2022

[1,2,3]Cheng-Min Zhang

1]National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China 2]School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China 3]School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China 4]NAOC-UKZN
Computational Astrophysics Centre, University of KwaZulu-Natal, Durban 4000,
South Africa 5]Key Laboratory of Radio Astronomy, Chinese Academy of Sciences,
Beijing 100101, China 6]Netherlands Institute for Radio Astronomy (ASTRON),
Postbus 2, 7990 AA Dwingeloo, the Netherlands 7]Astronomical Institute ‘Anton
Pannekoek’, Faculty of Science, University of Amsterdam, 1090 GE Amsterdam,
the Netherlands 8]Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, Urumqi, Xinjiang 830011, China 9]School of Physics and Electronic
Science, Guizhou Normal University, Guiyang 550001, China 10]School of Physics
and Electronic Science, Guizhou Education University, Guiyang 550018, China

# Luminosity distribution of fast radio bursts from CHIME/FRB Catalog 1 by the
updated Macquart relation

Xiang-Han Cui<EMAIL_ADDRESS>Di Li  Jian-Wei Zhang  Bo Peng  Wei-Wei Zhu
Richard Strom  Shuang-Qiang Wang  Na Wang  Qing-Dong Wu  De-Hua Wang  Yi-Yan
Yang  [  [  [  [  [  [  [  [  [  [

###### Abstract

Fast radio bursts (FRBs) are extremely strong radio flares lasting several
micro to milliseconds and come from unidentified objects at cosmological
distances, most of which are only seen once. Based on recently published data
in the CHIME/FRB Catalog 1 in the frequency bands 400-800 MHz, we analyze 125
apparently singular FRBs with low dispersion measure (DM) and find that the
distribution of their luminosity follows a lognormal form according to
statistical tests. In our luminosity measurement, the FRB distance is
estimated by using the Macquart relation which was obtained for 8 localized
FRBs, and we find it still applicable for 18 sources after adding the latest
10 new localized FRBs. In addition, we test the validity of the luminosity
distribution up to the Macquart relation and find that the lognormal form
feature decreases as the uncertainty increases. Moreover, we compare the
luminosity of these apparent non-repeaters with that of the previously
observed 10 repeating FRBs also at low DM, noting that they belong to
different lognormal distributions with the mean luminosity of non-repeaters
being two times greater than that of repeaters. Therefore, from the two
different lognormal distributions, different mechanisms for FRBs can be
implied.

###### keywords:

transients: fast radio burst - methods: statistical - stars: magnetars

##  1 Introduction

Fast radio bursts (FRBs) are very strong radio sparks lasting micro to
milliseconds in duration, which are mostly confirmed to be from objects at
cosmic distances, which perhaps noticed in 1980 (Linscott and Erkes, 1980).
The FRB phenomenon was systematically studied by Lorimer et al (2007), and
then Thornton et al (2013) discovered several more sources in 2013.
Subsequently, this field developed rapidly (Lorimer, 2018; Zhang, 2020;
Petroff et al, 2021), including the first localized repeating FRB 121102
(Spitler et al, 2016; Chatterjee et al, 2017), the first FRB-like signal-FRB
200428 from the Galactic soft gamma repeater (SGR) 1935+2154 (Bochenek et al,
2020; CHIME/FRB Collaboration et al, 2020; Lin et al, 2020; Li et al, 2021a)
and FRB 20200120E in M81 (Bhardwaj et al, 2021). In addition, with further
study, several research efforts showed that FRBs may be periodically active
(FRB 180916: 16 day and FRB 121102: 157 day) (Chime/Frb Collaboration et al,
2020; Rajwade et al, 2020) and emit periodic signals (FRB 20191221A: 216.8 ms,
FRB 20210206A: 2.8 ms, and FRB 20210213A: 10.7 ms) (The CHIME/FRB
Collaboration et al, 2021b). Meanwhile, thanks to the completion of advanced
radio instruments like the Canadian Hydrogen Intensity Mapping Experiment
(CHIME) (CHIME/FRB Collaboration et al, 2019a, b, 2020; Chime/Frb
Collaboration et al, 2020; The CHIME/FRB Collaboration et al, 2021a, b),
Australian Square Kilometre Array Pathfinder (ASKAP) (Shannon et al, 2018;
Kumar et al, 2019), and Five-hundred-meter Aperture Spherical radio Telescope
(FAST) (Li et al, 2018; Zhu et al, 2020; Luo et al, 2020b; Lin et al, 2020;
Niu et al, 2021; Li et al, 2021b), the data of FRBs, as well as their diverse
properties, have dramatically increased. For example, the 1652 and 849 bursts
related to FRB 121102 were observed by FAST (Li et al, 2021b) and Arecibo
(Jahns et al, 2022), respectively, and CHIME released the CHIME/FRB Catalog
1111https://www.chime-frb.ca/home (The CHIME/FRB Collaboration et al, 2021a)
recently.

Before the CHIME/FRB Catalog 1 came out in June of 2021 (The CHIME/FRB
Collaboration et al, 2021a), the FRB Catalogue
(FRBCAT)222https://www.frbcat.org/ was mostly used to study the statistical
properties of FRBs (Petroff et al, 2016). Up to now, 129 FRB sources were
published in the FRBCAT with various observation frequency bands from the
different radio telescopes. Although FRBCAT creates opportunities for us to
pursue the matter of FRBs from a variety of frequency bands, the difficulties
for statistical study are also confronted, such as the calibration of FRB data
from the different telescopes and the uncertain spectral index between the
diverse observational frequency bands (Cui et al, 2021a). Nowadays, after the
CHIME/FRB Catalog 1 became available, the above dilemma is alleviated because
data are only from the CHIME telescope at 400-800MHz (The CHIME/FRB
Collaboration et al, 2021a), and the amount of data has also increased
significantly, 535 FRB data from 492 sources, in which 462 apparent non-
repeaters (hereafter, referred to as non-repeaters) are newly observed (The
CHIME/FRB Collaboration et al, 2021a).

The CHIME/FRB collaborations and other researchers have made many statistical
analyses using the new database (The CHIME/FRB Collaboration et al, 2021a;
Rafiei-Ravandi et al, 2021; Chawla et al, 2022; Pleunis et al, 2021; Josephy
et al, 2021). Their conclusions further hint that the two groups of FRBs may
originate from various paths, or they are likely to come from the same origin
but different environments and conditions. However, the FRB luminosity has not
been discussed, perhaps because the source distance is hard to determine
without localization. In the former studies, luminosity function and
distribution of FRBs have been discussed, and many of them are impressive and
enlightening (Kumar et al, 2017; Li et al, 2017; Luo et al, 2018; Hashimoto et
al, 2020; Luo et al, 2020a). Among the evidence and results, the luminosity
distribution of FRBs is likely preferred to follow a power-law form or
Schechter function as a whole (Li et al, 2017; Luo et al, 2018; Lu and Piro,
2019; Luo et al, 2020a; Hashimoto et al, 2022), but this is still an open
question because there exist a possibility of lognormal distribution (Cui et
al, 2021a). Moreover, Li et al (2021b) recently found that the burst energy
distribution of a repeating FRB 121102 is not a single power-law but a bimodal
distribution, which may contain a lognormal component. Besides these, whether
the repeaters and non-repeaters share the same origin or physical properties
is also a controversial issue(Connor, 2019; Caleb et al, 2019). Some
statistical analyses suggested that the different distributions for repeating
and non-repeating FRBs may imply multiple origins or physical processes of
FRBs (Palaniswamy et al, 2018; Petroff et al, 2019; Fonseca et al, 2020; Cui
et al, 2021b; The CHIME/FRB Collaboration et al, 2021a), while others believed
that the difference is not so obvious or caused by the selection effects
(Connor et al, 2020; Gardenier et al, 2021). Therefore, according to these
unsettle questions (Kulkarni et al, 2014; Petroff et al, 2019; Cordes and
Chatterjee, 2019), by means of the new CHIME database, the statistical
explorations with various physical parameters between repeaters and non-
repeaters can go ahead.

