)},$

where $S^{*}$ is the backward shift operator on $H^{2}$ and $(S^{*})^{n}$ is
the $n$-fold composition of $S^{*}$. Since $\|S^{*}\|\leq 1$, we get

$\displaystyle\|f_{r}\|_{b}^{2}\leq\|f\|_{H^{2}}^{2}+\frac{\sum_{n\geq
0}r^{j}|c_{j}|^{2}}{1-r}\|f_{\sqrt{r}}\|_{H^{2}}^{2},$

where $\sum_{j\geq 0}r^{j}|c_{j}|^{2}<\infty$ since
$\phi\in\operatorname{Hol}(\mathbb{D})$. Finally, since
$\|f_{\sqrt{r}}\|_{H^{2}}^{2}\leq\|f\|_{H^{2}}^{2}$, we obtain

$\|f_{r}\|_{b}^{2}\leq C(\phi,r)\|f\|_{H^{2}}^{2},$

as desired, where $C(\phi,r):=1+\sum_{n\geq 0}r^{j}|c_{j}|^{2}/(1-r)$. It is
easy to see from the definition of the constant $C(\phi,r)$ that the function
$r\mapsto C(\phi,r)$ is increasing on $[0,1)$. ∎

We end this section by stating an estimate obtained in the core of the proofs
of Theorem 3.1 and Theorem 3.6 in [7]. This example plays a central role in
[7], and will be equally important for us here. The exact choices of $b$ and
$f\in\mathcal{H}(b)$ do not matter in our situation, so we will just state the
estimate as follows.

######  Theorem 3.8.

[7, Theorems 3.1 and 3.6]. There exist a non-extreme $b$ and a function
$f\in\mathcal{H}(b)$ such that $(f_{r})^{+}(0)$ is non-negative for all
$r\in[0,1)$ and, moreover, $(f_{r})^{+}(0)\rightarrow+\infty$ as $r\rightarrow
1^{-}$.

##  4\. Divergence of the logarithmic means

To prove that the logarithmic means diverge in the $\mathcal{H}(b)$-norm, we
use another expression of the logarithmic means in terms of the classical Abel
means or, equivalently, in terms of the dilates $f_{r}$ of a function
$f\in\operatorname{Hol}(\mathbb{D})$.

The first step toward this formula is the following lemma.

######  Lemma 4.1.

Let $b$ be non-extreme, let $f\in\mathcal{H}(b)$ and let $r\in(0,1)$. The
application $F:[0,r]\rightarrow\mathcal{H}(b)$, defined by $F(t):=f_{t}$, is
continuous from $[0,r]$ into $\mathcal{H}(b)$.

###### Proof.

By Corollary 3.6, for any $R>1$ there exists a finite positive constant $C>0$,
which depends on $R$ and $f$, such that

$\displaystyle\|s_{n}[f]\|_{b}\leq CR^{n}.$

By choosing $R$ so that $Rr<1$, we can ensure that the series

$(1-t)\sum_{n\geq 0}s_{n}[f]t^{n}\quad(0\leq t\leq r)$

converges in $\mathcal{H}(b)$-norm to some function $g^{(t)}\in\mathcal{H}(b)$
on $[0,r]$. The dilates $f_{t}$ can be expressed pointwise as

$\displaystyle f_{t}(z)=(1-t)\sum_{n\geq
0}s_{n}[f](z)t^{n}\quad(z\in\mathbb{D}),$

and therefore we must have that $g^{(t)}=f_{t}$, since convergence in
$\mathcal{H}(b)$ implies pointwise convergence.

The continuity of $F$ now follows from the scalar continuity of each map
$t\mapsto t^{n}(1-t)$ on $[0,r]$, $n\geq 0$. ∎

For the next lemma, we show that the integral formula (2.1) linking the
logarithmic means to the Abel means is also valid in $\mathcal{H}(b)$. We use
the Bochner integral as the definition of the vector-valued integral. For
background on this topic, we refer the reader to [9, Chapter 3, §1].

