# Using Artificial Neural Networks to extract the 21-cm Global Signal from the
EDGES data

Madhurima Choudhury 1, Atrideb Chatterjee 2, Abhirup Datta 1, Tirthankar Roy
Choudhury 2
1Discipline of Astronomy, Astrophysics Indian Institute of Technology Indore,
India
2National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, Pune 411007, India
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The redshifted 21-cm signal of neutral Hydrogen is a promising probe into the
period of evolution of our Universe when the first stars were formed (Cosmic
Dawn), to the period where the entire Universe changed its state from being
completely neutral to completely ionized (Reionization). The most striking
feature of this line of neutral Hydrogen is that it can be observed across an
entire frequency range as a sky-averaged continuous signature, or its
fluctuations can be measured using an interferometer. However, the 21-cm
signal is very faint and is dominated by a much brighter Galactic and extra-
galactic foregrounds, making it an observational challenge. We have used
different physical models to simulate various realizations of the 21-cm Global
signals, including an excess radio background to match the amplitude of the
EDGES 21-cm signal. First, we have used an artificial neural network (ANN) to
extract the astrophysical parameters from these simulated datasets. Then, mock
observations were generated by adding a physically motivated foreground model
and an ANN was used to extract the astrophysical parameters from such data.
The $R^{2}$ score of our predictions from the mock-observations is in the
range of 0.65-0.89. We have used this ANN to predict the signal parameters
giving the EDGES data as the input. We find that the reconstructed signal
closely mimics the amplitude of the reported detection. The recovered
parameters can be used to infer the physical state of the gas at high
redshifts.

###### keywords:

21-cm Global signal-cosmology- model-artificial neural networks

††pubyear: 2017††pagerange: Using Artificial Neural Networks to extract the
21-cm Global Signal from the EDGES data –Using Artificial Neural Networks to
extract the 21-cm Global Signal from the EDGES data

##  1 Introduction

The evolution of our Universe from the Cosmic Dawn (CD) and Epoch of
Reionization (EoR) is not very well understood. The HI 21-cm line promises to
be an excellent probe into these epochs (Furlanetto et al., 2006a; Morales &
Wyithe, 2010; Pritchard & Loeb, 2012a). The Global 21-cm experiments aim to
measure the sky-averaged signature of this redshifted line, by using ground-
based radio telescopes. Examples of such experiments are the Shaped Antenna
measurement of the background RAdio Spectrum (SARAS), (Patra et al., 2013;
Singh et al., 2017), the Large-Aperture Experiment to Detect the Dark Ages
(LEDA) (Greenhill & Bernardi, 2012), SCI-HI (Voytek et al., 2014), the
Broadband Instrument for Global Hydrogen Reionisation Signal (BIGHORNS)
(Sokolowski et al., 2015), and the Cosmic Twilight Polarimeter, CTP (Nhan et
al., 2018). The Experiment to Detect the Global EoR Signature (EDGES, Bowman
et al. (2018a)) has reported a possible detection of the sky-averaged HI 21-cm
Global signal from the Cosmic Dawn. However, this signal has an absorption
trough of about $0.5$K, which is twice the amplitude predicted by the standard
model of cosmology. If confirmed, this measured signal would give us a
completely new insight into the physics of the evolution of the Universe.
Following this detection, several models explaining this exceedingly deep
absorption trough has been explored. (Barkana, 2018a; Fraser et al., 2018;
Pospelov et al., 2018; Slatyer & Wu, 2018) had explained this with models with
excess cooling from nonstandard physics, including dark matter particles
scattering off baryons. (Fialkov et al., 2018; Fraser et al., 2018; Pospelov
et al., 2018; Ewall-Wice et al., 2018a; Feng & Holder, 2018) have explored
various possible 21-cm signals, varying the properties of the dark matter
particles, in addition to varying the astrophysical parameters. Ewall-Wice et
al. (2018a); Feng & Holder (2018); Fialkov & Barkana (2019), have explored
models which produce an excess radio background, in addition to the cosmic
radio background to explain the large amplitude of the absorption feature in
the EDGES detection. This radio excess could be, for example, due to
supernovae (Mirocha & Furlanetto, 2019; Jana et al., 2019) or primordial black
holes Ewall-Wice et al. (2018a) .

However, detecting the 21-cm signal is an observational challenge. When
observed from Earth, the 21-cm signal, which is of the order of a few hundreds
of $\mathrm{mK}$, is buried in a sea of very bright, dominating foregrounds.
These foregrounds are several orders of magnitude brighter than the signal. In
addition, there are effects of ionospheric distortion, radio frequency
interference (RFI) and the frequency response of instrument which make the
observed sky vary with time and other factors. One of the common techniques
for extracting the faint cosmological signal is by assuming that the
foreground is well-characterized, spectrally smooth and can be efficiently
removed from the total signal. The residual left would contain the 21-cm
signal including the signatures from the early phases of the formation of the
Universe.

There have been several works on the application of machine learning (ML)
techniques in the epoch of reionization. For example, Shimabukuro & Semelin
(2017) have used machine learning algorithms for extracting the parameters of
the 21-cm power spectrum, while Schmit & Pritchard (2018) have built an
emulator to generate several different realizations of 21-cm power spectrum
using artificial neural networks (ANN). Cohen et al. (2019) have emulated the
21-cm Global signal from the astrophysical parameters using ANN. Hassan et al.
(2019) have used convolutional neural networks (CNN) to identify reionization
sources from 21-cm maps. Deep learning models have been used to emulate the
radiative transfer during the epoch of reionization in Chardin et al. (2019).
In another work, Gillet et al. (2019) have recovered the astrophysical
parameters directly from 21-cm images, using deep learning with CNN. Jennings
et al. (2019) have compared machine learning techniques for predicting 21-cm
power spectrum from reionization simulations. While all the above-mentioned
works involve only the 21-cm signal of interest and reionization simulations,
Choudhury et al. (2020) had used artificial neural networks to extract the
21-cm Global signal parameters from mock observations which included effects
of foregrounds, instrumental effects and noise.

In this paper, we use Artificial Neural Networks (ANNs), to extract the
astrophysical parameters associated with the 21-cm Global signal from mock
observational data. A simple code has been used to generate several 21-cm
global signals by varying the input astrophysical parameters, which will be
discussed in the following sections. Adding foregrounds to these 21-cm
signals, we generate mock-observations which are then used as our training
set. Using this, we design an ANN to predict the signal parameters from the
total observed sky-signal with the added foreground. Finally, EDGES data has
been used as an input to this ANN and we have obtained the predicted signal
parameters. We then reconstruct the 21-cm signal using our code using the
predictions.

This paper is structured as follows: We begin with a brief overview of the
cosmological 21-cm signal in § 2. In § 3, we describe the physical models that
we have used to simulate the 21-cm Global signal. The foreground model used in
this work is described in detail in § 4. In § 5-6, we describe the methodology
and some basics of artificial neural networks. We present our results
obtained, in § 7 and use the ANN to predict the 21-cm signal from actual data,
in § 8. Finally, we discuss the implications of our predictions and summarize
our results.

##  2 Computing the 21-cm Global Signal

The hyperfine transition line of atomic hydrogen arises from the hyperfine
splitting of the ground state of the hydrogen due to the magnetic moment
interaction of the proton and electron. The differential brightness
temperature, $\delta T_{b}$, is measured relative to the Cosmic Microwave
Background (CMB) temperature, $T_{\gamma}$ and is given by (Furlanetto et al.,
2006b):

$\begin{split}\delta
T_{b}(\nu)&=\frac{T_{s}-T_{\gamma}}{1+z}(1-\exp^{-\tau_{\nu_{0}}})\\\ &\approx
27x_{HI}(1+\delta_{b})\left(\frac{\Omega_{b}h^{2}}{0.023}\right)\left(\frac{0.15}{\Omega_{m}h^{2}}\frac{1+z}{10}\right)^{1/2}\\\
&~{}~{}\left(1-\frac{T_{\gamma}(z)}{T_{s}}\right)\Big{[}\frac{\partial_{r}v_{r}}{(1+z)H(z)}\Big{]}^{-1},\end{split}$
(1)

where $x_{HI}$ is the neutral fraction of hydrogen, $\delta_{b}$ is the
fractional over-density of baryons, $\Omega_{b}$ and $\Omega_{M}$ are the
baryon and total matter density respectively, in units of the critical
density, $H(z)$ is the Hubble parameter and $T_{\gamma}(z)$ is the CMB
temperature at redshift z, $T_{s}$ is the spin temperature of neutral
hydrogen, and $\partial_{r}v_{r}$ is the velocity gradient along the line of
sight. The observed 21-cm Global signal is nothing but the sky averaged
differential brightness temperature, whose characteristic shape contains
information about the physical state of the Universe at early times, e.g., the
amount of UV radiation that can ionize hydrogen, the X-rays that can cause
heating of the gas, and $Ly-\alpha$ which causes Wouthuysen-Field coupling
(Wouthuysen, 1952; Field, 1959).

In our calculations, we neglect the peculiar velocity term and the density
fluctuation term in the Global signal (Eqn. 1), as it averages out to a linear
order and adds to a very small correction. So, Eqn. 1 can be re-written as:

${\delta T_{b}\approx
27(1-x_{i})\left(\frac{\Omega_{b}h^{2}}{0.023}\right)\left(\frac{0.15}{\Omega_{m}h^{2}}\frac{1+z}{10}\right)^{1/2}\left(1-\frac{T_{\gamma}}{T_{s}}\right)}$
(2)

This equation is primarily used to construct the Global signal.