For the study of FRB luminosity, the distance estimation is a key step, where
the Macquart relation between the dispersion measure and redshift (DM-z)
(Macquart et al, 2020) is most useful, which is acquired by the 8 localized
FRBs. Here, as the first step, to test the validity of the Macquart relation
with the updated 18 localized FRBs, by adding the 10 new ones (Heintz et al,
2020)333http://frbhosts.org/, we showed that the Macquart relation is still
applicable (details in Appendix A). Therefore, we employed the Macquart DM-z
relation and analyzed the new CHIME data to estimate the FRB luminosity,
processing the statistics that include the goodness of fit (Cui et al, 2021b),
Kolmogorov-Smirnov (K-S) test (Smirnov, 1948), and Mann-Whitney-Wilcoxon
(M-W-W) test (Mann and Whitney, 1947), Anderson–Darling (A-D) test (Anderson
and Darling, 1952), and Lilliefors test (Lilliefors, 1967).

The structure of our paper is organized as follows. In Section 2, we describe
the data selection and luminosity estimation. In Section 3, the FRB luminosity
distribution is fitted by the different types of functions, and the results of
statistical tests are given. In Section 4, we discuss the selection effects,
the error of Macquart DM-z relation, the applicability of lognormal
distribution, and the implications of statistical results. Finally, a brief
summary is exhibited at the end of the paper.

##  2 Data selection and estimation

In this section, we elaborate on the data selection of CHIME/FRB Catalog 1 and
the estimation of FRB distance, based on which the FRB luminosity is obtained.

###  2.1 Data selection

Our data are taken from the first release of CHIME/FRB Catalog 1 (The
CHIME/FRB Collaboration et al, 2021a) and its website444https://www.chime-
frb.ca/home. This catalog contains 474 non-repeaters and 18 repeaters, of
which 462 non-repeaters are published for the first time. However, due to
instrument responses, processing methods, and other potential effects, these
data need selection for non-repeaters for the later analyses, which were
mentioned in The CHIME/FRB Collaboration et al (2021a) and illustrated as
follows.

  * •

Events with the tab of ”excluded_flag = 1” should be removed, considering that
these data may be influenced by non-nominal telescope operation.

  * •

Signal-to-noise ratio (SNR) should be larger than 12, because events with SNR
$<$ 12 are sensitive to human bias and are more likely to be misclassified as
noise or radio frequency interference (RFI).

  * •

DM needs to meet the conditions that $\rm DM>1.5DM_{MW}$ and $\rm
DM>100\,pc\,cm^{-3}$, to ensure these events are truly extragalactic sources,
and where $\rm DM_{MW}$ we used is $\rm DM_{YMW16}$ model (Yao et al, 2017).

  * •

Scattering timescale for events needs less than 10 ms, due to the selection
effects bias against wide bursts.

  * •

Events detected in far side-lobes (indicated by the tab “Flux note” of Catalog
1) were rejected, as reliable flux or position measurements are not yet
available for these events.

  * •

Some sources have sub-bursts, so we only consider the first burst to avoid
repetitive counting.

After these criteria, 255 non-repeaters were reserved. However, the direct
distance data have not been given for sources. If we assume that the Macquart
relation (Macquart et al, 2020) of DM and redshift (DM-z) correlation is
correct, we can derive an empirical formula to roughly estimate the FRBs
distance. In addition, we also re-examine the Macquart DM-z relation by the
latest updated 18 localized FRBs and find that it still follows the previous
results, the detail of which is shown in Appendix A. Then, another problem
arises that the Macquart relation can only estimate the distance at low
redshift or DM (James et al, 2022). Meanwhile, in Catalog 1, the the inferred
power-law index for the cumulative fluence distribution of low DM (ranges in
$\rm 100-500\,pc\,cm^{-3}$) and high DM ($\rm DM>500\,pc\,cm^{-3}$) is
different (The CHIME/FRB Collaboration et al, 2021a), where DM is the data
that removes the Milky Way galaxy contribution. Besides that, for high DM
data, bright bursts are more likely to be recorded by receivers, which means
that the distant faint burst is incomplete for the full sample. Therefore,
finally 125 non-repeaters with low DM are selected in our sample, as well as
10 repeaters at low DM.

###  2.2 FRB Luminosity estimation

We employed the Macquart relation to estimate the distance of FRB, so the
first step is to clarify the DM we used. FRB’s DM is consist of 4 parts
(Cordes and Chatterjee, 2019), originating from the Milky Way galaxy and halo
($\rm DM_{MW}$), intergalactic medium in cosmic distance ($\rm DM_{IGM}$),
host galaxy ($\rm DM_{host}$) and surrounding medium ($\rm DM_{sur}$),
respectively. Because only 19 FRBs were localized, we know little information
about other host galaxies and their $\rm DM_{excess}$ ($\rm DM_{IGM}$, $\rm
DM_{host}$, and $\rm DM_{sur}$), where $\rm DM_{excess}$ is used to estimate
the upper limit of redshift and distance and $\rm DM_{MW}$ is subtracted
according to YMW model (Yao et al, 2017) but not NE2001 (Cordes and Lazio,
2002). Here, we do not estimate the DM contribution of the Milky Way galaxy
halo, because they are not well constrained by current observations so far
(Rafiei-Ravandi et al, 2021), which have a rough range of $\sim
10-100\,pc\,cm^{-3}$. Since our DMs are all larger than $100\,pc\,cm^{-3}$
after the above selections, we can ensure that these sources are from
extragalactic distances. Thus, based on our method, the luminosity we
calculate is the upper limit of the intrinsic value.

By using the basic knowledge of distance measures in cosmology (Hogg, 1999),
the proper distance ($d_{p}$) can be roughly estimated as $d_{p}=zc/H_{0}$,
where c is speed of light and $H_{0}$ is Hubble constant cited from Planck
Collaboration et al (2016) and Macquart et al (2020) ($H_{0}=\rm
67.74\,km\,s^{-1}\,Mpc^{-1}$). Based on the simplest assumption of a flat
universe ($\Omega_{k}=0$) and the definition of comoving distance ($d_{c}$),
the luminosity distance ($d_{L}$) is written as
$d_{L}=(1+z)d_{c}=(1+z)d_{p}/(1+z)=d_{p}$. Meanwhile, the data of Catalog 1
are only from CHIME at the frequency band of 400-800 MHz, so we do not need to
consider the impact of the different telescope calibrations on the data. Thus,
combined with the FRB flux ($S$), the upper limit of luminosity of non-
repeaters can be calculated as $L\sim Sd_{L}^{2}/(1+z)$. For the 10 repeaters,
we also estimated their distances by the above method, and the first flux
values of multiple observations are taken to represent this source.

##  3 Analysis and results

Figure 1:  Upper panel: normalized histogram and different fitted curves of
non-repeaters at low dispersion measure (DM) from CHIME/FRB Catalog 1. The
solid line is the curve of lognormal form and the dashed line is the curve of
power-law type. Middle and bottom panels are the residuals for the lognormal
and power-law curve, shown as solid points, which are calculated by the
normalized histogram data and curves. The dash-dotted lines refer to the cases
of residual = 0.

In our former work about the luminosity distribution of FRBs, we elucidated
that the lognormal distribution is better than the power-law type based on the
FRB data by CHIME and other telescopes (Cui et al, 2021a). However, our
conclusion was weakened due to the small amount of data and the difficulty of
calibrating between the different telescopes. Nowadays, thanks to the CHIME
Catalog 1, we can employ sufficient and more uniform data to do the further
statistical tests, where the 125 non-repeaters mentioned above are applied and
the histogram of their luminosity is plotted in Figure 1. Meanwhile, 5
statistical methods are used to judge our results, including goodness of fit,
K-S test, M-W-W test, A-D test, and Lilliefors test. The analyses of full
sample (255 non-repeaters) are shown in Appendix B.

To start our statistics, the test criteria need to be briefly introduced. The
test results are represented by the parameters of coefficient of determination
($R^{2}$) and p-values that are given under a 5% significance level, and the
procedures are as follows.

  * •

For the goodness of fit, the closer $R^{2}$ is to 1, the better the fit. On
the contrary, the fit is worse.

  * •

For the K-S test, M-W-W test, and Lilliefors test, the lower the p-value, the
greater the difference from the null hypothesis. If the p-value is less than
0.05, it indicates that this test rejects the null hypothesis.

  * •

For the A-D test, when the value of the statistic is larger than the critical
value, the null hypothesis can be rejected.

Firstly, we directly test the lognormal distribution by applying the K-S test,
Lilliefors test, and A-D test, and the null hypothesis is that the sample is
consistent with a lognormal form. The results are shown in Table 1. Meanwhile,
all p-values and statistics are larger than 0.05 and critical values,
respectively, which indicate that the lognormal form is consistent with both
repeaters and non-repeaters under these three tests.