######  Lemma 4.2.

Let $b$ be non-extreme and let $f\in\mathcal{H}(b)$, say $f(z)=\sum_{n\geq
0}a_{n}z^{n}$. Then, for each $r\in[0,1)$, we have $L_{r}[f]\in\mathcal{H}(b)$
and

$\displaystyle
L_{r}[f]=\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r}\frac{f_{t}}{1-t}\,dt.$

###### Proof.

Fix $r\in[0,1)$. From Lemma 4.1, the function $t\mapsto\frac{f_{t}}{1-t}$ is
continuous from $[0,r]$ into $\mathcal{H}(b)$, and so its Bochner integral is
well-defined.

By Corollary 3.6 again, for any $R>1$, we have that $\|s_{n}[f]\|_{b}\leq
CR^{n}$, where $C$ is a positive constant depending only on $R$ and $f$. Let
$R>1$ be chosen so that $Rr<1$.

Firstly, we compute an upper bound for the series defining $L_{r}[f]$:

$\displaystyle\frac{r}{\log\frac{1}{1-r}}\sum_{n\geq
0}\frac{\|s_{n}[f]\|_{b}}{n+1}r^{n}\leq
C\frac{r}{\log\frac{1}{1-r}}\sum_{n\geq
0}\frac{(rR)^{n}}{n+1}=C\frac{\log\frac{1}{1-rR}}{R\log\frac{1}{1-r}}.$

Therefore the series $\frac{r}{\log\frac{1}{1-r}}\sum_{n\geq
0}\frac{s_{n}[f]}{n+1}r^{n}$ converges absolutely in $\mathcal{H}(b)$ and it
defines a function in $\mathcal{H}(b)$, say $g^{(r)}\in\mathcal{H}(b)$. Since
convergence in $\mathcal{H}(b)$ implies pointwise convergence, we must also
have

$\displaystyle g^{(r)}(z)=\frac{r}{\log\frac{1}{1-r}}\sum_{n\geq
0}\frac{s_{n}[f](z)}{n+1}r^{n}\quad(z\in\mathbb{D}),$

which gives $g^{(r)}=L_{r}[f]\in\mathcal{H}(b)$.

Secondly, we have

$\displaystyle\frac{r}{\log\frac{1}{1-r}}\sum_{n\geq
0}\frac{s_{n}[f]}{n+1}r^{n}=\frac{1}{\log\frac{1}{1-r}}\sum_{n\geq
0}\int_{0}^{r}s_{n}[f]t^{n}\,dt.$

The series $\sum_{n\geq 0}s_{n}[f]t^{n}$ is absolutely and uniformly
convergent in $\mathcal{H}(b)$ on $[0,r]$. Therefore, the order of summation
and integration can be interchanged and we get

$\displaystyle L_{r}[f]=\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r}\sum_{n\geq
0}s_{n}[f]t^{n}\,dt=\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r}\frac{f_{t}}{1-t}\,dt,$

where the last equality comes from the fact that $f_{t}=(1-t)\sum_{n\geq
0}s_{n}[f]t^{n}$. ∎

Using this last lemma and the estimate in Corollary 3.7, we obtain the
following result. We consider the logarithmic means $L_{r}$ as linear
operators on $\mathcal{H}(b)$ defined by $L_{r}(f):=L_{r}[f]$ for each
$f\in\mathcal{H}(b)$.

######  Corollary 4.3.

Let $b$ be non-extreme. Then, for every $r\in[0,1)$, the linear map
$L_{r}:\mathcal{H}(b)\rightarrow\mathcal{H}(b)$ is a bounded linear operator
with $\|L_{r}\|\leq\sqrt{C(\phi,r)}$, where $C(\phi,r)$ is the same constant
as in Corollary 3.7. In fact we even have

$\displaystyle\|L_{r}(f)\|_{b}\leq\sqrt{C(\phi,r)}\|f\|_{H^{2}}\quad(f\in\mathcal{H}(b)).$
(4.1)