The spin temperature, $T_{s}$,is the most interesting quantity in the
expression for the differential brightness temperature which primarily
determines the intensity of the 21-cm radiation [See Eqn. 2]. The three
competing processes that determine $T_{s}$ are: (1) absorption of CMB photons
(as well as stimulated emission); (2) collisions with other hydrogen atoms,
free electrons, and protons; and (3) scattering of Lyman alpha photons through
excitation and de-excitation. The spin temperature is given by (Field, 1959;
Pritchard et al., 2015):

$T_{s}^{-1}=\frac{T_{\gamma}^{-1}+x_{c}T_{k}^{-1}+x_{\alpha}T_{\alpha}^{-1}}{1+x_{c}+x_{\alpha}},$
(3)

where $T_{\gamma}$ is the CMB temperature, $T_{k}$ is the kinetic gas
temperature, $T_{\alpha}$ is the temperature related to the existence of
ambient Lyman-alpha $(Ly\alpha)$ photons. $x_{c}$ and $x_{\alpha}$ are
respectively the collisional coupling and the $Ly\alpha$ coupling terms. For a
detailed review see Barkana & Loeb (2005).
For this work, we developed a semi-numerical code which can produce a Global
21-cm signal over the redshift range $6<z<50$ in less than a second
(Chatterjee et al., 2019). Our calculation closely follows that of Furlanetto
et al. (2006c) and Pritchard & Loeb (2012b). It takes the following
astrophysical parameters as input:

  1. 1.

star formation efficiency $f_{\star}$,

  2. 2.

the escape fraction $f_{esc}$ of ionizing photons,

  3. 3.

X-ray heating efficiency $f_{X,h}$,

  4. 4.

the number of Ly$\alpha$ photons $N_{\alpha}$ produced per baryon in the
relevant frequency range,

  5. 5.

efficiency parameter $f_{R}$ of the radio background (only for models with
excess radio background),

and provides $\delta T_{b}$ as a function of redshift (or observed frequency).
We discuss the parameters in more detail below.

###  2.1 The coupling coefficients

We will first discuss how the coupling coefficients are calculated.

  1. 1.

There are three different channels of collisions, e.g., Hydrogen-Hydrogen
(H-H), Hydrogen-electron (H-e) and Hydrogen-proton (H-p), which determine the
collisional coupling coefficient, $x_{c}$. The calculation of the coupling
coefficient in our numerical code is described in detail in Chatterjee et al.
(2019).

  2. 2.

The Ly$\alpha$ coupling coefficient, $x_{\alpha}$ can be computed once the
background Ly$\alpha$ flux, $J_{\alpha}$, is known (Furlanetto et al., 2006a)
(F06 hereafter). We approximate $J_{\alpha}$ following F06.

###  2.2 The gas kinetic temperature

In the redshift regime of our interest, the gas kinetic temperature, $T_{k}$,
is governed by two processes (i) Adiabatic cooling due to expansion of the
Universe and (ii) X-ray heating (F06). Given our lack of understanding in
modelling the X-ray background we take the simplest possible approach as
described in F06. We simply extrapolate the correlation between the star
formation rate (SFR), $\dot{M}_{*}$ and the X-ray luminosity, $L_{X}$, in the
present Universe to higher redshifts and write it as,

$L_{X}=3.4\times
10^{33}f_{X}\left(\frac{\dot{M}_{*}}{\text{M}_{\odot}~{}\text{yr}^{-1}}\right)\mbox{J~{}s}^{-1},$
(4)

where $f_{X}$ is an unknown normalization factor taking into account the
difference between local and high-$z$ observations. Defining $f_{coll}$ to be
the collapsed fraction of dark matter halos with mass greater than $m_{min}$,
the X-ray emissivity can be expressed as,

$\epsilon_{X}=7.5\times 10^{3}\text{K}\,nk_{B}H(z)\times
f_{X,h}\left(\frac{f_{\star}}{0.1}\frac{df_{coll}/dz}{0.01}\frac{1+z}{10}\right),$
(5)

where n is the total number of gas particles, $k_{B}$ is the Boltzmann’s
constant and $f_{X,h}=f_{X}\times f_{h}$, where $f_{h}$ is the heating
fraction of the X-rays (rest of them ionize the IGM) .

Although it is possible to use a fixed value for the ionization fraction
$x_{i}\simeq 10^{-4}$ (Bharadwaj & Ali, 2004) for calculating the 21-cm signal
in the redshift range we are interested in, we still compute the evolution of
$x_{i}$ with redshift using the following equation (F06),

$\frac{{\rm d}x_{i}}{{\rm
d}t}=\zeta(z)\frac{df_{coll}}{dt}-x_{i}~{}\alpha_{B}~{}{\cal C}~{}n_{H,{\rm
com}}~{}(1+z)^{3},$ (6)

where $n_{H,{\rm com}}$ is the comoving number density of Hydrogen,
$\alpha_{B}$ is the (case B) recombination rate coefficient and ${\cal C}$ is
the clumping factor of the IGM. $\zeta(z)$ is defined as,

$\zeta=A_{\mathrm{H}e}f_{\star}f_{esc}\text{N}_{\text{ion}}$ (7)

where $A_{\mathrm{H}e}=1.21$ is a correction factor for helium, $f_{\rm esc}$
is the escape fraction of ionizing photons, $\text{N}_{\text{ion}}$ is the
number of ionizing photons per baryon produced in stars. We modelled the
clumping factor as ${\cal C}=1+43z^{-1.71}$ (Haardt & Madau, 2012; Pawlik et
al., 2009).

(a) Traditional set

(b) Exotic Set

(c) Total set

Figure 1: The sample 21-cm signals generated by using the (a)traditional
models and (b)the models with excess radio background. The differential
brightness temperature, $\delta T_{b}$, is plotted versus frequencies(MHz).
This set of sample 21cm signals are used for building the training datasets
(c). The colorbar on the right represents the optical depth.

###  2.3 Models with excess radio background

It has now been well accepted that matching the amplitude of the EDGES signal
would either require making the gas colder, using exotic physics such as new
dark matter interactions (Barkana, 2018b; Fraser et al., 2018; Pospelov et
al., 2018; Slatyer & Wu, 2018) or invoking a radiation temperature larger than
the CMB temperature (Fraser et al., 2018; Pospelov et al., 2018; Ewall-Wice et
al., 2018b; Feng & Holder, 2018). In this work, we take the latter approach as
it got some support by observations of an excess radio background in the local
Universe from ARCADE-2 experiment (Fixsen et al., 2011).

The excess radio background is modelled assuming the local radio-SFR
($L_{R}-\dot{M}_{*}$) relation at $150$ MHz (Gürkan et al., 2018) holds at
higher redshift and can be written as

$L_{R}=10^{22}f_{R}\left(\frac{\dot{M}_{*}}{1\text{M}_{\odot}\text{yr}^{-1}}\right)\mbox{J~{}s}^{-1}\mbox{Hz}^{-1},$
(8)

where $f_{R}$ is a free parameter taking into account any differences between
the local observations and the observations of the high redshift Universe.
This relation is extrapolated to higher frequencies by assuming a spectral
index of $-0.7$ (Gürkan et al., 2018). The globally averaged radio luminosity
at 150 MHz per unit comoving volume at redshift $z$ is then given by,

$\epsilon_{R,150}(z)=f_{R}\times
10^{22}f_{\star}{\bar{\rho}}_{\text{m}}\frac{{\rm d}f_{coll}}{{\rm d}t}\,{\rm
J~{}s^{-1}Hz^{-1}Mpc^{-3}},$ (9)

therefore the corresponding 21-cm radiation flux can be computed as (Ciardi &
Madau, 2003)

$\displaystyle F_{R}(z)=$
$\displaystyle\left(\frac{1420}{150}\right)^{-0.7}\frac{c(1+z)^{3.7}}{4\pi}f_{R}$
(10) $\displaystyle\times
10^{22}f_{\star}{\bar{\rho}}_{\text{m}}\int_{z}^{\infty}\frac{{\rm
d}f_{coll}}{{\rm d}z^{\prime}}\frac{1}{(1+z^{\prime})^{0.7}}{\rm
d}z^{\prime},$

where ${\bar{\rho}}_{\text{m}}$ is the mean density of matter in the Universe.
We convert this flux into a radio brightness temperature $T_{R}$. This results
in a total background temperature given by $T_{\gamma}(z)=T_{R}(z)+T_{\rm
CMB}(z)$.

Although an excess radio background can enhance the amplitude of the
absorption signal, a redshift independent value creates an radio excess which
is much higher than that observed by ARCADE-2 experiment, hence this excess
radio background has been turned off at $z=16$ (however for numerical
stability we model this as a tanh function) as discussed in Chatterjee et al.
(2019). Physically, the rise and suppression of the radio background in a
small redshift range could be explained by a transient population of metal-
free PopIII stars (Mebane et al., 2020) or a rapid heating by cosmic rays Jana
et al. (2019).

Figure 2: The training dataset for the neural network. Here, we have plotted
the total signal (21-cm signal along with foregrounds) which is used for
training. As expected, the 21-cm Global signature is lost entirely when the
foregrounds are added. The y-axis represents the total sky-signal in mK and on
the x-axis, the corresponding frequency channels are shown.

##  3 Foregrounds

The major observational challenge while detecting the faint global 21-cm
signal from CD/EoR is the bright foregrounds in the sky. Radio emission from
our galaxy and other extra-galactic sources are several orders of magnitude
brighter than the cosmological signal of interest. The expected global 21-cm
signal is about four orders of magnitude weaker than the foreground emission.
For all ground-based observations, there is a significant amount of corruption
due to the variations in the Earth’s atmospheric conditions as well. In
addition, the instrument response further corrupts the signal. The bandwidth
corresponding to the observations of the redshifted 21-cm signal overlaps with
the bandwidth used for telecommunication, including the FM-radio band. All
such man-made terrestrial contributions constitute the Radio Frequency
Interference (RFI). The bright foregrounds, RFI, instrumental calibration
errors, combined together poise significant challenges to the Global 21-cm
experiments. Hence, several sophisticated simulations and modelling are
necessary to understand the effect of these corruption terms on the possible
signal extraction methods. It also makes it critical to have an accurate model
for the foregrounds at these radio frequencies, and a well calibrated
instrument. A simple representation of the foregrounds as a polynomial in
$\mathrm{\log(\nu)-\log T}$, is given in Pritchard & Loeb (2010); Bernardi et
al. (2015). In our work, we use the foreground model as in Bowman et al.
(2018b), which we call the Bowman-Rogers (BR) foreground model, Which we
explained briefly in the following section.