Table 1: Statistical test results of lognormal distribution for repeaters and
non-repeaters at low DM

FRBs | Number | K-S test | Lilliefors test | A-D testa
---|---|---|---|---
Non-repeaters | 125 | 0.872 | 0.605 | (0.333, 0.764)
Repeaters | 10 | 0.640 | 0.154 | (0.519, 0.684)

Notes. a The results of A-D test are shown in the form of statistic and
critical values (statistic, critical values) for a 5% significance level.

Then, we use the lognormal and power-law types to fit the luminosity
distribution (in Figure 1) and utilize the statistical methods to test the
fitting closeness, including the goodness of fit, K-S test, and M-W-W test. To
avoid deviations caused by different coefficients, we normalized the data and
fitted curves and performed statistical analysis on them. From Figure 1, we
can see that lognormal curve is closer to the data histogram comparing with
power-law curve, and the fitting residuals of lognormal are also smaller than
that of power-law, where power-law form is mentioned as Schechter function
(Luo et al, 2018). From Table 2, the goodness of fit of lognormal
($R_{log}^{2}$ = 0.99989) is higher than that of power-law ($R_{power}^{2}$ =
0.820). Moreover, further tests of the K-S and M-W-W on the fitting property
also show that the lognormal conforms better than the power-law. Although the
goodness of fit of power-low is high, it is still smaller than that of
lognormal, and p-values of power-law are all smaller than 0.05 in Table 2.
Therefore, from the distribution pattern and the above statistical test
results, a preliminary conclusion is that the lognormal form is preferred for
luminosity distribution.

Table 2: Statistical test results of different luminosity distribution for non-repeaters at low DM Different types | Goodness of fit | K-S test | M-W-W test
---|---|---|---
Lognormal | 0.99989 | 0.630 | 0.795
Power-law | 0.820 | $3.70\times 10^{-3}$ | $7.78\times 10^{-3}$

However, we noticed that the above analysis is based on the strict Macquart
relation between DM and z. Considering that this relation is estimated from
broad scattered data points, DM and z relation is not an exactly proportional
one-to-one correspondence (Luo et al, 2020a; James et al, 2022). That is, a DM
may correspond to a distribution of z rather than a single value. So, we
introduce various errors to the DM-z transition for 125 non-repeaters to
simulate this uncertainty. The value of error injection is z, which is from
10% to 100% with a 10% interval in the uniform and normal error pattern,
respectively. Specifically, error pattern means that the transition error of
DM-z may belong to a particular distribution (uniform or normal), and its
statistical characteristics (mean, upper and lower limits for uniform
distribution, and mean and variance for normal distribution) are affected by
error percentage (10%-100%). Therefore, for each error percentage and error
pattern, we perform the Monte Carlo sampling 300 times based on two
assumptions: the first is that the probability and shape of error in each data
set is the same in one sampling; secondly, the sampling range for each data
set depends on its own characteristics. For example, for the data with z = 0.5
and a 20% normal error shape, we take 300 times sampling within normal
distribution with a mean of 0.5 and a variance of 0.1 ($0.5\times 20\%=0.1$).
It should be noted that for the situation of large errors like 100%, values of
0 may occur during the sampling process, for which we only use values greater
than 0 for analysis.

Then, we find the z with maximum (z-max) deviations that are compared with
their averages in each group. After that, the values of luminosity are
calculated by using the above z-maxs drawn in Figure 2 (uniform) and Figure 3
(normal), and we also plot the original lognormal fitting curve (the red line
in Figure 1) in each sub-figure. Meanwhile, K-S, M-W-W, and Lilliefors tests
are applied for each group’s luminosity to check whether they are still
consistent with the lognormal form. Nevertheless, in fact, this only performs
statistical tests on one set of data, which may cause fluctuations. Thus, to
eliminate the randomness of sampling, we do a 100 times loop for the above
sampling, and take the harmonic mean values for each $p_{ks}$, $p_{mww}$, and
$p_{lilliefors}$ individually to represent the test results under the
corresponding error percentages and patterns, which is listed in Table 3. The
further discussions are shown in Section 4.2.

Table 3: Statistical test results (p-values) of luminosity distribution with
various error shapes and error percentages

Statistical tests | Error shapes | Error percentages
---|---|---
10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100%
K-S test | Uniform | 0.748 | 0.746 | 0.780 | 0.800 | 0.796 | 0.762 | 0.697 | 0.499 | 0.241 | 0.0128
Normal | 0.794 | 0.811 | 0.637 | 0.204 | 0.168 | 0.225 | 0.0422 | 0.0416 | 0.0334 | 0.0782
M-W-W test | Uniform | 0.889 | 0.878 | 0.870 | 0.868 | 0.855 | 0.848 | 0.839 | 0.784 | 0.651 | 0.303
Normal | 0.884 | 0.873 | 0.804 | 0.602 | 0.528 | 0.539 | 0.434 | 0.402 | 0.411 | 0.447
Lilliefors test | Uniform | 0.363 | 0.323 | 0.391 | 0.374 | 0.421 | 0.340 | 0.235 | 0.0509 | 0.00949 | 0.00144
Normal | 0.416 | 0.448 | 0.108 | 0.00975 | 0.00683 | 0.00967 | 0.00328 | 0.00247 | 0.00255 | 0.00334

Figure 2: Luminosity histogram of non-repeaters at low dispersion measure (DM)
in uniform error shape with changing error percentages from 10% to 100%, from
sub-figure a to sub-figure j. In each sub-figure, the cross-hatched histogram
is the maximum deviation and the empty one means the minimum deviation of 300
times sampling in the corresponding error percentage. The line represents the
best fitting curve without considering the error of the Macquart relation,
which is an unnormalized fitting curve in Figure 1. Figure 3: Luminosity
histogram of non-repeaters at low dispersion measure (DM) in normal error
shape with changing error percentages from 10% to 100%, from sub-figure a to
sub-figure j. Other annotations are consistent with those in Figure 2. Figure
4: The distinction between the repeaters and non-repeaters at low dispersion
measure (DM) in the aspect of luminosity. The top panel (sub-figure a) is the
CDF of repeaters and non-repeaters for luminosity. The dashed (solid) line is
for repeaters (non-repeaters). The bottom panel (sub-figure b) is the
histogram of luminosity for two samples. The cross-hatched (empty) histogram
means the repeaters (non-repeaters). The dashed (solid) line represents the
mean value of repeaters (non-repeaters).

Finally, we analyze the data of repeaters and non-repeaters. Because the
luminosity distribution of repeaters had been discussed in our previous work
and it also showed the lognormal (Cui et al, 2021a), we will not make the
redundant analysis of the luminosity distribution for the repeaters here. But,
it is necessary to compare the distributions of repeaters and non-repeaters
again, by considering that there were only 12 non-repeaters in the comparison
of the previous work. Therefore, we compare the two samples of repeaters (10)
and non-repeaters (125) at low DM in the aspect of luminosity, the
distributions of which are plotted in Figure 4, and we employ the K-S and
M-W-W tests on these two samples. The results are obtained as $p_{ks}$ =
0.0120 and $p_{mww}$ = 0.00943, indicating that the two samples may follow the
different statistical distributions. These results infer that, although their
distribution types are the same, the specific statistical characteristics are
different.

##  4 Discussions and conclusions

Four aspects will be discussed in this section: selection effects, the
applicability of the lognormal for FRB luminosity, the magnetar origin for
FRBs, and the difference between the repeaters and non-repeaters. To begin
with, we need to clarify the possible selection effect in our sample,
discussing whether the lognormal distribution will be varied as the error
percentage of DM-z relation changes. Then, we discuss what kinds of magnetars
can produce the FRB luminosity as obtained, and the models for the repeaters
and non-repeaters.

###  4.1 Selection effects

The detection position from far side-lobes and off-axis beams will effectively
affect the performance of the telescope. Although most of these data have been
removed after selection, we cannot guarantee that all flux data are under the
same instrument response due to the current positional uncertainties of the
CHIME/FRB Catalog 1 events. But we believe that this may be a minority case
and has less impact on the conclusions. Besides that, we should note that the
data of burst flux in CHIME/FRB Calalog 1 are biased, such as in high-DM and
wide bursts with long scattering time, so The CHIME/FRB Collaboration et al
(2021a) injected simulate bursts to quantify this effect. However, we do not
use this method but apply the data selection of low-DM and short scattering
events to study the FRBs population. Therefore, it should be remarked that our
analyses are only applicable to the current selection method.