###### Proof.

Let $r\in[0,1)$ and $f\in\mathcal{H}(b)$. Then, from Corollary 3.7, we have

$\displaystyle\frac{1}{\log\left(\frac{1}{1-r}\right)}\int_{0}^{r}\frac{\|f_{t}\|_{b}}{1-t}\,dt\leq\frac{\|f\|_{H^{2}}}{\log\left(\frac{1}{1-r}\right)}\int_{0}^{r}\frac{\sqrt{C(\phi,t)}}{1-t}\,dt,$

and, since $t\mapsto C(\phi,t)$ is increasing on $[0,r)$, the above expression
is

$\displaystyle\leq\frac{C(\phi,r)\|f\|_{H^{2}}}{\log\left(\frac{1}{1-r}\right)}\int_{0}^{r}\frac{1}{1-t}\,dt\leq\sqrt{C(\phi,r)}\|f\|_{H^{2}}.$

In combination with Lemma 4.2, this establishes (4.1). Since
$\|f\|_{H^{2}}\leq\|f\|_{b}$, it follows that $L_{r}$ is a bounded operator on
$\mathcal{H}(b)$ with $\|L_{r}\|\leq\sqrt{C(\phi,r)}$. ∎

Now we can exploit the integral formula of $L_{r}[f]$ to express
$(L_{r}[f])^{+}$ in terms of a certain integral involving $(f_{r})^{+}$.

######  Theorem 4.4.

For any $f\in\mathcal{H}(b)$ and $r\in(0,1)$, we have that

$\displaystyle(L_{r}[f])^{+}=\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r}\frac{(f_{t})^{+}}{1-t}\,dt.$
(4.2)

###### Proof.

Define $F(t):=\frac{f_{t}}{1-t}$. This is a continuous function from $[0,r]$
into $\mathcal{H}(b)$, by Lemma 4.1. The fact that $F(t)$ is continuous from
$[0,r]$ into $\mathcal{H}(b)$, combined with the formula for the norm in
$\mathcal{H}(b)$ in terms of $f$ and $f^{+}$, imply that the mapping
$t\mapsto\frac{(f_{t})^{+}}{1-t}$ is continuous from $[0,r]$ into $H^{2}$.
Therefore, it is Bochner-integrable on $[0,r]$. Thus, since
$T_{\overline{a}}:H^{2}\rightarrow H^{2}$ and
$T_{\overline{b}}:H^{2}\rightarrow H^{2}$ are bounded operators, we get

$\displaystyle
T_{\overline{a}}\left(\left(\log\frac{1}{1-r}\right)^{-1}\int_{0}^{r}\frac{(f_{t})^{+}}{1-t}\,dt\right)$
$\displaystyle=\left(\log\frac{1}{1-r}\right)^{-1}\int_{0}^{r}\frac{T_{\overline{a}}(f_{t})^{+}}{1-t}\,dt$
$\displaystyle=\left(\log\frac{1}{1-r}\right)^{-1}\int_{0}^{r}\frac{T_{\overline{b}}f_{t}}{1-t}\,dt$
$\displaystyle=T_{\overline{b}}\left(\left(\log\frac{1}{1-r}\right)^{-1}\int_{0}^{r}\frac{f_{t}}{1-t}\,dt\right).$

Thus, by the uniqueness of $(L_{r}[f])^{+}$, we get formula (4.2). ∎

Now, we can prove our main result.

###### Proof of Theorem 1.1.

Choose $b$ and $f$ as in Theorem 3.8. Let $A>0$ and choose $r_{0}\in(0,1)$ so
that

$\displaystyle(f_{t})^{+}(0)\geq A\quad(r_{0}<t<1).$

Convergence in $H^{2}$ implies pointwise convergence on $\mathbb{D}$.
Therefore, by Theorem 4.4, the following equality holds:

$\displaystyle(L_{r}[f])^{+}(0)=\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r}\frac{(f_{t})^{+}(0)}{1-t}\,dt.$

Fix $r\in[0,1)$. Splitting the integral in two parts, from $0$ to $r_{0}$ and
from $r_{0}$ to $r$, we have