###  3.1 Bowman-Rogers(BR) foreground model

The observed sky brightness can be written as (following Hills et al. (2018)):

$\rm{T_{sky}=(T_{BG}+T_{21}+T_{FG})e^{-\tau_{ion}}+T_{e}(1-e^{-\tau_{ion}}}),$
(11)

where, $\rm T_{BG}$ represents the background radiation which is arising from
higher redshifts, $\rm T_{21}$ is the 21-cm signal feature, $\rm T_{FG}$ is
the foreground radiation, $\rm\tau_{ion}$ is the opacity of the ionosphere and
$\rm T_{e}$ is the opacity weighted temperature of the electrons in the
ionosphere. Bernardi et al. (2015) use a log-polynomial expansion which
describes the foreground emission as a function of frequency, around a central
frequency or reference frequency $\rm\nu_{c}$,

$\log\mathrm{T_{FG}(\nu)=\sum^{N-1}_{n=0}c_{n}[\log(\nu/\nu_{c})]^{n}.}$ (12)

A linearized version of the physically-motivated foreground model, as used in
Bowman et al. (2018a); Hills et al. (2018), along with the ionospheric terms,
which we call the Bowman-Rogers (BR) foreground model $\rm{T_{FG(BR)}}$, is
given by:

$\begin{split}\mathrm{T_{FG(BR)}}=~{}&b_{0}\cdot\left(\frac{\nu}{\nu_{c}}\right)^{-2.5}+b_{1}\cdot\left(\frac{\nu}{\nu_{c}}\right)^{-2.5}\ln\left(\frac{\nu}{\nu_{c}}\right)\\\
&+b_{2}\cdot\left(\frac{\nu}{\nu_{c}}\right)^{-2.5}\cdot\left[\ln\left(\frac{\nu}{\nu_{c}}\right)\right]^{2}\\\
&+b_{3}\cdot\left(\frac{\nu}{\nu_{c}}\right)^{-4.5}+b_{4}\cdot\left(\frac{\nu}{\nu_{c}}\right)^{-2},\end{split}$
(13)

where, the coefficients $\rm b_{0},b_{1},b_{2},b_{3},b_{4}$ are the parameters
of the foreground model and $\rm\nu_{c}$ is a reference frequency which is
taken to be 80 MHz for this work. We have used $\rm b_{i}$’s as the variables
for the coefficients, instead of $\rm a_{i}$’s as used in Hills et al. (2018);
Bowman et al. (2018a). We use this model for generating the foregrounds and
construct the mock observations. We fit this model to a sky-averaged total
spectra and find that the fitted values corresponding to a sub-band
$\rm\approx(52.7-98.4)$ MHz are,
$\rm\\{b_{0},~{}b_{1},~{}b_{2},~{}b_{3},~{}b_{4}\\}=\\{-13491.45,-16643.00,-11561.28,147.30,14826.74\\}$.
Using these best-fit values, we vary the parameters of this foreground model
by $\rm 1-5\%$ to generate the mock foreground data which is used to build the
training sets(described in § 5-6.

Figure 3: Correlation map of the various parameters used for training. We can
observe from the plot that the astrophysical parameters, for example $\rm
f_{esc}$ and $\rm f_{*}$ are highly correlated. The foreground parameters of
this model are comparatively less correlated with the signal parameters.

##  4 Overview of ANN

In this section, we briefly introduce the concepts of ANN. The functional unit
of an ANN is called neuron. A simple neural network consists of three basic
layers: an input layer, one or more hidden layers, and an output layer. In a
feed-forward, fully connected neural network, each neuron in a layer is
connected to every neuron in the next layer and the flow of information is
unidirectional. Each of these connections are associated with a weight and a
bias. At the output layer, after each forward-pass a cost function or an error
function is computed. This cost function is optimized, during the training
process by back-propagating the errors, iteratively. A detailed description of
the basic algorithm used in artificial neural networks is discussed in
Choudhury et al. (2020). In our feed forward network, we have used an ANN with
4 Dense layers (using the Sequential Model from the keras api, Python). We
have 1024 neurons in the input layer, corresponding to the 1024 frequency
channels between $\rm 50-91~{}\mathrm{MHz}$. We have used standard scikitlearn
(Pedregosa et al., 2011) and keras modules to build our network. We choose the
number and size of the hidden layers for the best network performance. The
number of neurons in the output layer corresponds to the number of output
parameters which we want to predict. The detailed structure of the neural
network used in our work is described in the following sections.

Parameters | Ranges |
---|---|---
$\rm f_{X}$ | 0.51-20.0 |
$\rm f_{Xh}$ | 0.050-0.399 |
$\rm f_{R}$ | 0 | (traditional)
| 2000-18000 | (exotic)
$\rm f_{*}$ | 0.0030-0.0099 |
$\rm f_{esc}$ | 0.06-0.19 |
$\rm N_{\alpha}$ | 9000-80000 |
Table 1: Parameter range randomly sampled to generate the 21-cm signals for
the training set using our code.

##  5 Building the training datasets

Using the models described in §2, we generate several 21-cm Global signals by
varying the signal parameters:
$~{}\mathrm{\\{f_{X},f_{X,h},f_{R},f_{*},f_{esc},N_{\alpha}\\}}$. The ranges
of parameters used are tabulated in Tab. 1. We explicitly construct two sets
of 21-cm Global signals: one set including only the traditional models,
limiting the electron optical depth, $\rm\tau\approx 0.05$ and allowing a
maximum absorption trough of $\rm-250\mathrm{mK}$ [Fig.1]. The other set
includes the models which account for the X-ray heating and an additional
excess radio background. The amplitude of the absorption trough is limited to
a maximum of $\rm-800\mathrm{mK}$ with $\rm\tau\approx 0.05$. We take into
account the frequency range $\rm 50-91$MHz, across 1024 channels to build the
total 21-cm signals set, which is then used to construct the training data. We
construct this set in this manner to ensure that both the varieties of shapes
of signals contribute equally to the training process, and we do not get a
biased result. We call this set of signals the Global signal training set, and
use an artificial neural network to predict the astrophysical parameters,
without any added foregrounds in § 8.1

The mock-data training set, consists of $\mathrm{T_{21}}$ along with the BR
foreground, $\mathrm{T_{FG(BR)}}$ as given by Eqn. 13. Each sample in the
mock-data training set (see Fig. 2) is given by:

$\mathrm{T_{tot}(\nu)=T_{21}(\nu)+T_{FG(BR)}(\nu)}$ (14)

where, all the temperatures are in mK. The output parameter set consists of 6
signal parameters, $\mathrm{{f_{X},f_{X,h},f_{R},f_{*},f_{esc},N_{\alpha}}}$
and 5 foreground parameters: $\mathrm{b_{0},b_{1},b_{2},b_{3},b_{4}}$. Fig. 3
shows the correlation between the various parameters, which we are using for
training the network. We see that the astrophysical parameters, as expected,
are strongly correlated as compared to the foreground parameters. The ranges
of the parameters explored are listed in the Tab. 1.

##  6 Building the test sets

For this work, we construct two test sets to check the performance of our
network. (i) Test Set A: Consists of 1000 sample 21-cm signals from both
categories, along with the BR foregrounds and noise corresponding to 1000
hours of observation. (ii) Test Set B: Includes one sample 21-cm signal which
contains the excess radio background and one traditional 21-cm signal (without
any excess radio background). Both the samples are contaminated by the BR
foregrounds and noise, corresponding to 1000 hours of observation. The test
sets are constructed as:

$\mathrm{T_{test}=T_{21}+T_{FG(BR)}+\sigma_{t}}$ (15)

Here, $\rm\sigma_{t}$ is the thermal noise corresponding to a bandwidth,
$\rm\Delta\nu$ and an observation time, $\rm\tau$. From the standard
radiometer equation, $\rm\sigma_{t}$ can be re-written for $\rm N_{t}$ hours
of observation:

$\displaystyle\sigma_{t}(\nu)$ $\displaystyle=$
$\displaystyle\frac{T_{sys}(\nu)}{\sqrt{\Delta\nu\cdot\tau}}$ $\displaystyle=$
$\displaystyle\frac{T_{FG}(\nu)}{\sqrt{\Delta\nu\cdot 10^{6}\cdot 3600\cdot
N_{t}}}$

After the network has been saved, we use these test sets to predict the output
parameters (6 signal and 5 foreground parameters). The test set A consisting
of 1000 sample 21-cm signals, is used to compute the performance metric of the
network, while test set B is used to check the accuracy of the signal
parameter extraction.

##  7 $\mathrm{R^{2}}$ Score

We choose $R^{2}$-score as a metric to assess and compare the performance of
the networks. The coefficient of determination, $\rm R^{2}$, is calculated for
each of the parameters from the predictions of test set A. The $\rm R^{2}$
score is defined as:

$\mathrm{R^{2}=\frac{\Sigma(y_{pred}-\overline{y}_{ori})^{2}}{\Sigma(y_{ori}-\overline{y}_{ori})^{2}}=1-\frac{\Sigma(y_{pred}-y_{ori})^{2}}{\Sigma(y_{ori}-\overline{y}_{ori})^{2}}}$
(16)

where, $\overline{y}_{\mathrm{ori}}$ is the average of the original or true
parameter, and the summation is over the entire test set. The score $\rm
R^{2}=1$, implies a perfect inference of the parameters, while $\rm R^{2}$ can
vary between 0 and 1.

##  8 Results

###  8.1 Training without foregrounds

Firstly, we consider only the Global signal training set (as shown in Fig.
1(c). We use the Sequential API from keras to build an ANN, with 1024 input
neurons, corresponding to the 1024 frequency channels, two hidden layers of
512 and 32 neurons respectively, activated by the ’sigmoid’ activation
function. the output layer has 6 output neurons, to predict the astrophysical
parameters. The input is preprocessed by using the ’StandardScaler’ function
available in sklearn, while the signal parameters are scaled by the
’MinMaxScaler’. We use ’adagrad’ (Duchi et al., 2011) as the optimizer and
’mean squared error’ as the loss function. Once the network is trained,
validated and tested, we save the model. To constitute the test set for this
case, we add thermal noise, $\rm\sigma_{t}$ corresponding to 1000 hours of
observation, Which can be written as:

$\rm{T_{test}^{\prime}=T_{21}+\sigma_{t}}.$ (17)

We calculate the R2-scores for each parameter from the test set predictions.
The predicted values of the parameters are shown in Fig. 4. From the R2-
scores we can observe that the parameters $\rm N_{\alpha}$ and $\rm f_{R}$
have the highest R2 scores of 0.9666 and 0.9239, respectively. However, $\rm
N_{\alpha}$ is plotted in logarithmic scale, and $f_{R}$ is scaled by a factor
of 1000. These two parameters are not as highly correlated as the $\rm
f_{X}-f_{Xh}$ and $\rm f_{*}-f_{\rm esc}$ pairs.