Due to repeaters having been observed several times, they may obtain better
measured DM data and estimated fluence. These effects probably cause
differences in selection functions compared with non-repeaters samples, we
cannot use this statistical result to determine whether a particular source
will be repeated in the future. Thus, the difference between them may stem
from selection effects like beaming and propagation, or the intrinsic
characteristics of FRBs are various (Pleunis et al, 2021), and our following
discussion mainly focuses on the possibility of intrinsic features.

###  4.2 Applicability of the lognormal distribution

Because our study on the FRB luminosity depends on the distance estimation by
the Macquart DM-z relation, which needs further discussions as below. To
verify the Macquart relation, we firstly use the new data of 18 localized FRBs
to confirm its validity as previously declaimed (as shown in Figure 5 in
Appendix A). We obtain that the slope (k) by using a linear function for
fitting ($k_{18}\sim 1028$ ) is almost parallel with the value of Macquart
relation ($k_{m}\sim 973$), and the difference between the two slops is no
more than 6%. Under the constraints of 10 new data of localized FRBs, the
Macquart relation does not change significantly, and the further analysis is
listed in Appendix. Therefore it is reasonable that we employ this DM-z
relation to estimate the distance to calculate the FRB luminosity. Then, the
approximated isotropic luminosity data of FRBs based on the CHIME/FRB Catalog
1 are acquired under the Macquart relation, which follows the lognormal
distribution and rejects the power-law type.

However, considering that the Macquart relation is not a strict one-to-one
conversion from DM to z (distance), the lognormal distribution of FRB
luminosity perhaps to be deformed with the increase of error for DM-z
correlation. Here, with the different error shapes (uniform and normal type)
and error percentages (from 10% to 100%), the variations of the lognormal
characteristics are represented by statistical test results ($p_{ks}$,
$p_{mww}$, and $p_{lilliefors}$), which means that the lognormal distribution
may be constrained with a certain scope of application. According to Table 3,
for the different error percentages, the lognormal characteristics are
gradually decreasing under K-S, M-W-W, and Lilliefors test results. Although
the majority of p-values are greater than 0.05, this clear downward trend
indicates a weakening of lognormal feature.

Specifically, for the uniform error shape, when the percentages reach 100% in
the K-S test and 90% in the Lilliefors test, the corresponding p-values are
lower than 0.05. For the normal error shape, the p-values smaller than 0.05
have appeared while the percentages go to 70% and 40% in K-S and Lilliefors
test, respectively. Therefore, the lognormal feature is threatened in the
above situations. The reason for the different p-values under multiple tests
lies in the different emphasis of each method. For example, the K-S is tests
the maximum distance between an empirical distribution function and a
cumulative distribution function, while the M-W-W analyzes by the median.
Lilliefors test is an improved K-S test, which is appropriate when the test
distribution must be specified by parameters estimated from the data
(Lilliefors, 1967). This also explains why $p_{lilliefors}$ always precedes
$p_{ks}$ less than 0.05. If we give the criterion that they pass all three
tests in both error shapes, they will meet the lognormal distribution. Thus,
it can be inferred that when the estimation error of the Macquart relation
between DM and z is less than 40%, the lognormal luminosity distribution is
credible. On the contrary, the lognormal feature is not obvious.

It should be clarified that the errors we are discussing here do not come from
a particular uncertainty, but representing all possible aspects in DM-z
transition. This is intended to simulate and test whether the form of
luminosity distribution changes when uncertainty is introduced, rather than to
describe and discuss the type and cause of specific error. Indeed, the
selective effect and the observational bias may significantly affect DM-z
conversion, so James et al (2022) gave the detailed analysis for possible
errors and concluded that works may produce erroneous results when assuming a
1-1 DM-z relation, which is consistent with our original intention to add
errors in DM-z conversion. Meanwhile, since our curve fitting is only for the
situation that the error is not considered, the fitting result may have a
slight change after the error is introduced, but this will not affect our
final conclusion, because the statistical test values are given for the
different errors.

###  4.3 Magnetar origin of FRBs

Next, we discuss the upper limit of the intrinsic luminosity that we obtained
in Figure 1. As concluded in our former work (Cui et al, 2021a) and combined
with new CHIME data, the lognormal distribution is more supportive of the
magnetar origin models (Popov and Postnov, 2010; Lyutikov and Popov, 2020),
and it is also valid in terms of the magnitude of the luminosity. In
particular, for 125 non-repeaters, the maximum luminosity ($L_{max}$) is about
$2.86\times 10^{45}\,\rm erg\,s^{-1}$ and the minimum luminosity ($L_{min}$)
is about $6.83\times 10^{42}\,\rm erg\,s^{-1}$ with the mean value of
$2.42\times 10^{44}\,\rm erg\,s^{-1}$. Some giant flares from
soft-$\gamma$-ray repeaters (SGRs) that are about $10^{44}\,\rm erg\,s^{-1}$
in hard X-ray or soft $\gamma$-ray band (Mazets et al, 1979; Hurley et al,
1999; Mereghetti et al, 2005). If we consider a relatively high radio
efficiency of 0.1 between the radio and X-ray luminosity, then the above SGR
luminosity may conform to the FRBs’ luminosity. However, if the relative low
radio emission efficiency is about $10^{-5}$, like SGR 1935+2154 (Margalit et
al, 2020), then the radio luminosity of SGR will not meet the above FRBs. Only
the rare huge-giant flare can satisfy with the obtained FRB luminosity
(Lyutikov, 2017), like the spike luminosity of about $10^{47}\,\rm
erg\,s^{-1}$ in the X-ray band (Hurley et al, 2005; Palmer et al, 2005), or
the newborn millisecond magnetars may produce such huge-giant flares that have
not been observed.

However, for the $L_{max}$ of FRBs being much higher than the inferred radio
luminosity of the giant flares of SGRs as observed, it may be necessary to
consider that the special magnetars have the super-strong magnetic fields
(Beloborodov, 2017), higher than the known values of $10^{15}\,\rm G$ (Duncan
and Thompson, 1992; Kaspi and Beloborodov, 2017). If their magnetic fields
increase by half or one order of magnitude to $10^{16}\,\rm G$, the total
released magnetic energy will increase by one or two orders of magnitude, as
$E_{tot}\sim B^{2}V/8\pi\sim 10^{49}\,\rm erg$, where $B\sim 10^{16}\,\rm G$
is the surface magnetic field of a young magnetar and $V\sim 10^{18}cm^{3}$ is
the volume of the magnetar. If we assume the radio emission efficiency to be
about $10^{-4}$ and lasting time of FRBs to be 10 milliseconds, then the
corresponding FRB luminosity of non-repeater will rise to $10^{47}\,\rm
erg\,s^{-1}$. Therefore, this indicates that the cosmic FRBs are likely to
come from the rare and violent bursts or interactions of magnetars.

###  4.4 Difference of repeaters and non-repeaters

The possible distinction between the repeaters and non-repeaters is briefly
discussed in this subsection. Although the luminosity distribution of
repeaters is also lognormal (Cui et al, 2021a), the two groups have different
statistical distributions. The average luminosity of repeaters ($3.85\times
10^{43}\,\rm erg\,s^{-1}$) is about three times lower than that of non-
repeaters ($1.31\times 10^{44}\,\rm erg\,s^{-1}$). This implies that they may
come from a similar origin but different environments or emission mechanisms,
such as the magnetars with the mediate magnetic field strengths or special
structures. For example, the magnetic field strength of repeater source is
about $10^{14-15}\,\rm G$, while that of non-repeater may be higher than
$10^{15}\,\rm G$. In terms of the FRB origin models, there are two promising
candidate forms: violent outburst from magnetar and supergiant pulse from
strong magnetic neutron star (Popov et al, 2018; Katz, 2020; Zhang, 2020).
Specifically, the huge-giant flares from ultra-strong magnetar ($\sim
10^{16}\,\rm G$) (Lyubarsky, 2014; Murase et al, 2016; Beloborodov, 2017) may
be one of the explanations for a single burst of non-repeaters, and extremely
high luminosity ($\sim 10^{47}\,\rm erg\,s^{-1}$) could lead to the long burst
interval that it is considered as a non-repeater. On the other aspect, the
supergiant pulses from the extragalactic neutron stars (Cordes and Wasserman,
2016; Connor et al, 2016) may be a good description for repeaters.
Consequently, our results further support the opinion of multiple origins of
FRBs, which is also implied in previous studies based on CHIME/FRB catalog 1
in different parameters (Pleunis et al, 2021) and another database (Cui et al,
2021b).