$\displaystyle(L_{r}[f])^{+}(0)\geq\frac{\log\frac{1-r_{0}}{1-r}}{\log\frac{1}{1-r}}A+\frac{1}{\log\frac{1}{1-r}}\int_{0}^{r_{0}}\frac{(f_{t})^{+}(0)}{1-t}\,dt.$

Taking the $\liminf$ as $r\rightarrow 1^{-}$, we get

$\displaystyle\liminf_{r\rightarrow 1^{-}}(L_{r}[f])^{+}(0)\geq A.$

Since $A$ was arbitrary, $\liminf_{r\rightarrow
1^{-}}(L_{r}[f])^{+}(0)=\infty$. By the expression of the norm of $L_{r}[f]$
in $\mathcal{H}(b)$, it follows that

$\displaystyle\|L_{r}[f]\|_{b}\geq|(L_{r}[f])^{+}(0)|$

which implies that $\liminf_{r\rightarrow 1^{-}}\|L_{r}[f]\|_{b}=\infty$. This
concludes the proof of the theorem. ∎

##  5\. Scalar-inclusion and summability in Banach spaces

Before analyzing the consequences of Theorem 1.1 on other power-series
methods, we take a little detour to prove an abstract theorem on scalar-
inclusion of two sequence-to-function summability methods. This theorem will
be the main ingredient to prove Corollary 1.2.

######  Theorem 5.1.

Let $K$ and $H$ be two regular sequence-to-function summability methods. Let
$X$ and $Y$ be Banach spaces, and let $S:X\rightarrow Y$ and
$S_{n}:X\rightarrow Y$ $(n\geq 0)$ be bounded linear operators. Suppose that:

  * •

$S_{n}(x)\to S(x)$ for all $x\in W$, where $W$ is a dense subset of $X$;

  * •

$(S_{n}(x))_{n\geq 0}$ is $K$-summable to $S(x)$ for all $x\in X$;

  * •

$K$ is scalar-included in $H$.

Then $(S_{n}(x))_{n\geq 0}$ is $H$-summable to $S(x)$ for all $x\in X$.

###### Proof.

Let $(k_{n})_{n\geq 0},(h_{n})_{n\geq 0}:[0,R)\to\mathbb{C}$ be the functions
defining the summability methods $K$ and $H$ respectively. We need to prove
that, for each $x\in X$:

  1. (i)

$\sum_{n\geq 0}h_{n}(r)S_{n}(x)$ converges in $Y$ for all $r\in[0,R)$;

  2. (ii)

$\|\sum_{n\geq 0}h_{n}(r)S_{n}(x)-S(x)\|_{Y}\to 0$ as $r\to R^{-}$.

We begin with (i). Fix $r\in[0,R)$. Given $x\in X$, the sequence
$(S_{n}(x))_{n\geq 0}$ is $K$-summable to $S(x)$. By linearity and continuity,
for each $\phi\in Y^{*}$, the sequence $(\phi(S_{n}(x)))$ is $K$-summable to
$\phi(S(x))$. As $K$ is scalar-included in $H$, it follows that
$(\phi(S_{n}(x)))$ is also $H$-summable to $\phi(S(x))$. In particular, the
series $\sum_{n\geq 0}h_{n}(r)\phi(S_{n}(x))$ converges in $\mathbb{C}$. Hence

$\sup_{m\geq
0}\Bigl{|}\sum_{n=0}^{m}h_{n}(r)\phi(S_{n}(x))\Bigr{|}<\infty\quad(x\in
X,~{}\phi\in Y^{*}).$

In other words

$\sup_{m\geq
0}\Bigl{|}\phi\Bigl{(}\sum_{n=0}^{m}h_{n}(r)S_{n}(x)\Bigr{)}\Bigr{|}<\infty\quad(x\in
X,~{}\phi\in Y^{*}).$

Applying the Banach–Steinhaus theorem twice (once for $\phi$ and once for
$x$), we obtain that

$M:=\sup_{m\geq 0}\Bigl{\|}\sum_{n=0}^{m}h_{n}(r)S_{n}\Bigr{\|}<\infty,$ (5.1)

where now the norm is the operator norm.