(a) $\rm f_{R}/10^{3}$

(b) $\rm log~{}f_{*}$

(c) $\rm f_{esc}$

(d) $\rm f_{X}$

(e) $\rm f_{Xh}$

(f) $\rm log~{}N_{\alpha}$

Figure 4: Each of the plots above is a scatter plot of the predicted values of
the signal parameters, where the test test set consists of only 21-cm Global
signals and noise for the case where the network is trained by only 21cm
Global signals. The central dashed line through the origin depicts the true
values. The R2 scores for each parameter is calculated and mentioned in the
top corner in each plot.

(a) $\rm f_{R}/10^{3}$

(b) $\rm f_{*}\cdot f_{esc}$

(c) $\rm f_{X}\cdot f_{Xh}$

(d) $\rm N_{\alpha}/10^{3}$

Figure 5: The plots above show the predicted values of the parameters for the
test set consisting of only the 21-cm Global signal and noise. The network is
trained only with 21cm Global signals (without adding foregrouds). We have
combined the parameter-pairs which are highly correlated into single
parameters. The first plot on the top panel shows the parameter $\rm
f_{R}/1000$, and the second one is the combined parameter $\rm f_{*}\cdot
f_{esc}$. The bottom panel shows the plots of the parameters $\rm f_{X}\cdot
f_{Xh}$ and $\rm N_{\alpha}/1000$. The dashed straight line in each plot
represents the true values of the parameters. The calculated R2-scores are
mentioned in each plot.

In order to check whether the network performance improves, we combined the
highly correlated parameters into a single parameter. So now the four output
parameters would be $\rm f_{R},f_{*}\cdot f_{esc},f_{x}\cdot
f_{xh},N_{\alpha}$. We use a neural network with 1024 input neurons, two
hidden layers with 256 and 133 neurons, respectively. The hidden layers
neurons are activated by a ’tanh’ function, and we use the ’adagrad’
optimizer. The output layer consists of 4 neurons. We use the same test set as
described above, and plot the predictions in Fig 5. The R2-scores obtained are
around $0.88-96$, for the combined parameters. These are much higher as
compared to the previous case, where we had considered all the 6 signal
parameters separately. This is expected, as we are well aware of the high
correlation between these parameters.

Figure 6: Evolution of the loss function for the network when there are no
added foregrounds. The MSE is calculated at the end of every epoch, and
optimized. The blue curve is the loss function for the traiing set which
saturates around 80 epochs. The Loss function for the test set, closely
follows that of the training set.

###  8.2 Training after adding Foregrounds

Observations of the 21-cm signal would be totally dominated by foregrounds for
all ground-based experiments. In this section, we have considered the training
data to be the mock-data, as described by Eqn. 14. We use the sequential API
form keras, to build the neural network. For the training dataset (as shown in
Fig. 2), we build a 4 layer neural network. The input layer has 1024 neurons
corresponding to the number of the input frequency channels. There are 2
hidden layers with 256 and 64 neurons respectively. We use the sigmoid
activation function in the hidden layers. The output layer consists of 11
neurons corresponding to the 6 signal parameters and the 5 foreground
parameters. We use the ‘adam’ (Kingma & Ba, 2014) optimizer, and mean squared
error as the loss function. Once the network is trained and the model is
saved, we use it to predict the parameters of the signal from the constructed
test datasets.

As described in §6, the test set A contains 1000 mock observations(including
foreground) unseen to the network. This set is used to evaluate the
performance of our network by calculating the $\rm R^{2}$ scores. We plot the
original versus the predicted values of the signal parameters along with the
computed $R^{2}$ score in Fig. 8. We have tabulated these values of $\rm
R^{2}$ for each of the signal parameters in Tab. 2. The signal parameters are
predicted with $\rm R^{2}$-scores ranging from 0.65 to 0.81.

Figure 7: Evolution of the loss function for the network trained by the mock-
observations, which includes the signal and the foreground.

(a) $\rm f_{X}$

(b) $\rm f_{Xh}$

(c) $\rm N_{\alpha}$

(d) $\rm f_{star}$

(e) $\rm f_{R}$

(f) $\rm f_{esc}$

Figure 8: Each plot above shows the predictions by the network for each parameter, when Test Set A (which contains the 21cm-signals along with foreground and noise) is input. The plots show the original versus predicted values for the signal and foreground parameters. The dashed line through the origin in each plot, represents the true value of the inputs.  Parameters |  $\rm R^{2}$-score(BR-FG)
---|---
$\rm f_{*}$ | 0.728
$\rm f_{esc}$ | 0.650
$\rm f_{Xh}$ | 0.723
$\rm f_{X}$ | 0.816
$\rm f_{R}$ | 0.899
$\rm N_{\alpha}$ | 0.731
Table 2: Listing the $\rm R^{2}$-scores for the all the signal parameters.

(a)

(b)

$\rm f_{X}$ | $\rm f_{Xh}$ | $\rm f_{R}/10^{4}$ | $\rm f_{*}$ | $\rm f_{esc}$ | $\rm N_{\alpha}$
---|---|---|---|---|---
38.091 | 3e-02 | 0.0 | 1.756e-02 | 6.948e-02 | 1.6263e+05
32.484 | 1.29e-02 | 0.0 | 2.141e-02 | 6.754e-02 | 1.1419e+05
(c)

$\rm f_{X}$ | $\rm f_{Xh}$ | $\rm f_{R}/10^{4}$ | $\rm f_{*}$ | $\rm f_{esc}$ | $\rm N_{\alpha}$
---|---|---|---|---|---
19.0506 | 0.372 | 1.5932 | 5.158e-03 | 0.177 | 4.2925e+05
13.5237 | 0.310 | 1.1755 | 5.788e-03 | 0.160 | 2.5537e+05
(d)

Figure 9: Reconstructed 21-cm Global signals from the predictions of the ANN
when Test set B (which includes two 21-cm signal along with foreground and
noise) is input to the ANN. The plot on the left shows the original and the
reconstructed signals for the traditional signal input. The plot on the right
depicts the case where the input signal is the one with the excess radio
background. The original (top row) and predicted (bottom row) values of the
parameters are listed in the tables below the plots, respectively. These
predicted parameters are used to reconstruct the signal using our model.

Test set B, consisting of the two known sample signals, (one from the
traditional set and the other from the exotic set, along with foregrounds) is
then input into the saved network. We use the predicted values of the signal
parameters by the network to reconstruct the 21-cm signals using our code. We
have shown the original and predicted parameters corresponding to both the
samples in the test set B, in a table included below the Fig. 9, along with
the plots of the original and reconstructed signals. We can see that the
reconstructed signal is similar in shape, to the original signals. The
parameter $f_{R}$ is zero for the traditional signal input and $\sim~{}11000$
for the model with excess radio background. Also,it is to be noted that the
reconstructed signals span the entire frequency range 25-200 MHz, which we can
generate using our code, with the input astrophysical parameters.

(a)

Parameters | $\rm f_{*}$ | $\rm f_{esc}$ | $\rm f_{Xh}$ | $\rm f_{X}$ | $\rm f_{R}$ | $\rm N_{\alpha}$
---|---|---|---|---|---|---
Predicted values | 0.006 | 0.139 | 0.3 | 16.24 | 1e4 | 9728
(b)

Figure 10: The reconstructed signal along with the EDGES signal. We can see
that by training our ANN with models, which include EDGES-like large
absorption troughs apart from the traditional models, we get the predicted
parameters which are used to reconstruct the signal. We can see that the
magnitude of the signal is comparable to the EDGES detection, but the
flattened nature of the EDGES signal cannot be reconstructed by the signal
reconstructed by our models.

###  8.3 EDGES data as the test input

The possible detection of the absorption trough in the sky-averaged signal by
EDGES, is our only available real observations, currently. We input the
observed sky signal from the EDGES data
111http://loco.lab.asu.edu/edges/edges-data-release/ into the saved network.
We pre-process and input the data corresponding to the sub-band 52.7-98.4 MHz
to the saved ANN. The network predicts the values of the signal parameters,
which is used to reconstruct the signal using our model. We plot the
reconstructed signal along with the EDGES signal in Fig. 10. The signal
parameters predicted by the network are tabulated below the same figure. We
see that the amplitude of the signal almost matches with the EDGES signal. The
predicted value of the $\rm f_{R}$ parameter, implies that there is a strong
excess radio background, which is responsible for the deep absorption trough
of the signal.

We find that although the reconstructed signal correctly predicts the location
and depth of the EDGES signal, the detailed match between the two is not good.
This is not surprising as the models where the star formation follows the
collapsed fraction of the dark matter halos are unable to account for the
flat-bottomed part of the absorption profile and also the rapid rise in the
signal in the epoch of heating (Ewall-Wice et al., 2018a). Obtaining a better
match would require somewhat non-trivial evolution of the star formation rate,
e.g., using PopIII stars (Ewall-Wice et al., 2018a; Chatterjee et al., 2020).

##  9 Summary and Discussions

In this work, we have used ANN to estimate the 21-cm signal parameters from
synthetic datasets mimicking mock-observations for the first time. Our network
is capable of predicting the parameters with $R^{2}$-scores ranging between
0.65 (for $\rm f_{esc}$) and 0.89 (for $\rm f_{R}$), when the signal was
corrupted by the foregrounds and thermal noise. When we consider the case
where there are no foregrounds, the $\rm R^{2}$score for the combined-
parameters ranges between $0.88$ (for $\rm f_{X}\cdot f_{X,h}$) and 0.96 (for
$\rm f_{*}\cdot f_{esc}$). The corresponding plots showing the predicted and
the true values of the parameters are shown in Fig. 8(for the case where
foregrounds are added) and in Fig. 4-5(where there are no foregrounds). As we
are using a single neural network to predict multiple parameters, it is
difficult to compare the efficiency of prediction of each parameter
individually. The overall accuracy of the neural network deteriorates when the
training set is modified by adding foregrounds. As the 21-cm signal parameters
are closely associated with the physics of the evolution of the signal, the
prediction of the signal parameters gives us a direct physical interpretation
of the signal. The mock observations input to the ANN as test sets, gives us a
measure of the reliability of the network.