In the end, we need to clarify that the luminosity of two samples may have an
overlap part, which means that some non-repeaters may be repeated in the
future, like FRB 171019 (Kumar et al, 2019). In other words, the merger of the
binary system (Totani, 2013; Kashiyama et al, 2013; Mingarelli et al, 2015)
and catastrophic collision events (Geng and Huang, 2015; Dai et al, 2016) may
also be mixed in the non-repeater sample. However, the proportion of these
one-time collision events should be a small part, because the luminosity
distribution of non-repeaters shows a single lognormal distribution.
Therefore, the non-repeater samples can also be divided into two groups: the
true non-repeaters and repeaters as a single observed burst. Finally, because
of the statement in Section 4.1 and a fewer data of only 10 repeaters, the low
p-values need to be treated carefully, and its statistical conclusion might be
tentative. Thus we look forward to having more such data to uncover the
mysteries of FRBs.

A brief summary of conclusions:

  * •

We added the latest 10 localized FRBs to reconfirm the Macquart relation,
indicating that this relation is still credible for the known 18 localized
FRBs.

  * •

The luminosity of repeaters and non-repeaters are calculated based on the
distance estimated by the Macquart relation, and their distribution conforms
with the lognormal with the different mean values and derivations. However,
the lognormal features decrease as the transition error of DM-z increases
until the error reaches 40%, at where the lognormal features are weakened.

  * •

These implies that the two samples possibly come from similar origins, such as
magnetar or strong magnetic NS.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grant No. 11988101, No. U1938117, No. U1731238, No. 11703003, No. 11725313,
and 12163001), the International Partnership Program of Chinese Academy of
Sciences grant No. 114A11KYSB20160008, the National Key R&D Program of China
No. 2016YFA0400702, and the Guizhou Provincial Science and Technology
Foundation (Grant No. [2020]1Y019). The data underlying this article are
available in the references below: (1) The data of repeating and non-repeaters
are taken from the database of CHIME/FRB Catalog 1, available at
https://www.chime-frb.ca/home. (2) The information of localized FRB is
provided at the web http://frbhosts.org/

## Appendix A: Verification of the Macquart relation with updated 18 localized
FRBs

When the Macquart relation was given, only 8 localized FRBs were considered.
Now, we add the newly localized 10 FRBs, and a total of 18 data points are
taken into account to verify the Macquart relation. Since the M81 is too close
to us (Bhardwaj et al, 2021), FRB 20200120E is not listed in the above 18
data. The fitted line of 18 data is given, with the goodness of fit as 0.75.
As shown in Figure 5, our fitted line (solid) is almost parallel to that of
the Macqurat relation (dashed), and the deviation of two slopes is less than
6%. This indicates that the DM-z relation is still available, at least for the
case that z is less than 0.7. While the obvious difference between the two
lines is reflected as a fact that our fitting has an intercept value of 84.43
$\rm pc\,cm^{-3}$ with the vertical axis. A possible reason for this gap is
that the different DM data have been used. Our $\rm DM_{excess}$ contain the
$\rm DM_{host}$ and $\rm DM_{sur}$, but the $\rm DM_{cosmic}$ in the Macquart
relation does not. Meanwhile, the contribution of the Milk way galaxy halo
does not consider in our analysis either. Therefore, the gap value of 84.43
$\rm pc\,cm^{-3}$ may infer the DM in the host galaxy, surrounding medium,
halo, or both of them.

Figure 5: Diagram of dispersion measure (DM) and redshift (z). The solid line
represents the best fit curve of 18 localized FRBs, and the dashed line stands
for the Macquart relation. The squares represent 8 localized FRBs given by
Macquart et al (2020), and the 10 circles are the newly added data.

## Appendix B: Analysis of full DM data

Under our data selection, we only analyze the repeaters and non-repeaters at
low DM in the main text, so we discuss the full data here, including FRBs at
both high and low DM. Same as the above, we test lognormal feature for these
date under K-S, Lilliefors, and A-D tests in Table 4. The results are
consistent with the low DM data. Meanwhile, the goodness of fit, K-S and M-W-W
tests are also applied on the fitted curves, and the results are shown in
Table 5 which further support the lognormal distribution of luminosity for
non-repeaters.

Besides that, we plot full DM data in Figure 6 to compare whether repeaters
and non-repeaters have the same distribution with mean values of the two
samples (repeaters: $1.23\times 10^{44}\,erg\,s^{-1}$, non-repeater:
$5.48\times 10^{44}\,erg\,s^{-1}$). Although the repeaters and non-repeaters
still show the different distributions ($p_{ks}=6.17\times 10^{-4}$ and
$p_{mww}=5.70\times 10^{-5}$) like in the case of low DM selection, the mean
values of all data are inconsistent with the data at low DM, indicating that
the high and low DM may be different. Furthermore, the maximum and minimum
luminosity of non-repeaters is $8.00\times 10^{45}\,\rm erg\,s^{-1}$ and
$6.83\times 10^{42}\,\rm erg\,s^{-1}$, respectively, which has a wider
distribution range than the sample at low DM. Therefore, these imply that the
reasons for these differences are possibly due to the observational effects,
data processing methods, or even their intrinsic physical properties, and we
need further study to figure out the puzzles.

Overall, no matter whether we use the low DM or full DM data, it does not
impact our main conclusions on the luminosity distribution form of non-
repeaters, which is lognormal type is better for describing them.

Table 4: Statistical test results of lognormal distribution for repeaters and
non-repeaters for full DM

FRBs | Number | K-S test | Lilliefors test | A-D testa
---|---|---|---|---
Non-repeaters | 255 | 0.452 | 0.111 | (0.481, 0.775)
Repeaters | 18 | 0.716 | 0.238 | (0.397, 0.687)

Notes. a The results of A-D test are shown in the form of statistic and
critical values (statistic, critical values).

Table 5: Statistical test results of different luminosity distribution for all non-repeaters Different types | Goodness of fit | K-S test | M-W-W test
---|---|---|---
Lognormal | 0.99994 | 0.851 | 0.670
Power-law | 0.614 | $4.57\times 10^{-4}$ | $3.29\times 10^{-3}$
Figure 6: The distinction for all repeaters and non-repeaters (both low and
high dispersion measure (DM)) in the aspect of luminosity. The top panel (sub-
figure a) is the CDF of repeaters and non-repeaters for luminosity. The dashed
(solid) line is for repeaters (non-repeaters). The bottom panel (sub-figure b)
is the histogram of luminosity for two samples. The cross-hatched (empty)
histogram means the repeaters (non-repeaters). The dashed (solid) line
represents the mean value of repeaters (non-repeaters).

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## Statements & Declarations

### Funding

This work was supported by the National Natural Science Foundation of China
(Grant No. 11988101, No. U1938117, No. U1731238, No. 11703003 and No.
11725313), the International Partnership Program of Chinese Academy of
Sciences grant No. 114A11KYSB20160008, the National Key R&D Program of China
No. 2016YFA0400702, and the Guizhou Provincial Science and Technology
Foundation (Grant No. [2020]1Y019).

### Data availability

The dataset of repeaters and non-repeaters is available in the CHIME/FRB
Catalog 1 repository, https://www.chime-frb.ca/home. The dataset of localized
FRBs is available in the FRB HOST DATABASE, http://frbhosts.org/.

### Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

### Author Contributions

All authors contributed to the study conception and design. Xianghan Cui and
Chengmin Zhang wrote the main manuscript text, Di Li and Jianwei Zhang provied
analytical methods, Di Li, Bo Peng, Weiwei Zhu. and Richard Strom illustrated
physical properties, Shuangqiang Wang, Na Wang and Qingdong Wu gave
suggestions on figures, and Dehua Wang and Yangyi Yan modify the manuscript.
All authors participated in the discussion and read the manuscript.