Given $x\in X$ and $\epsilon>0$, choose $w\in W$ such that
$\|x-w\|_{X}<\epsilon/M$. As $S_{n}(w)\to S(w)$ and $H$ is a regular
summability method, the sequence $(S_{n}(w))$ is $H$-summable to $S(w)$. In
particular, the series $\sum_{n\geq 0}h_{n}(r)S_{n}(w)$ converges in $Y$. It
follows that $\|\sum_{n=m_{1}}^{m_{2}}h_{n}(r)S_{n}(w)\|_{Y}<\epsilon$ for all
large enough $m_{1},m_{2}$. For all such $m_{1},m_{2}$, we then have

$\displaystyle\Bigl{\|}\sum_{n=m_{1}}^{m_{2}}h_{n}(r)S_{n}(x)\Bigr{\|}_{Y}$
$\displaystyle\leq\Bigl{\|}\sum_{n=m_{1}}^{m_{2}}h_{n}(r)S_{n}(x-w)\Bigr{\|}_{Y}+\Bigl{\|}\sum_{n=m_{1}}^{m_{2}}h_{n}(r)S_{n}(w)\Bigr{\|}_{Y}$
$\displaystyle\leq\Bigl{\|}\sum_{n=m_{1}}^{m_{2}}h_{n}(r)S_{n}\Bigr{\|}\|x-w\|_{X}+\epsilon$
$\displaystyle\leq 2M(\epsilon/M)+\epsilon=3\epsilon.$

This shows that the series $\sum_{n=0}^{\infty}h_{n}(r)S_{n}(x)$ is Cauchy,
and therefore it converges in $Y$, thereby completing the proof of (i).

Now we turn to (ii). For each $r\in[0,R)$, define an operator $S^{H}_{r}:X\to
Y$ by

$S^{H}_{r}(x):=\sum_{n\geq 0}h_{n}(r)S_{n}(x)\quad(x\in X).$

By (i) the series converges, so $S^{H}_{r}$ is well-defined and linear.
Furthermore, it follows easily from (5.1) that $S^{H}_{r}$ is a bounded linear
operator from $X$ into $Y$. As we saw in (i), for each $x\in X$ and $\phi\in
Y^{*}$, the sequence $\phi(S_{n}(x))$ is $H$-summable to $\phi(S(x))$, in
other words

$\phi(S^{H}_{r}(x))\to\phi(S(x))\quad(r\to R^{-}).$ (5.2)

We want to prove that $S_{r}^{H}(x)\to S(x)$ as $r\to R^{-}$. To do so, let
$(r_{j})_{j\geq 0}$ be a sequence in $[0,R)$ such that $r_{j}\to R^{-}$
($j\to\infty$). We will prove that $S_{r_{j}}^{H}(x)\rightarrow S(x)$ as
$j\to\infty$. By (5.2), we have $\phi(S^{H}_{r_{j}}(x))\to\phi(S(x))$ as
$j\to\infty$ for each $x\in X$ and $\phi\in Y^{*}$. By the Banach–Steinhaus
theorem, again applied twice, it follows that

$N:=\sup_{j\geq 0}\|S^{H}_{r_{j}}\|<\infty.$

Given $x\in X$ and $\epsilon>0$, choose $w\in W$ such that

$\|x-w\|_{X}<\epsilon/\max\\{N,\|S\|\\}.$

By regularity of $H$, we have

$S^{H}_{r_{j}}(w)\to S(w)\quad(j\to\infty).$

Hence $\|S^{H}_{r_{j}}(w)-S(w)\|_{Y}<\epsilon$ for all $j$ sufficiently large.
For all such $j$, we then have

$\displaystyle\|S^{H}_{r_{j}}(x)-S(x)\|_{Y}$
$\displaystyle\leq\|S^{H}_{r_{j}}(x-w)\|_{Y}+\|S^{H}_{r_{j}}(w)-S(w)\|_{Y}+\|S(w-x)\|_{Y}$
$\displaystyle\leq N(\epsilon/N)+\epsilon+\|S\|(\epsilon/\|S\|)=3\epsilon.$