The reported EDGES(low-band)-detection with the exceedingly deep absorption
trough, could give us a totally new insight on the evolution of the Universe.
The parameter $\rm f_{R}$ determines the magnitude of the radio-excess as
described in our model in § 2.3. From the predictions from our ANN for the
simulated mock data sets (which included one traditional signal and one signal
with the excess radio background), we observed that the value of $\rm f_{R}$
contributed most significantly in determining the shape of reconstructed
signal. A high value of $\rm f_{R}$ implies a very bright radio background,
resulting in a signal with a large absorption trough, while a $\rm f_{R}=0$
turns off the excess radio background, and generates a traditional signal. Our
network is able to predict the other astrophysical parameters with $\rm
R^{2}$-scores ranging between $\rm 0.65-0.816$, when the input consists of the
21-cm signal along with foregrounds and noise. The parameters $\rm f_{esc}$
and $\rm f_{*}$ have high negative correlation, possibly explaining their
lower $\rm R^{2}$-scores. The parameters $\rm f_{X}$ and $\rm f_{X,h}$
determines the location of the minima of the absorption trough, where X-ray
heating starts dominating. When we input the EDGES data as the test data, we
obtain the signal parameters as the predictions of the network, which is
further used to reconstruct the signal (Fig. 10).

The use of ANN to extract the signal parameters efficiently from the total
observed sky signal directly without modelling and subtracting the
foregrounds, is a novel technique. However, our neural network is limited to
the parameter space that we have chosen, and the model foregrounds used. We
have used a very specific model of foreground as prescribed by Bowman et al.
(2018a), which is a physically motivated model including the effects of
ionosphere. We plan to expand the flexibility of the network by including
different models of foreground and introducing instrumental effects in a
future work. These would make the mock-observations more realistic. We observe
how crucial the foreground modelling is, in order to retrieve the signal
parameters efficiently. Artificial neural networks can very efficiently
retrieve the signal parameters when the foreground is removed.

In contrast to the existing techniques of parameter estimation, ANN can
extract features from data by constructing functions which associate the input
with the output parameters. They do not require a specified prior, though the
training sets can be considered to play a similar role as the prior in
Bayesian techniques. The use of ML expedites the computational process
considerably. The training process for the networks we have used for 6000
samples have taken around 12 mins, when trained on a personal computer.
Publicly available codes, for example, ARES(Accelerated Reionization Era,
(Mirocha et al., 2012; Mirocha, 2014) and 21cmFAST (Mesinger et al., 2011) are
also widely used to generate the 21cm Global Signals and reionization
simulations. In our code, we have simply introduced an excess-radio background
and have not accounted for other exotic phenomena, to generate EDGES-like
signals. In this way, we could use a common set of parameters for both the
traditional and EDGES-like sets of signals, which enabled us to use a single
neural network to extract signal parameters from both types of input signals,
even when foregrounds and thermal noise were added. Our code is very simple
and is sensitive to each of the astrophysical parameters which it takes as
input. We would include a variety of more sophisticated physical models, as
incorporated in ARES and 21cmFAST to generate various reionization simulations
in future works. We would like to design a complete signal extraction tool
which would be trained with several different models of the signal, foreground
and instrument response, such that it could be used as a robust tool to
predict the various astrophysical parameters associated with the signal.

##  10 Acknowledgements

MC acknowledges the support of DST for providing the INSPIRE fellowship
(IF160153).AD acknowledges the support of EMR-II under CSIR. AC & TRC
acknowledge support of the Department of Atomic Energy, Government of India,
under project no. 12-R&D-TFR-5.02-0700.

##  11 Data Availability

We have used the publicly available EDGES data release, which can be found at
http://loco.lab.asu.edu/edges/edges-data-release/.

## References

  * Barkana (2018a) Barkana R., 2018a, Nature, 555, 71
  * Barkana (2018b) Barkana R., 2018b, Nature, 555, 71
  * Barkana & Loeb (2005) Barkana R., Loeb A., 2005, ApJ, 624, L65
  * Bernardi et al. (2015) Bernardi G., McQuinn M., Greenhill L. J., 2015, ApJ, 799, 90
  * Bharadwaj & Ali (2004) Bharadwaj S., Ali S. S., 2004, MNRAS, 352, 142
  * Bowman et al. (2018a) Bowman J. D., Rogers A. E. E., Monsalve R. A., Mozdzen T. J., Mahesh N., 2018a, Nature, 555, 67
  * Bowman et al. (2018b) Bowman J. D., Rogers A. E. E., Monsalve R. A., Mozdzen T. J., Mahesh N., 2018b, Nature, 555, 67
  * Chardin et al. (2019) Chardin J., Uhlrich G., Aubert D., Deparis N., Gillet N., Ocvirk P., Lewis J., 2019, MNRAS, 490, 1055
  * Chatterjee et al. (2019) Chatterjee A., Dayal P., Choudhury T. R., Hutter A., 2019, MNRAS, 487, 3560
  * Chatterjee et al. (2020) Chatterjee A., Dayal P., Choudhury T. R., Schneider R., 2020, arXiv e-prints, p. arXiv:2003.05911
  * Choudhury et al. (2020) Choudhury M., Datta A., Chakraborty A., 2020, MNRAS, 491, 4031
  * Ciardi & Madau (2003) Ciardi B., Madau P., 2003, ApJ, 596, 1
  * Cohen et al. (2019) Cohen A., Fialkov A., Barkana R., Monsalve R., 2019, arXiv e-prints, p. arXiv:1910.06274
  * Duchi et al. (2011) Duchi J., Hazan E., Singer Y., 2011, J. Mach. Learn. Res., 12, 2121–2159
  * Ewall-Wice et al. (2018a) Ewall-Wice A., Chang T.-C., Lazio J., Doré O., Seiffert M., Monsalve R. A., 2018a, ApJ, 868, 63
  * Ewall-Wice et al. (2018b) Ewall-Wice A., Chang T.-C., Lazio J., Doré O., Seiffert M., Monsalve R. A., 2018b, ApJ, 868, 63
  * Feng & Holder (2018) Feng C., Holder G., 2018, ApJ, 858, L17
  * Fialkov & Barkana (2019) Fialkov A., Barkana R., 2019, arXiv e-prints,
  * Fialkov et al. (2018) Fialkov A., Barkana R., Cohen A., 2018, Phys. Rev. Lett., 121, 011101
  * Field (1959) Field G. B., 1959, ApJ, 129, 536
  * Fixsen et al. (2011) Fixsen D. J., et al., 2011, ApJ, 734, 5
  * Fraser et al. (2018) Fraser S., et al., 2018, Physics Letters B, 785, 159
  * Furlanetto et al. (2006a) Furlanetto S. R., Oh S. P., Pierpaoli E., 2006a, Phys. Rev. D, 74, 103502
  * Furlanetto et al. (2006b) Furlanetto S. R., Oh S. P., Briggs F. H., 2006b, Phys. Rep., 433, 181
  * Furlanetto et al. (2006c) Furlanetto S. R., Oh S. P., Briggs F. H., 2006c, Phys. Rep., 433, 181
  * Gillet et al. (2019) Gillet N., Mesinger A., Greig B., Liu A., Ucci G., 2019, MNRAS, 484, 282
  * Greenhill & Bernardi (2012) Greenhill L. J., Bernardi G., 2012, arXiv e-prints, p. arXiv:1201.1700
  * Gürkan et al. (2018) Gürkan G., et al., 2018, MNRAS, 475, 3010
  * Haardt & Madau (2012) Haardt F., Madau P., 2012, ApJ, 746, 125
  * Hassan et al. (2019) Hassan S., Liu A., Kohn S., La Plante P., 2019, MNRAS, 483, 2524
  * Hills et al. (2018) Hills R., Kulkarni G., Meerburg P. D., Puchwein E., 2018, Nature, 564, E32
  * Jana et al. (2019) Jana R., Nath B. B., Biermann P. L., 2019, MNRAS, 483, 5329
  * Jennings et al. (2019) Jennings W. D., Watkinson C. A., Abdalla F. B., McEwen J. D., 2019, MNRAS, 483, 2907
  * Kingma & Ba (2014) Kingma D. P., Ba J., 2014, preprint,  (arXiv:1412.6980)
  * Mebane et al. (2020) Mebane R. H., Mirocha J., Furlanetto S. R., 2020, MNRAS, 493, 1217
  * Mesinger et al. (2011) Mesinger A., Furlanetto S., Cen R., 2011, MNRAS, 411, 955
  * Mirocha (2014) Mirocha J., 2014, MNRAS, 443, 1211
  * Mirocha & Furlanetto (2019) Mirocha J., Furlanetto S. R., 2019, MNRAS, 483, 1980
  * Mirocha et al. (2012) Mirocha J., Skory S., Burns J. O., Wise J. H., 2012, ApJ, 756, 94
  * Morales & Wyithe (2010) Morales M. F., Wyithe J. S. B., 2010, ARA&A, 48, 127
  * Nhan et al. (2018) Nhan B. D., Bordenave D. D., Bradley R. F., Burns J. O., Tauscher K., Rapetti D., Klima P. J., 2018, arXiv e-prints, p. arXiv:1811.04917
  * Patra et al. (2013) Patra N., Subrahmanyan R., Raghunathan A., Udaya Shankar N., 2013, Experimental Astronomy, 36, 319
  * Pawlik et al. (2009) Pawlik A. H., Schaye J., van Scherpenzeel E., 2009, MNRAS, 394, 1812
  * Pedregosa et al. (2011) Pedregosa F., et al., 2011, Journal of Machine Learning Research, 12, 2825
  * Pospelov et al. (2018) Pospelov M., Pradler J., Ruderman J. T., Urbano A., 2018, Physical Review Letters, 121, 031103
  * Pritchard & Loeb (2010) Pritchard J. R., Loeb A., 2010, Phys. Rev. D, 82, 023006
  * Pritchard & Loeb (2012a) Pritchard J. R., Loeb A., 2012a, Reports on Progress in Physics, 75, 086901
  * Pritchard & Loeb (2012b) Pritchard J. R., Loeb A., 2012b, Reports on Progress in Physics, 75, 086901
  * Pritchard et al. (2015) Pritchard J., et al., 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), p. 12
  * Schmit & Pritchard (2018) Schmit C. J., Pritchard J. R., 2018, MNRAS, 475, 1213
  * Shimabukuro & Semelin (2017) Shimabukuro H., Semelin B., 2017, MNRAS, 468, 3869
  * Singh et al. (2017) Singh S., et al., 2017, ApJ, 845, L12
  * Slatyer & Wu (2018) Slatyer T. R., Wu C.-L., 2018, Phys. Rev. D, 98, 023013
  * Sokolowski et al. (2015) Sokolowski M., et al., 2015, Publ. Astron. Soc. Australia, 32, e004
  * Voytek et al. (2014) Voytek T. C., Natarajan A., Jáuregui García J. M., Peterson J. B., López-Cruz O., 2014, ApJ, 782, L9
  * Wouthuysen (1952) Wouthuysen S. A., 1952, AJ, 57, 31


∎

11institutetext: S. Bahadur 22institutetext: Department of Mathematics &
Astronomy, University of Lucknow, Lucknow, India
22email<EMAIL_ADDRESS>33institutetext: Varun 44institutetext:
Department of Mathematics & Astronomy, University of Lucknow, Lucknow, India
44email<EMAIL_ADDRESS>
# Higher order Hermite-Fejér Interpolation on the unit circle.