# Approximation in weighted holomorphic Besov spaces in $\mathbb{C}^{n}$

Ali Abkar
Department of Pure Mathematics, Faculty of Science,
Imam Khomeini International University, Qazvin 34149, Iran

Email<EMAIL_ADDRESS>

Abstract. We study certain weighted Bergman and weighted Besov spaces of
holomorphic functions in the polydisk and in the unit ball. We seek Mergelyan-
type conditions on the non-radial weight function to guarantee that the
dilations of a given function tend to the same function in norm; in
particular, we seek conditions on the non-radial weights to ensure that the
analytic polynomials are dense in the space.

Keywords: Bergman space, analytic Besov space, dilation, non-radial weight,
angular weight

MSC2020: 46E15, 46E20, 30H25, 32A36

##  1 Introduction

In the study of Banach spaces of analytic functions on sub-domains of
$\mathbb{C}^{n}$, an important question is to know whether the polynomials are
dense in the space. In the complex plane, the first result on the
approximation by polynomials in the space of $p$-th power area integrable
functions on simply connected domains was obtained by Torsten Carleman in his
1923 paper [6]. Indeed, Carleman approximated functions in the Bergman space
of the domain, but at the time the phrase ”Bergman space” was not yet coined.
Carleman’s result was then extended to more general regions by O.J. Farrell
(see [7], [8]). Farrell proved that for each $p$-th power area integrable
function $f$, there is a sequence of polynomials $p_{n}$ such that $p_{n}\to
f$ in norm. We should remark that the Taylor polynomials of a given function
do not necessarily approximate the function in norm (see [13], and [16]).
Instead, one tries to show that the dilations $f_{r}(z)=f(rz)$ for $0\leq
r<1$, converge to $f$ in norm. It is easily understood that the dilations
$f_{r}$ that are defined on bigger domains are more well-behaved and tractable
than the function $f$ itself.

In this paper we are concerned with approximation in weighted Bergman and
weighted Besov spaces in several complex variables. We mean by the polydisk
the subset

$\mathbb{D}^{n}=\\{z=(z_{1},...,z_{n})\in\mathbb{C}^{n}:|z_{j}|<1,\,\,1\leq
j\leq n\\}$

of the $n$-dimensional complex space, and by the unit ball the set

$\mathbb{B}_{n}=\\{z=(z_{1},...,z_{n})\in\mathbb{C}^{n}:|z|^{2}=|z_{1}|^{2}+\cdots+|z_{n}|^{2}<1\\}.$

Let $\operatorname{Hol}(\mathbb{D}^{n})$ denote the space of holomorphic
functions on $\mathbb{D}^{n}$. For $0<p<\infty$, the Bergman space on the
polydisk $\mathbb{D}^{n}$ is defined as

$A^{p}_{\alpha}(\mathbb{D}^{n})=\operatorname{Hol}(\mathbb{D}^{n})\cap
L^{p}(\mathbb{D}^{n},dV_{\alpha})$

where $dV_{\alpha}(z)=dA_{\alpha}(z_{1})\cdots dA_{\alpha}(z_{n})$, and

$dA_{\alpha}(z_{k})=(1-|z_{k}|^{2})^{\alpha}dx_{k}\,dy_{k}.$

In general, $\alpha$ is bigger than $-1$, but in this paper we shall assume
that $\alpha\geq 0$. If $\alpha=0$, we write $dV(z)=dA(z_{1})\cdots
dA(z_{n})$.

Let $w$ be a positive function on $\mathbb{D}^{n}$. For $0<p<\infty$, the
weighted Bergman space $A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})$ consists of
holomorphic functions $f$ on the polydisk such that the integral

$\|f\|^{p}_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}=\int_{\mathbb{D}^{n}}|f(z)|^{p}w(z)dV_{\alpha}(z)$

is finite. In a similar fashion, let $\operatorname{Hol}(\mathbb{B}_{n})$
denote the space of holomorphic functions on the unit ball. The Bergman space
on the unit ball is defined as

$A^{p}(\mathbb{B}_{n})=\operatorname{Hol}(\mathbb{B}_{n})\cap
L^{p}(\mathbb{B}_{n},dv)$

where $dv$ is the normalized volume measure on $\mathbb{B}_{n}$. Indeed, the
weighted Bergman space $A^{p}_{w}(\mathbb{B}_{n},dv)$ consists of holomorphic
functions $f$ in the unit ball such that

$\|f\|^{p}_{A^{p}_{w}(\mathbb{B}_{n},dv)}=\int_{\mathbb{B}_{n}}|f(z)|^{p}w(z)dv(z)<\infty.$

Let $0<r<1$ and $z=(z_{1},...,z_{n})$. We mean by $rz=(rz_{1},...,rz_{n})$ and

$\frac{z}{r}=\left(\frac{z_{1}}{r},...,\frac{z_{n}}{r}\right).$

In this paper, we seek conditions on the weight function $w$ (defined on
$\mathbb{D}^{n}$ or $\mathbb{B}_{n}$) to guarantee that the dilations $f_{r}$
converge to $f$ in norm. Recall that if the weight function is radial, the
problem is well-known; the difficulty arises when we consider non-radial
weights. Since $f_{r}$ are holomorphic on a bigger domain, they can be
approximated by polynomials so that $f$ can be approximated by polynomials too
(provided that the weighted space contains the polynomials). We shall consider
weights that satisfy the following Mergelyan-type condition: there is a non-
negative integer $k$ and an $r_{0}\in(0,1)$ such that

$r^{k}w\left(\frac{z}{r}\right)\leq Cw(z),\quad\,r_{0}\leq r<1,\,|z|<r.$ (1)

This condition on the weight function is universal; it does not depend on the
underlying region. In the next section we shall provide examples of non-radial
weights satisfying this condition. We will prove some approximation theorems
both for Bergman spaces and for Besov spaces. We should remark that the
definition of Besov spaces of holomorphic functions of several variables is
slightly tricky; we shall present two different definitions for this spaces;
the first definition that appears in Section §3 is taken from [17]. The second
definition, based on the notion of radial derivatives, is discussed in Section
§4. To the best of our knowledge, the latter definition is new and were unable
to trace it back in the literature. Finally, in section 5, we introduce the
concept of angular weights on polydisk. Theses are weights that just depend on
the arguments of $z_{1},...,z_{n}$. Similar approximation results are
established for angular weights. To the best of the author’s knowledge, the
definition of angular weights are somewhat new, and have not yet been
discussed in the literature.

Polynomial approximation has many applications in operator theory of function
spaces; see the papers [5], [11]. Although approximation theory for non-radial
weights has been discussed in earlier papers like [9], [10], but there are
still many problems to be settled; see the comprehensive account on this topic
in [14]. We close this section by mentioning several results in one variable
case, see for instance [1], [2], [3], [4].

##  2 The Bergman spaces

It is a classical theorem that if the weight function on the unit disk is
radial, that is $w(z)=w(|z|)$, then the polynomials are dense in the weighted
Bergman space $A^{2}_{w}(\mathbb{D},dA)$. This was proved by S. Mergelyan [12]
under the assumption that

$\int_{0}^{1}rw(r)dr<\infty.$

The same result is of course true for the Bergman spaces
$A^{p}_{w}(\mathbb{D},dA)$, for $0<p<\infty$. We assume now that $w$ is an
arbitrary weight function (not necessarily radial) for which

$\int_{\mathbb{D}}w(z)dA(z)<\infty.$

We shall see that under some mild condition on $w$, the polynomials are dense
in the weighted Bergman spaces $A^{p}(\mathbb{D},d\mu_{\alpha})$. We begin
with the following theorem.

###### Theorem 2.1

Let $0<p<\infty$ and $d\mu_{\alpha}(z)=w(z)dV_{\alpha}(z)$ be a finite
positive measure on the polydisk such that $w$ satisfies the condition (1).
Then the polynomials are dense in the weighted Bergman space
$A^{p}(\mathbb{D}^{n},d\mu_{\alpha})$.