We conclude that $S^{H}_{r_{j}}(x)\to S(x)$ as $j\rightarrow\infty$,
completing the proof of (ii). ∎

##  6\. Consequences for other power-series methods

Our final goal is to prove Corollary 1.2. Recall from §2 that the generalized
Abel means of order $\alpha>-1$ applied to a function
$f\in\operatorname{Hol}(\mathbb{D})$ are

$\displaystyle A^{\alpha}_{r}[f](z):=(1-r)^{1+\alpha}\sum_{n\geq
0}\binom{n+\alpha}{\alpha}s_{n}(z)r^{n}\quad(z\in\mathbb{D},\,r\in[0,1))\text{.}$

Let $f\in\mathcal{H}(b)$ and let $r\in[0,1)$. By Corollary 3.6, for any $R>1$,
we have $\|s_{n}[f]\|_{b}\leq CR^{n}$ for some constant $C$ depending only on
$R$ and $f$. From this, it is easy to see that the series defining
$A^{\alpha}_{r}[f]$ converges absolutely to some function
$g^{(r)}\in\mathcal{H}(b)$ and, since convergence in $\mathcal{H}(b)$ implies
pointwise convergence, we get $A^{\alpha}_{r}[f]=g^{(r)}\in\mathcal{H}(b)$.
Thus, we may view the generalized Abel mean as a linear operator on
$\mathcal{H}(b)$ defined by

$\displaystyle
A^{\alpha}_{r}(f):=A^{\alpha}_{r}[f]\quad(f\in\mathcal{H}(b),\,r\in[0,1)).$

By the closed-graph theorem, it is a bounded linear operator on
$\mathcal{H}(b)$.

To use Theorem 5.1, we need the following relation between the generalized
Abel methods and the logarithmic method. This result can be found in [3, §5].

######  Theorem 6.1.

All generalized Abel methods of order $\alpha$, with $\alpha>-1$, are scalar-
included in the logarithmic method.

We are now ready to prove Corollary 1.2.

###### Proof of Corollary 1.2.

Let $b$ be as in Theorem 1.1. Suppose, if possible, that, for every
$f\in\mathcal{H}(b)$,

$A_{r}^{\alpha}(f)\rightarrow f,\quad(r\rightarrow 1^{-}).$

To apply Theorem 5.1, let $X=Y=\mathcal{H}(b)$, let $W$ be the set of
polynomials, let $R=1$, let $K$ and $H$ be the generalized Abel method of
order $\alpha>-1$ and the logarithmic method respectively, let $S_{n}:=s_{n}$,
the operator that maps each function to the $n$-th partial sum of its Taylor
expansion, and let $S:=I$ be the identity on $\mathcal{H}(b)$.

By Theorem 6.1, the Abel method of order $\alpha>-1$ is scalar-included in the
logarithmic method. The logarithmic method is also a regular sequence-to-
function summability method because the function
$\frac{1}{r}\log(\tfrac{1}{1-r})\rightarrow\infty$ as $r\rightarrow 1^{-}$.
Moreover, by Theorem 3.3, the set of polynomials $W$ is dense in
$\mathcal{H}(b)$ since $b$ is non-extreme.

It remains to verify the first condition of Theorem 5.1, that is,
$s_{n}(p)\rightarrow p$ for any $p\in W$. This is clear since, for $p\in W$,
with

$p(z)=\sum_{n=0}^{N}a_{n}z^{n}\quad(z\in\mathbb{D}),$

we have $s_{n}(p)=s_{n}[p]=p$ if $n\geq N$.

Therefore, by Theorem 5.1, we infer that $(s_{n}(f))_{n\geq 0}$ is summable by
the logarithmic method for every $f\in\mathcal{H}(b)$. This contradicts
Theorem 1.1. Therefore, there exists a function $f\in\mathcal{H}(b)$ such that
$A_{r}^{\alpha}(f)\not\rightarrow f$ as $r\rightarrow 1^{-}$. ∎

In fact, Corollary 1.2 generalizes to a whole family of power series methods.
This generalization is a consequence of the following inclusion theorem due to
Borwein.