Higher order Hermite-Fejér Interpolation.

Swarnima Bahadur  Varun

(Received: date / Accepted: date)

###### Abstract

The aim of this paper is to study the approximation of functions using a
higher-order Hermite-Fejér interpolation process on the unit circle. The
system of nodes is composed of vertically projected zeros of Jacobi
polynomials onto the unit circle with boundary points at $\pm 1$. Values of
the polynomial and its first four derivatives are fixed by the interpolation
conditions at the nodes. Convergence of the process is obtained for analytic
functions on a suitable domain, and the rate of convergence is estimated.

###### Keywords:

Unit circle Non-uniform nodes Jacobi Polynomial Rate of Convergence Lagrange
Interpolation Hermite-Fejér interpolation

##  1 Introduction

Approximation of continous functions can be done using different methods by
constructing algebraic or trigonometric polynomials. Hermite interpolation
attracted the attention of many researchers in the last century.
Hermite interpolation 10 : It is the process of finding a polynomial which
coincides with the continous function at certain pre-assigned points, called
the nodes of interpolation, and its successsive derivatives coinciding with
arbritarily chosen numbers.

An important step was taken when Fejér 7  in 1916 proved a theorem where the
values of the derivatives in the Hermite scheme were equal to zero.

Fejér’s theorem : If $f\in C[-1,1]$, then $H_{n}(f,x)$ converges to $f(x)$
uniformly on [-1,1] as $n$ tends to infinty. Interpolation polynomials
$H_{n}(f,x)$ is defined by

$H_{n}(f,x)=\sum_{k=1}^{n}f(x_{kn})(1-x_{kn}x)\Bigg{(}\frac{T_{n}(x)}{n(x-x_{kn})}\Bigg{)}^{2},$

where $x_{kn}$ are the zeros of the Chebyshev polynomial of the first kind.
$H_{n}(f,x)$ satisfies the below given conditions where $k=1,2,...,n$.

$\displaystyle H_{n}(f,x_{kn})=f(x_{kn})\quad and\quad
H_{n}^{\prime}(f,x_{kn})=0.$

Mills 9  in his paper highlights Hermite and Hemite Fejér interpolation as
important techniques in the approximation theory. Fejér’s theorem has been
extended to more general nodal systems. For example, in 2001, Daruis and
González-Vera 6  extended Fejér’s result to the unit circle by considering the
nodal system constituted by the complex $n^{th}$ roots of unity. They proved
that the sequence of Hermite-Fejér interpolation polynomials uniformly
converge for continous functions on the unit circle.

Berriochoa, Cachafeiro and García-Amor 2  extended the Fejér’s second theorem
to the unit circle. Then Berriochoa, Cachafeiro, Díaz, and Martínez Brey 3
obtained the supremum norm of the error of interpolation for analytic
functions and computed the order of convergence of Hermite-Fejér interpolation
on the unit circle considering the same set of nodes as of 6 .

Apart from the uniform nodal system (where nodes are equally spaced on the
unit circle), Hermite-Fejér interpolation on the unit circle have been also
studied on some non-uniformly distributed nodes on the unit circle (see 1  and
5 ).
Higher order Hermite-Fejér interpolation:  It is a process of finding a
polynomial which coincides with a continous function at the nodes of the
interpolation and the derivatives upto $r^{th}$ order $(r>1)$ are null at the
nodal points.

A considerable number of papers on higher order Hermite-Fejér interpolation
processes on real nodes have been published (see 11  and 14 ). This motivated
us to consider a higher order Hermite-Fejér interpolation problem on non-
uniform set of complex nodes on the unit circle. Let us denote nodal system
containing the zeros of the Jacobi polynomial $P_{n}^{(\alpha,\beta)}(x)$ by
Gauss Jacobi point system. Let us also define two sets
$\mathbb{T}=\\{z:|z|=1\\}$ and $\mathbb{D}=\\{z:|z|<1\\}$.

In the present paper, we consider a Hermite-Fejér interpolation problem on the
nodal system constituted of $\pm 1$ and the projections of the Gauss Jacobi
point system vertically onto the unit circle by the transformation
$x=\dfrac{1+z^{2}}{2z}$. The aim of this paper is to extend the Hermite-Fejér
interpolation on the unit circle problem on all the above said projected nodes
upto the fourth derivative and prove the following convergence theorem:

###### Theorem 1.1

Let $f(z)$ is a function continous on $\mathbb{T}\cup\mathbb{D}$ and analytic
on $\mathbb{D}$. The sequence of interpolatory polynomial $\\{Q_{n}(z)\\}$
satisfies the below relation

$\mid Q_{n}(z)-f(z)\mid\,=\,\textbf{O}\big{(}\omega(f,n^{-1})\log n\big{)}$
(1)

where $\omega(f,n^{-1})$ represents the modulus of continuity of the function
$f(z)$ and O notation refers to as $n\rightarrow\infty$ .

The paper has been organised in following manner. Preliminaries are given in
section 2. Section 3 covers the interpolation problem and explicit
representation of the interpolatory polynomial. Section 4 is devoted to
finding estimates and the proof of theorem 1.1 has been assigned section 5.

##  2 Preliminaries

The differential equation satisfied by $P_{n}^{(\alpha,\beta)}(x)$ is

$(1-x^{2})P_{n}^{(\alpha,\beta)^{{}^{\prime\prime}}}(x)+[\beta-\alpha-(\alpha+\beta+2)x]P_{n}^{(\alpha,\beta)^{{}^{\prime}}}(x)+n(n+\alpha+\beta+1)P_{n}^{(\alpha,\beta)}(x)=0.\\\
$

Using the Szegő transformation $x=\dfrac{1+z^{2}}{2z}$,

$\begin{split}(z^{2}-1)^{4}P_{n}^{(\alpha,\beta)^{{}^{\prime\prime}}}(x)+4z(z^{2}-1)&\big{[}\\{(\alpha+\beta+2)z^{2}+1\\}(z^{2}-1)-2z^{3}(\beta-\alpha)\big{]}P_{n}^{(\alpha,\beta)^{{}^{\prime}}}(x)\\\
-16z^{6}n(n+\alpha+\beta+1)P_{n}^{(\alpha,\beta)}(x)&=0.\\\ \end{split}$ (2)

Let ${Z}_{n}$ be set of nodes

$\begin{split}{Z}_{n}=\\{z_{0}=1,z_{2n+1}=-1,z_{k}=x_{k}+iy_{k}=\cos\theta_{k}+i\,\sin\theta_{k}\,;\,z_{n+k}=\overline{z_{k}};\\\
\,\,k=1,2,3,...,n\,;\,x_{k},y_{k}\in R\\},\\\ \end{split}$

which are obtained by projecting vertically the Gauss Jacobi point system on
the unit circle together with $\pm 1$.

The polynomial defined on $Z_{n}$ are given by (3),

Figure 1: An arbitrary point $z$ and the nodal system $Z_{n}$

$R(z)=(z^{2}-1)W(z),$ (3)

where

$W(z)=\displaystyle\prod_{k=1}^{2n}(z-z_{k})=K_{n}P_{n}^{(\alpha,\beta)}\Bigg{(}\frac{1+z^{2}}{2z}\Bigg{)}z^{n},$
(4)
$K_{n}=2^{2n}n!\frac{\Gamma(\alpha+\beta+n+1)}{\Gamma(\alpha+\beta+2n+1)}.$

The fundamental polynomials of Lagrange interpolation on the zeros of $R(z)$
are given by

$L_{k}(z)=\frac{R(z)}{(z-z_{k})R^{{}^{\prime}}(z_{k})},\hskip
28.45274ptk=0,1,...,2n+1.$ (5)

We can write $z=x+iy$, where $x,y\in R$. If $z\in\mathbb{T}$, then

$\mid z^{2}-1\mid=2\sqrt{1-x^{2}}$ (6)

and

$\mid z-z_{k}\mid=\sqrt{2}\,\sqrt{1-xx_{k}-\sqrt{1-x^{2}}\sqrt{1-x_{k}^{2}}}.$
(7)

In order to evaluate the estimates of the fundamental polynomials formed in
section 3, we will be using below results.

For $-1\leq x\leq 1$, we have

$(1-x^{2})^{1/2}\mid{P_{n}^{(\alpha,\beta)}(x)}\mid=O(n^{\alpha-1}),$ (8)
$\mid{P_{n}^{(\alpha,\beta)}(x)}\mid=O(n^{\alpha}),$ (9)
$\mid{P_{n}^{(\alpha,\beta)^{\prime}}(x)}\mid=O(n^{\alpha+2}),$ (10)
$\mid{P_{n}^{(\alpha,\beta)^{\prime\prime}}(x)}\mid=O(n^{\alpha+4}).$ (11)

Considering set of nodes $Z_{n}$, where $x_{k}=\cos\theta_{k}\,,k=1,2,...,n$
are the zeros of ${P_{n}^{(\alpha,\beta)}(x)}$, then

$(1-x_{k}^{2})^{-1}\sim\Bigg{(}\frac{k}{n}\Bigg{)}^{-2},$ (12)
$\mid{P_{n}^{(\alpha,\beta)}(x_{k})}\mid\sim
k^{-\alpha-\frac{1}{2}}n^{\alpha},$ (13)
$\mid{P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})}\mid\sim
k^{-\alpha-\frac{3}{2}}n^{\alpha+2}.$ (14)

For more details, refer pg.164-166 of 13 .