Proof. Let $f\in A^{p}(\mathbb{D}^{n},d\mu_{\alpha})$, and consider the
dilatations $f_{r}(z)=f(rz)$ for $0\leq r<1$, and $z\in\mathbb{D}^{n}$. By a
change of variables, we see that for non-negative $\alpha$ and big enough $r$
($r\geq r_{0}$, see the condition (1) above) we have

$\displaystyle\int_{\mathbb{D}^{n}}|f_{r}(z)|^{p}d\mu_{\alpha}(z)$
$\displaystyle=\frac{1}{r^{2n+k}}\int_{r\mathbb{D}\times\cdots\times
r\mathbb{D}}|f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\prod_{k=1}^{n}\Big{(}\frac{r^{2}-|z_{k}|^{2}}{r^{2}}\Big{)}^{\alpha}dV(z)$
$\displaystyle\leq\frac{C}{r^{2n+k}}\int_{r\mathbb{D}\times\cdots\times
r\mathbb{D}}|f(z)|^{p}w(z)dV(z)<\infty.$

We now use the dominated convergence theorem to conclude that

$\limsup_{r\to
1^{-}}\int_{{\mathbb{D}}^{n}}|f_{r}|^{p}d\mu_{\alpha}(z)\leq\int_{{\mathbb{D}}^{n}}|f|^{p}d\mu_{\alpha}(z).$

This inequality together with Fatou’s lemma implies that for each measurable
subset $E\subseteq\mathbb{D}^{n}$,

$\lim_{r\to
1^{-}}\int_{E}|f_{r}|^{p}d\mu_{\alpha}(z)=\int_{E}|f|^{p}d\mu_{\alpha}(z).$
(2)

By the continuity property of integral on its domain, there is a $\delta>0$
such that

$\mu(E)<\delta\implies\int_{E}|f|^{p}d\mu_{\alpha}(z)<\epsilon.$

It follows from Egorov’s theorem that there is a subset
$E\subset\mathbb{D}^{n}$ such that $\mu_{\alpha}(\mathbb{D}^{n}\setminus
E)<\delta$ and $f_{r}\to f$ uniformly on $E$. We now write

$\displaystyle\int_{{\mathbb{D}}^{n}}|f_{r}-f|^{p}d\mu_{\alpha}$
$\displaystyle=\int_{E}|f_{r}-f|^{p}d\mu_{\alpha}+\int_{\mathbb{D}\setminus
E}|f_{r}-f|^{p}d\mu_{\alpha}$
$\displaystyle\leq\int_{E}|f_{r}-f|^{p}d\mu_{\alpha}+2^{p}\int_{\mathbb{D}^{n}\setminus
E}(|f_{r}|^{p}+|f|^{p})d\mu_{\alpha}.$ (3)

Due to the uniform convergence of $f_{r}$ to $f$ on $E$, the first integral on
the right-hand side of (2) tends to zero as $r\to 1^{-}$. As for the second
integral, we use (2) to obtain

$\displaystyle 0\leq\liminf_{r\to
1^{-}}\int_{\mathbb{D}^{n}}|f_{r}-f|^{p}d\mu_{\alpha}(z)$
$\displaystyle\leq\limsup_{r\to
1^{-}}\int_{\mathbb{D}^{n}}|f_{r}-f|^{p}d\mu_{\alpha}(z)$ $\displaystyle\leq
2^{p+1}\int_{\mathbb{D}^{n}\setminus E}|f|^{p}d\mu_{\alpha}$
$\displaystyle\leq 2^{p+1}\epsilon,$

from which it follows that

$\lim_{r\to 1^{-}}\int_{\mathbb{D}^{n}}|f_{r}-f|^{p}d\mu_{\alpha}(z)=0.$ (4)

Given $\epsilon>0$, there is $0<r_{0}<1$ such that
$\|f-f_{r_{0}}\|_{A^{p}(\mathbb{D}^{n},d\mu_{\alpha})}<\epsilon$. Since
$f_{r_{0}}$ is analytic in a bigger domain than the unit polydisk, we may use
Mergelyan’s theorem to approximate $f_{r_{0}}$ uniformly on $\mathbb{D}^{n}$
by a polynomial $Q$. Since $\mu_{\alpha}$ is a finite positive measure, we
obtain

$\displaystyle\|f_{r_{0}}-Q\|^{p}_{A^{p}(\mathbb{D}^{n},d\mu_{\alpha})}$
$\displaystyle=\int_{\mathbb{D}^{n}}|f_{r_{0}}-Q|^{p}d\mu_{\alpha}$
$\displaystyle\leq\epsilon^{p}\int_{\mathbb{D}^{n}}d\mu_{\alpha}(z):=\epsilon^{p}C,$

from which it follows that

$\|f-Q\|_{A^{p}(\mathbb{D}^{n},d\mu_{\alpha})}\leq\epsilon(1+C^{1/p}).$

This proves the theorem. $\square$

###### Remark 2.2

The condition (1) can be weakened in the following way: there is a non-
negative function $g$ such that for sufficiently large $r$ we have
$r^{k}w(z/r)\leq g(z)$ and

$\int_{\mathbb{D}^{n}}|f(z)|^{p}g(z)dV_{\alpha}(z)<\infty.$

Again we use the dominated convergence theorem to obtain

$\limsup_{r\to
1^{-}}\int_{E}|f_{r}|^{p}d\mu_{\alpha}(z)\leq\int_{E}|f|^{p}d\mu_{\alpha}(z).$

###### Theorem 2.3

Let $0<p<\infty$ and $d\mu_{\alpha}(z)=w(z)dV_{\alpha}(z)$ be a finite
positive measure on the polydisk such that for some non-negative integer $k$,
the function

$r\mapsto r^{k}w\left(\frac{z}{r}\right),\quad|z|<r,$

is increasing in $r$. Then the polynomials are dense in
$A^{p}(\mathbb{D}^{n},wdV_{\alpha})$.

Proof. It suffices to show that each function $f\in
A^{p}(\mathbb{D}^{n},d\mu_{\alpha})$ satisfies

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}^{n}}|f_{r}|^{p}d\mu_{\alpha}(z)\leq\int_{\mathbb{D}^{n}}|f|^{p}d\mu_{\alpha}(z).$

To do this, we make a change of variables to get

$\int_{\mathbb{D}^{n}}|f_{r}(z)|^{p}d\mu_{\alpha}(z)=\frac{1}{r^{2n+k}}\int_{r\mathbb{D}\times\cdots\times
r\mathbb{D}}|f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\prod_{j=1}^{n}\Big{(}\frac{r^{2}-|z_{j}|^{2}}{r^{2}}\Big{)}^{\alpha}dV(z).$

Note that the functions

$r\mapsto r^{k}w\left(\frac{z}{r}\right),\quad
r\mapsto\Big{(}\frac{r^{2}-|z_{j}|^{2}}{r^{2}}\Big{)}^{\alpha}$

are increasing in $r$ (by the assumption and the fact that $\alpha$ is non-
negative), so that the monotone convergence theorem applies to yield

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}^{n}}|f_{r}(z)|^{p}d\mu_{\alpha}(z)=\int_{\mathbb{D}^{n}}|f(z)|^{p}d\mu_{\alpha}(z).$

Now, we use an argument as in the proof of Theorem 2.1 to get the result.
$\square$

###### Example 2.4

(a). Let $\alpha>-1$ and
$w(z)=\prod_{k=1}^{n}(\alpha+1)(1-|z_{k}|^{2})^{\alpha}$.

$r^{k}w\left(\frac{z}{r}\right)=\prod_{k=1}^{n}(\alpha+1)r^{k-2\alpha}(r^{2}-|z_{k}|^{2})^{\alpha}\leq
w(z),$

provided that $k\geq 2\alpha+1$. Now Theorem 2.1 can be applied.
(b). It is easily seen that the Gaussian weight

$w(z)=e^{-\beta|z|^{2}}=e^{-\beta(|z_{1}|^{2}+\cdots+|z_{n}|^{2})},\quad\beta>0,$

satisfies the condition of the above theorem for $k=0$, since for $0<r<r$,

$\displaystyle w\left(\frac{z}{r}\right)$
$\displaystyle=e^{\frac{-\beta}{r^{2}}(|z_{1}|^{2}+\cdots+|z_{n}|^{2})}$
$\displaystyle\leq e^{-\beta(|z_{1}|^{2}+\cdots+|z_{n}|^{2})}$
$\displaystyle=w(z).$

The same is true for the non-radial weight

$w(z)=e^{-\beta(|x_{1}|^{2}+\cdots+|x_{n}|^{2})},\quad\beta>0,$

where $z=(z_{1},...,z_{n}),\,z_{k}=x_{k}+iy_{k},\,1\leq k\leq n$.
Note that in some instances the function $r\mapsto w(z/r)$ is not increasing,
but we may find some $k$ for which $r\mapsto r^{k}w(z/r)$ is increasing for
$r$ bigger than $|z|$. For example, if we take $w(z)=\exp(|z|)$, then