######  Theorem 6.2.

[3, Theorem A] Let $p(x)=\sum_{n\geq 0}p_{n}x^{n}$ be a power series with a
radius of convergence $R_{p}>0$. Let $q(x)=\sum_{n\geq 0}q_{n}x^{n}$ be
another power series with a radius of convergence $R_{q}=R_{p}$. Suppose that
there exist an integer $N$, a finite signed measure $\mu$ on the interval
$[0,1]$ and a number $\delta\in(0,1]$ such that, for all $n\geq N$,

  1. (1)

$p_{n}=q_{n}\int_{0}^{1}t^{n}\,d\mu(t)$;

  2. (2)

$\int_{0}^{1}t^{n}\,d\mu(t)\geq\delta\int_{0}^{1}t^{n}\,|d\mu(t)|$.

Then the power-series method $(q)$ is scalar-included in the power-series
method $(p)$.

Consider the power-series method defined by the power series
$p(r):=\sum_{n\geq 0}p_{n}r^{n}$ with a radius of convergence $R_{p}=1$. We
suppose that, for the coefficients $(p_{n})_{n\geq 0}$, there exist an integer
$N$, a finite signed measure $\mu$ on $[0,1]$ and a number $\delta\in(0,1]$
such that, for every $n\geq N$,

  1. (A)

$\frac{1}{n+1}=p_{n}\int_{0}^{1}t^{n}\,d\mu(t)$;

  2. (B)

$\frac{1}{n+1}\geq\delta p_{n}\int_{0}^{1}t^{n}\,|d\mu(t)|$.

As an explicit example, the coefficients $p_{n}:=\binom{n+\alpha}{\alpha}$
with $\alpha>-1$ satisfy these requirements (see [3] for details).

Recall that, applied to the partial sums of the Taylor expansion of
$f(z)=\sum_{n\geq 0}a_{n}z^{n}$, the means defined by the power-series method
$(p)$ are

$\displaystyle P_{r}[f]:=\frac{1}{p(r)}\sum_{n\geq
0}p_{n}s_{n}[f]r^{n}\quad(0\leq r<1).$

Using formula (3.1) again, we can prove that the series defining $P_{r}[f]$
converges, in $\mathcal{H}(b)$, to some function $g^{(r)}\in\mathcal{H}(b)$
and that $P_{r}[f]=g^{(r)}\in\mathcal{H}(b)$. Also, by the closed-graph
theorem, the linear operator $P_{r}:\mathcal{H}(b)\rightarrow\mathcal{H}(b)$,
defined by $P_{r}(f):=P_{r}[f]$, is bounded for each $r\in[0,1)$.

######  Theorem 6.3.

Let $p(r):=\sum_{n\geq 0}p_{n}r^{n}$ be a power series with a radius of
convergence $R_{p}=1$. Suppose that the coefficients $(p_{n})_{n\geq 0}$
satisfy the conditions (A) and (B) above, with $N\geq 0$ an integer, $\mu$ a
finite signed measure on $[0,1]$ and $\delta\in(0,1]$. Then there exist a non-
extreme point $b\in H^{\infty}$ and a function $f\in\mathcal{H}(b)$ such that
$P_{r}[f]\not\rightarrow f$ in $\mathcal{H}(b)$ as $r\rightarrow 1^{-}$.

###### Proof.

The proof is similar to Corollary 1.2. The only difference is that we use
Theorem 6.2 instead of Theorem 6.1 to show that the power-series method
defined by $(p)$ is scalar-included in the logarithmic method. ∎

There are plenty of other summability methods which are included in the
logarithmic method that we did not treat here. For example, in [4], it was
shown that the generalized Borel method $B^{\alpha,\beta}$, with its means
defined by

$B_{x}^{\alpha,\beta}[f]:=\sum_{n\geq N}\frac{s_{n}[f]}{\Gamma(\alpha
n+\beta)}x^{\alpha n+\beta-1},$

where $x>0$, $\alpha>0$, $\beta\in\mathbb{R}$ and $N$ is a non-negative
integer such that $\alpha N+\beta>1$, is scalar-included in the logarithmic
method. With similar techniques, we can prove that
$B^{\alpha,\beta}_{x}[f]\in\mathcal{H}(b)$ for each $f\in\mathcal{H}(b)$ and,
by using Theorem 5.1, we can prove that there exist a non-extreme $b$ and a
function $f\in\mathcal{H}(b)$ such that
$B^{\alpha,\beta}_{x}[f]\not\rightarrow f$ in $\mathcal{H}(b)$ as
$x\rightarrow\infty$.