Let $f(z)$ be continous on $\mathbb{T}\cup\mathbb{D}$ and analytic on
$\mathbb{D}$. Then, there exists a polynomial $F_{n}(z)$ of degree less than
$(2n+2)(r+1)$ satisfying Jackson’s inequality.8

$\mid f(z)-F_{n}(z)\mid\,\leq\,C\,\omega(f,n^{-1}),$ (15)

where $\omega(f,n^{-1})$ represents the modulus of continuity of the function
$f(z)$ and $C$ is independent of $n$ and $z$.

##  3 The problem and explicit representation of interpolatory polynomial

Here, we are interested in determining the convergence of interpolatory
polynomial $Q_{n}(z)$ of degree less than $(2n+2)(r+1)$ on the distinct set of
nodes $\\{z_{k}\\}^{2n+1}_{k=0}$ with Hermite conditions at all points
satisfying

$\begin{cases}Q_{n}{(z_{k})}=\alpha_{k},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\
k=0,1,...,2n+1\\\ Q_{n}^{(r)}{(z_{k})}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\
k=0,1,...,2n+1,\,r=1,2,3,4\,,\\\ \end{cases}$ (16)

where $\alpha_{k}$ and $\beta_{k}$ are complex constants.

###### Theorem 3.1

We shall write $Q_{n}(z)$ satisfying (16)

$Q_{n}(z)=\sum_{k=0}^{2n+1}f(z_{k})A_{0k}(z),$ (17)

where $A_{0k}(z)$ is a polynomial of degree less than $(2n+2)(r+1)$ satisfying
the conditions given in (18).

For $j,k=0,1,...,2n+1$ ,

$\begin{cases}A_{0k}(z_{j})=\delta_{kj},\\\
A_{0k}^{(r)}(z_{j})=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;r=1,2,3,4\,\,,\\\
\end{cases}$ (18)

where

$A_{0k}(z)=[L_{k}(z)]^{5}+\sum_{p=1}^{4}c_{pk}A_{pk}(z),$ (19)
$A_{pk}(z)=[R(z)]^{p}(L_{k}(z))^{5-p},$ (20)
$c_{1k}=-\,\frac{5\,L_{k}^{\prime}(z_{k})}{R^{\prime}(z_{k})},$ (21)

$c_{2k}=-\,\frac{5}{2!\,[R^{\prime}(z_{k})]^{2}}\,\big{[}L_{k}^{\prime\prime}(z_{k})+10\,[L_{k}^{\prime}(z_{k})]^{2}\big{]},$
(22)

$c_{3k}=-\,\frac{5}{3!\,[R^{\prime}(z_{k})]^{3}}\,\big{[}-18\,L_{k}^{\prime\prime}(z_{k})\,L_{k}^{\prime}(z_{k})\,+\,L_{k}^{\prime\prime\prime}(z_{k})\,-\,198[L_{k}^{\prime}(z_{k})]^{3}\big{]},$
(23)
$\begin{split}c_{4k}=&-\frac{5}{4!\,[R^{\prime}(z_{k})]^{4}}\bigg{[}L_{k}^{\prime\prime\prime\prime}(z_{k})-24L_{k}^{\prime\prime\prime}(z_{k})\,L_{k}^{\prime}(z_{k})\\\
&+[L_{k}^{\prime\prime}(z_{k})]^{2}-156[L_{k}^{\prime}(z_{k})]^{2}L_{k}^{\prime\prime}(z_{k})+2544[L_{k}^{\prime}(z_{k})]^{4}\bigg{]}.\\\
\end{split}$ (24)

###### Proof

Let $A_{0k}(z)$ be written as

$A_{0k}(z)=[L_{k}(z)]^{5}+\sum_{p=1}^{4}c_{pk}[R(z)]^{p}(L_{k}(z))^{5-p}.$
(25)

At $z=z_{j}$, where $j=0,1,...,2n+1$ ,

$A_{0k}(z_{j})=[L_{k}(z_{j})]^{5}+\sum_{p=1}^{4}c_{pk}[R(z_{j})]^{p}(L_{k}(z_{j}))^{5-p}.$

Using (3), we have $R(z_{j})=0$ and from (5), we have

$A_{0k}(z_{j})=\delta_{kj}.$ (26)

Clearly, the first set of condition in (18) is satisfied.
In order to determine the $c_{pk}$’s, we use the second set of conditions of
(18).
On differentiating $A_{0k}(z)$ in (25) one time with respect to $z$, we get

$A_{0k}^{\prime}(z)=5L_{k}^{\prime}(z)[L_{k}(z)]^{4}+c_{1k}[R(z)(L_{k}(z))^{4}]^{\prime}+\Big{[}\sum_{p=2}^{4}c_{pk}[R(z)]^{p}(L_{k}(z))^{5-p}\Big{]}^{\prime}.$
(27)

Clearly, at $z=z_{j}\,\,(j\neq k)$, we have $A_{0k}^{\prime}(z)=0.$
At $z=z_{k}$, $A_{0k}^{\prime}(z)$ must be equal to zero. We have

$\displaystyle 5\,L_{k}^{\prime}(z_{k})+c_{1k}\,R^{\prime}(z_{k})=0,$

which provides (21). In a similar manner, differentiating (25) two, three and
four times with respect to $z$ gives (22), (23) and (24) respectively by using
conditions given in (18).

##  4 Estimation of the fundamental polynomials

In order to find the estimates, we intend to represent the constants $c_{pk}$
in general form as given under ($p$=0,1,2,3,4)

$c_{pk}=\frac{5}{p![R^{\prime}(z_{k})]^{p}}\sum_{s=0}^{[\frac{p}{2}]}\sum_{r=s}^{p-s}e_{psr}[L_{k}^{(s)}(z_{k})]^{r}L_{k}^{(p-sr)}(z_{k}),$
(28)

where $e_{psr}$ are the constants independent of $n$ and $z$ and
$\bigg{[}\dfrac{p}{2}\bigg{]}$ denotes greatest integer function Also,
$L_{k}^{(s)}(z_{k})$ denotes $s^{th}$ derivative of $L_{k}(z)$ at $z=z_{k}.$

###### Lemma 1

Let $L_{k}(z)$ be given by (5), then for $z\in\mathbb{T}\cup\mathbb{D}$, we
have

$\mid
L_{k}(z)\mid=\textit{{O}}\Bigg{(}\frac{1}{k^{-\alpha+\frac{3}{2}}}\bigg{)}$
(29)

###### Proof

For $k=1,2,...,2n$

$L_{k}(z)=\frac{R(z)}{(z-z_{k})R^{{}^{\prime}}(z_{k})}.$ (30)

Taking modulus on the both sides,

$\displaystyle\mid L_{k}(z)\mid$ $\displaystyle=$ $\displaystyle\frac{\mid
R(z)\mid}{\mid z-z_{k}\mid\mid R^{{}^{\prime}}(z_{k})\mid},$ $\displaystyle=$
$\displaystyle\frac{\mid(z^{2}-1)W(z)\mid}{\mid
z-z_{k}\mid\mid\\{2zW(z)+(z^{2}-1)W^{\prime}(z))\\}_{z=z_{k}}\mid}.$

Since $z_{k}^{\prime}s$ are the zeros of $W(z)$, using (4), we get

$\displaystyle\mid L_{k}(z)\mid$ $\displaystyle=$
$\displaystyle\frac{\Big{|}(z^{2}-1)K_{n}P_{n}^{(\alpha,\beta)}\Big{(}\frac{1+z^{2}}{2z}\Big{)}z^{n}\Big{|}}{\mid
z-z_{k}\mid\Big{|}(z_{k}^{2}-1)\Big{\\{}K_{n}P_{n}^{(\alpha,\beta)}\Big{(}\frac{1+z^{2}}{2z}\Big{)}z^{n}\Big{\\}}^{\prime}_{z=z_{k}}\Big{|}}$
$\displaystyle=$
$\displaystyle\frac{\Big{|}(z^{2}-1)P_{n}^{(\alpha,\beta)}\Big{(}\frac{1+z^{2}}{2z}\Big{)}z^{n}\Big{|}}{\mid
z-z_{k}\mid\Big{|}(z_{k}^{2}-1)\Big{\\{}nz_{k}^{n-1}P_{n}^{(\alpha,\beta)}(x_{k})+z_{k}^{n}P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})\Big{(}\frac{z_{k}^{2}-1}{2z_{k}^{2}}\Big{)}\Big{\\}}\Big{|}}$
$\displaystyle=$
$\displaystyle\frac{2|z^{2}-1||P_{n}^{(\alpha,\beta)}(x)||z|^{n}}{\mid
z-z_{k}\mid|z_{k}|^{n-2}|z_{k}^{2}-1|^{2}|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|}.$

Using (6) and (7), we get

$\displaystyle\mid L_{k}(z)\mid$ $\displaystyle=$
$\displaystyle\frac{2.2\sqrt{1-x^{2}}|P_{n}^{(\alpha,\beta)}(x)||z|^{n}}{4({1-x_{k}^{2}})\sqrt{2}\,\sqrt{1-xx_{k}-\sqrt{1-x^{2}}\sqrt{1-x_{k}^{2}}}|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{1-x^{2}}|P_{n}^{(\alpha,\beta)}(x)||z|^{n}\,\sqrt{1-xx_{k}+\sqrt{1-x^{2}}\sqrt{1-x_{k}^{2}}}}{\sqrt{2}({1-x_{k}^{2}})\,\sqrt{(1-xx_{k})^{2}-({1-x^{2}})({1-x_{k}^{2}})}|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{1-x^{2}}|P_{n}^{(\alpha,\beta)}(x)||z|^{n}\,\sqrt{1-xx_{k}+\sqrt{1-x^{2}-x_{k}^{2}+x^{2}x_{k}^{2}}}}{\sqrt{2}({1-x_{k}^{2}})\,\mid
x-x_{k}\mid|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-x^{2}}|P_{n}^{(\alpha,\beta)}(x)||z|^{n}\,\sqrt{1-xx_{k}+\sqrt{(1-xx_{k})^{2}-(x-x_{k})^{2}}}}{\sqrt{2}({1-x_{k}^{2}})\,\mid
x-x_{k}\mid|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|}$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{1-x^{2}}|P_{n}^{(\alpha,\beta)}(x)|\,\sqrt{1-xx_{k}}}{({1-x_{k}^{2}})\,\mid
x-x_{k}\mid|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|}.$

For $\mid x-x_{k}\mid\geq\frac{1}{2}\mid 1-x_{k}^{2}\mid$, we get

$\displaystyle\mid L_{k}(z)\mid$ $\displaystyle\leq$ $\displaystyle
C\frac{\sqrt{1-x^{2}}|P_{n}^{(\alpha,\beta)}(x)|\,}{({1-x_{k}^{2}})^{3/2}\,|P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})|},$

where $C$ is constant independent of $n$ and $z$. Using (8), (12) and (14), we
have

$\displaystyle\mid L_{k}(z)\mid$ $\displaystyle=$
$\displaystyle\textit{{O}}\Bigg{(}\frac{1}{k^{-\alpha+\frac{3}{2}}}\Bigg{)}.$
(31)

Similarly, for $\mid x-x_{k}\mid\leq\frac{1}{2}\mid 1-x_{k}^{2}\mid$, we get
the same result as (31).
For $k=0$ and $k=2n+1$, we have

$\mid L_{0}(z)\mid=\mid L_{2n+1}(z)\mid=\textit{{O}}(1).$ (32)

From (31) and (32), we have Lemma 1.

###### Lemma 2

Let $c_{pk}$ be given by (28), then

$\mid
c_{pk}\mid=\textit{{O}}\Bigg{(}\frac{1}{k_{n}^{p}\,n^{p(\alpha-1)}\,k^{\,{p}/{2}-p\alpha}}\Bigg{)}.$
(33)

###### Proof

From (3) and (4), we have

$R^{\prime}(z_{k})=\Bigg{(}\frac{K_{n}}{2}\Bigg{)}z_{k}^{n-2}(z_{k}^{2}-1)^{2}P_{n}^{(\alpha,\beta)^{\prime}}(x_{k}).$

Taking modulus on both the sides, we get

$\displaystyle\mid R^{\prime}(z_{k})\mid=\Bigg{(}\frac{K_{n}}{2}\Bigg{)}\mid
z_{k}\mid^{n-2}\,\mid(z_{k}^{2}-1)\mid^{2}\,\mid
P_{n}^{(\alpha,\beta)^{\prime}}(x_{k})\mid.$

From (6), (12) and (14), we have

$\mid
R^{\prime}(z_{k})\mid=\textit{{O}}(K_{n}\,k^{-\alpha+\frac{1}{2}}n^{\alpha}).$
(34)

Similarly, from (2), (5) and (6), we have

$\mid L_{k}^{(s)}(z_{k})\mid=\textit{{O}}(n^{s}).$ (35)

Using (34) and (35) in (28), we have Lemma 2.

###### Lemma 3

Let $A_{0k}(z)$ be given by (19) and $c_{pk}$ given by (28), then for
$z\in\mathbb{T}\cup\mathbb{D}$,

$\sum\limits_{k=0}^{2n+1}\mid A_{0k}(z)\mid\;=\textit{{O}}(\log n),$ (36)

where $-1<\alpha\leq\dfrac{1}{2}.$

###### Proof

From (3) and (4), we have

$\displaystyle
R(z)=(z^{2}-1)K_{n}P_{n}^{(\alpha,\beta)}\Bigg{(}\frac{1+z^{2}}{2z}\Bigg{)}z^{n}.$
(37)

Taking modulus on both the sides and using (6) and (8), we have

$\mid R(z)\mid=\textit{{O}}(K_{n}n^{\alpha-1}).$ (38)

For $\mid x_{k}-x\mid\,\geq\dfrac{1}{2}\mid 1-x_{k}^{2}\mid$ and from (19) and
(20), we have

$\displaystyle\sum\limits_{k=0}^{2n+1}\mid A_{0k}(z)\mid$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{2n+1}\mid
L_{k}(z)\mid^{5}+\sum\limits_{k=0}^{2n+1}\sum_{p=1}^{4}\mid c_{pk}\mid\,\mid
R(z)\mid^{p}\,\mid L_{k}(z)\mid^{5-p}.$

Using (38), Lemma 1 and Lemma 2, we get

$\displaystyle\sum\limits_{k=0}^{2n+1}\mid A_{0k}(z)\mid$ $\displaystyle=$
$\displaystyle\textit{{O}}\Bigg{(}\sum\limits_{k=0}^{2n+1}\frac{1}{k^{-5\alpha+\frac{15}{2}}}+\sum\limits_{k=0}^{2n+1}\sum_{p=1}^{4}\,\frac{1}{k^{-5\alpha-p+\frac{15}{2}}}\Bigg{)},$
$\displaystyle=$
$\displaystyle\textit{{O}}\Bigg{(}\sum\limits_{k=0}^{2n+1}\sum_{p=0}^{4}\,\frac{1}{k^{-5\alpha-p+\frac{15}{2}}}\Bigg{)}.$

Now, we get

$\sum\limits_{k=0}^{2n+1}\mid A_{0k}(z)\mid=\textit{{O}}(\log
n),\qquad\quad\qquad\qquad\Bigg{\\{}-1<\alpha\leq\frac{13}{10}-\frac{p}{5}\Bigg{\\}}.$
(39)

Similarly, for $\mid x_{k}-x\mid\leq\frac{1}{2}\mid 1-x_{k}^{2}\mid$, we get
the same result.
Since range of $\alpha$ with $p=4$ lies in the intersection of all the cases.
Hence, the lemma follows.

##  5 Proof of theorem 1.1

Let $f(z)$ be a function that is continous on $\mathbb{T}\cup\mathbb{D}$ and
analytic on $\mathbb{D}$. Since $Q_{n}(z)$ is the uniquely determined
polynomial of degree less than $(2n+2)(r+1)$ and the polynomial $F_{n}(z)$
satisfying equation (15) can be expressed as

$F_{n}(z)=\sum\limits_{k=0}^{2n+1}F_{n}(z_{k})A_{k}(z).$ (40)

Then

$\mid Q_{n}(z)-f(z)\mid\,\leq\,\mid Q_{n}(z)-F_{n}(z)\mid+\mid
F_{n}(z)-f(z)\mid.$ (41)

Using (17) and (40), we have

$\displaystyle\mid Q_{n}(z)-f(z)\mid\,$ $\displaystyle\leq$
$\displaystyle\,\underbrace{\sum\limits_{k=0}^{2n+1}\mid
f(z_{k})-F_{n}(z_{k})\mid\mid A_{k}(z)\mid}_{N_{1}}+\underbrace{\mid
F_{n}(z)-f(z)\mid}_{N_{2}}.$

We have

$\mid Q_{n}(z)-f(z)\mid\,\leq\,N_{1}+N_{2}.$ (42)

From (15) and (36), we have

$\displaystyle N_{1}=\textit{{O}}\big{(}\omega(f,n^{-1})\log n\big{)}.$ (43)

From (15), we have

$\displaystyle N_{2}=\textit{{O}}\big{(}n\,\omega(f,n^{-1})\big{)}.$ (44)

Using (43) and (44) in (42), we get

$\displaystyle\mid Q_{n}(z)-f(z)\mid\,=\,\textit{
{O}}\big{(}\omega(f,n^{-1})\log n\big{)}.$

Hence, Theorem 1.1 follows.

Data Availibility: Data sharing not applicable to this article as no data-sets
were generated or analysed during the current study.

Author contributions:
Conceptualisation: S. Bahadur, Varun ; Writing-Original Draft: Varun

Conflicts of Interest: The authors declare no conflict of interest.

## References

  * (1) Bahadur, S.: (0,0,1) interpolation on the unit circle. International Journal of Mathematical Analysis. 5, 1429-1434 (2011)
  * (2) Berriochoa, E., Cachafeiro, A., García-amor, J. M.: An extension of Fejér’s condition for Hermite interpolation. Complex Analysis and Operator Theory. 6, 651–664 (2012)
  * (3) Berriochoa, E., Cachafeiro, A., Díaz, J., Martínez Brey, E.: Rate of convergence of Hermite- Fejér interpolation on the unit circle. Journal of Applied Mathematics. Article ID 407128, 8 pages (2013) https://doi.org/10.1155/2013/407128
  * (4) Berriochoa, E., Cachafeiro, A., Díaz, J.: Hermite Interpolation on the Unit Circle Considering up to the Second Derivative. ISRN Mathematical Analysis. (2014) http://dx.doi.org/10.1155/2014/808519
  * (5) Chen, W., Sharma, A.: Lacunary interpolation on some non-uniformly distributed nodes on the unit circle. Annales Universitatis Scientiarum Budapestinensis. 16, 69-82 (2004)
  * (6) Daruis, L., González-vera, P.: A Note on Hermite -Fejér Interpolation for the Unit Circle. Applied Mathematic Letters. 14, 997-1003 (2001)
  * (7) Fejér, L.: Über Interpolation. Gött. Nachr. 6, 66–91 (1916)
  * (8) Jackson, D.: Ueber die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung. Göttingen. (1911)
  * (9) Mills, T. M.: Some techniques in Approximation theory. Math. Scientist. 5, 105-120 (1980)
  * (10) Prasad, J.: On Hermite and Hermite-Fejér interpolation of higher order. Demonstratio Mathematica. 26, 413-425 (1993)
  * (11) Sung, H.S., Ko, D. H., Sakai, R.: Lp Convergence of Higher order Hermite or Hermite-Fejér Interpolation polynomials with exponential-type weights ( II ). Global Journal of Pure and Applied Mathematics. 13, 7401–7426 (2017)
  * (12) Szabados, J., Vértesi, P.: Interpolation of Functions. World Scientific Publishers. (1990)
  * (13) Szegő, G.: Orthogonal Polynomials. Amer. Math. Soc. Coll. 23, (1975)
  * (14) Xiang, S., He, G.: The Fast Implementation of Higher Order Hermite–Fejér Interpolation. SIAM Journal on Scientific Computing. 37, A1727–A1751 (2015)