##  7\. An open problem

We have studied some classes of sequence-to-function summability methods in
this article and we proved that none of them works in general for de
Branges–Rovnyak spaces. Moreover, in [11], the authors showed that there exist
a Hilbert space of analytic functions $\mathcal{H}$ continuously embedded in
$\operatorname{Hol}(\mathbb{D})$ and a function $f\in\mathcal{H}$ such that
$f$ lies outside the closed linear span of $\left\\{s_{n}[f]\,:\,n\geq
0\right\\}$. In particular, no summability methods applied to the partial sums
$s_{n}[f]$ would work to make the Taylor series summable to $f$ in
$\mathcal{H}$. This motivates the following problem.

######  Problem 7.1.

Does there exist a non-extreme function $b$ and a function
$f\in\mathcal{H}(b)$ such that $f$ lies outside the closed linear span of
$\left\\{s_{n}[f]\,:\,n\geq 0\right\\}$?

## Acknowledgement

We are grateful to the anonymous referee for reading the article carefully and
for making several helpful suggestions to improve the exposition.

## References

  * [1] J. Boos.  Classical and modern methods in summability.  Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000\.  Assisted by Peter Cass, Oxford Science Publications.
  * [2] D. Borwein.  On a scale of Abel-type summability methods.  Proc. Cambridge Philos. Soc., 53:318–322, 1957.
  * [3] D. Borwein.  On methods of summability based on power series.  Proc. Roy. Soc. Edinburgh Sect. A, 64:342–349, 1957.
  * [4] D. Borwein and B. Watson.  On the relation between the logarithmic and Borel-type summability methods.  Canad. Math. Bull., 24(2):153–159, 1981.
  * [5] N. Chevrot, D. Guillot, and T. Ransford.  De Branges-Rovnyak spaces and Dirichlet spaces.  J. Funct. Anal., 259(9):2366–2383, 2010.
  * [6] P. L. Duren.  Theory of $H^{p}$ spaces.  Pure and Applied Mathematics. Academic Press, New York and London, 1970\.
  * [7] O. El-Fallah, E. Fricain, K. Kellay, J. Mashreghi, and T. Ransford.  Constructive approximation in de Branges-Rovnyak spaces.  Constr. Approx., 44(2):269–281, 2016.
  * [8] G. H. Hardy.  Divergent series.  Oxford, at the Clarendon Press, 1949.
  * [9] E. Hille and R. S. Phillips.  Functional analysis and semi-groups.  American Mathematical Society, Providence, R. I., 1974.  Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications, Vol. XXXI.
  * [10] M. Jevtić, D. Vukotić, and M. Arsenović.  Taylor coefficients and coefficient multipliers of Hardy and Bergman-type spaces, volume 2 of RSME Springer Series.  Springer, Cham, 2016.
  * [11] J. Mashreghi, P.-O. Parisé, and T. Ransford.  Failure of approximation of odd functions by odd polynomials.  Constr. Approx. to appear.
  * [12] J. Mashreghi and T. Ransford.  Linear polynomial approximation schemes in banach holomorphic function spaces.  Analysis and Mathematical Physics, 9:899–905, 2019.
  * [13] A. Robinson.  On functional transformations and summability.  Proc. London Math. Soc. (2), 52:132–160, 1950.
  * [14] D. Sarason.  Sub-Hardy Hilbert spaces in the unit disk.  The University of Arkensas Lecture Notes in the Mathematical Sciences. Wiley-Interscience, 1994.