$\frac{d}{dr}w\left(\frac{z}{r}\right)=-\frac{|z|}{r^{2}}\exp\left(\frac{|z|}{r}\right)<0$

while

$\frac{d}{dr}\left[rw(\frac{z}{r})\right]=\left(1-\frac{|z|}{r}\right)\exp\left(\frac{|z|}{r}\right)>0,\quad|z|<r.$

Note also that $r^{2}e^{|z|/r}$ is increasing for $r>1/2$ since

$\frac{d}{dr}(r^{2}e^{|z|/r})=2re^{|z|/r}+r^{2}(\frac{-1}{r^{2}}e^{|z|/r})=(2r-1)e^{|z|/r}>0.$

##  3 The Besov spaces

In the unit disk, the weighted Dirichlet space
$\mathcal{D}^{p}_{w},\,1<p<\infty$, consists of analytic functions in the unit
disk for which

$\|f\|_{\mathcal{D}^{p}_{w}}=\left(|f(0)|^{p}+\int_{\mathbb{D}}|f^{\prime}(z)|^{p}w(z)dA(z)\right)^{1/p}$

is finite. These spaces are particular cases of certain Banach spaces of
analytic functions, namely, the weighted analytic Besov spaces. The weighted
analytic Besov space $\mathcal{B}_{w}^{p}$ consists of analytic functions $f$
in the unit disk for which the integral

$\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|f^{\prime}(z)|^{p}w(z)dA(z)$

is finite. The norm of a function in the weighted Besov space is given by

$\|f\|_{\mathcal{B}^{p}_{w}}=\left(|f(0)|^{p}+\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|f^{\prime}(z)|^{p}w(z)dA(z)\right)^{1/p}.$

In higher dimensions, the definition of Besov spaces is more subtle. Let
$dv(z)$ be the normalized volume measure in the unit ball $\mathbb{B}_{n}$ of
$\mathbb{C}^{n}$, and let

$d\tau(z)=\frac{dv(z)}{(1-|z|^{2})^{n+1}}$

be the Möbius invariant measure in $\mathbb{B}_{n}$. For $0<p<\infty$, the
Besov space $\mathcal{B}^{p}$ consists of all holomorphic functions
$f\in\mathbb{B}_{n}$ such that

$(1-|z|^{2})^{N}\frac{\partial^{m}f}{\partial z^{m}}(z)\in
L^{p}(\mathbb{B}_{n},d\tau),\quad|m|=N,$

where $N$ is any fixed positive integer satisfying $pN>n$. According to [17,
Theorem 6.1], the definition of $\mathcal{B}^{p}$ is independent of the choice
of $N$. The ”norm” is defined by

$\|f\|_{\mathcal{B}^{p}}^{p}=\sum_{|m|\leq
N-1}\left|\frac{\partial^{m}f}{\partial
z^{m}}(0)\right|^{p}+\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|(1-|z|^{2})^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{p}d\tau(z).$

When $p=2,\,\mathcal{B}^{2}$ plays the role of the Dirichlet space and is
denoted by $\mathcal{D}^{2}$.

###### Theorem 3.1

Let $d\mu(z)=w(z)d\tau(z)$ be a finite positive measure on the unit ball such
that $w$ satisfies the condition (1). Then the polynomials are dense in
$\mathcal{D}^{2}_{w}(\mathbb{B}_{n},d\mu)$.

Proof. As for approximation, it is enough to work with the following
expression for norm (the constant term does not play any role in
approximation):

$\|f\|^{2}_{\mathcal{D}^{2}_{w}}=\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|(1-|z|^{2})^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}w(z)d\tau(z).$

Note that

$\|f_{r}\|^{2}_{\mathcal{D}^{2}_{w}}=\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|(1-|z|^{2})^{N}\frac{\partial^{m}f_{r}}{\partial
z^{m}}(z)\right|^{2}w(z)d\tau(z).$

We now replace $z$ by $z/r$ to obtain

$\displaystyle\|f_{r}\|^{2}_{\mathcal{D}^{2}_{w}}$
$\displaystyle=\sum_{|m|=N}\frac{1}{r^{k}}\int_{r\mathbb{B}_{n}}\left|\Big{(}\frac{r^{2}-|z|^{2}}{r^{2}}\Big{)}^{N}r^{|m|}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}r^{k}w\left(\frac{z}{r}\right)\frac{r^{2}}{(r^{2}-|z|^{2})^{n+1}}dv(z)$
$\displaystyle\leq
C\frac{r^{2+2|m|}}{r^{k+4N}}\sum_{|m|=N}\int_{r\mathbb{B}_{n}}\left|\big{(}1-|z|^{2}\big{)}^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}w(z)\frac{dv(z)}{(r^{2}-|z|^{2})^{n+1}}$
$\displaystyle=C\frac{r^{2+2|m|}}{r^{k+4N}}\sum_{|m|=N}\int_{\mathbb{B}_{n}}\chi(r\mathbb{B}_{n})(z)\left|\big{(}1-|z|^{2}\big{)}^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}w(z)\frac{dv(z)}{(r^{2}-|z|^{2})^{n+1}}$
$\displaystyle\leq
C\frac{r^{2+2|m|}}{r^{k+4N}}\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|\big{(}1-|z|^{2}\big{)}^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}w(z)\frac{dv(z)}{(r^{2}-|z|^{2})^{n+1}}.$

Since the function

$r\mapsto\frac{1}{(r^{2}-|z|^{2})^{n+1}},\quad|z|<r,$

is monotone decreasing in $r$, the last integral converges ar $r\to 1^{-}$, so
that a version of the dominated convergence theorem can be applied to the
first integral:

$\displaystyle\limsup_{r\to 1^{-}}\|f_{r}\|^{2}_{\mathcal{D}^{2}_{w}}$
$\displaystyle=\limsup_{r\to
1^{-}}\sum_{|m|=N}\frac{1}{r^{k}}\int_{r\mathbb{B}_{n}}\left|\Big{(}\frac{r^{2}-|z|^{2}}{r^{2}}\Big{)}^{N}r^{|m|}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}r^{k}w\left(\frac{z}{r}\right)\frac{r^{2}}{(r^{2}-|z|^{2})^{n+1}}dv(z)$
$\displaystyle=\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|(1-|z|^{2})^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{2}w(z)\frac{dv(z)}{(1-|z|^{2})^{n+1}}$
$\displaystyle=\|f\|^{2}_{\mathcal{D}^{2}_{w}}.$

The rest of the proof goes in the same way as in the proof of Theorem 2.1; we
just replace $f$ and $f_{r}$, by $\frac{\partial^{m}f}{\partial z^{m}}$ and
$\frac{\partial^{m}f_{r}}{\partial z^{m}}$, respectively, to obtain

$\lim_{r\to 1^{-}}\|f_{r}-f\|^{2}_{\mathcal{D}^{2}_{w}}=\lim_{r\to
1^{-}}\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|(1-|z|^{2})^{N}\left(\frac{\partial^{m}f}{\partial
z^{m}}(z)-\frac{\partial^{m}f_{r}}{\partial
z^{m}}(z)\right)\right|^{2}w(z)d\tau(z)=0.$

On the other hand, each dilation $f_{r}$ can be approximated by polynomials
(see [17, Theorem 6.2]), therefore, $f$ can be approximated by polynomials.
$\square$

We now state a similar statement for the analytic Besov spaces.

###### Theorem 3.2

Let $d\mu(z)=w(z)d\tau(z)$ be a finite positive measure on the unit ball such
that $w$ satisfies the condition (1). Then the polynomials are dense in
$\mathcal{B}^{p}_{w}(\mathbb{B}_{n},d\mu)$.

Proof. We begin by inspecting the expression

$\|f_{r}\|^{p}_{\mathcal{B}^{p}_{w}}=\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|(1-|z|^{2})^{N}\frac{\partial^{m}f_{r}}{\partial
z^{m}}(z)\right|^{p}w(z)d\tau(z).$

Similar to the preceding case, we have

$\displaystyle\|f_{r}\|^{p}_{\mathcal{B}^{p}_{w}}$
$\displaystyle=\sum_{|m|=N}\frac{1}{r^{k}}\int_{r\mathbb{B}_{n}}\left|\Big{(}\frac{r^{2}-|z|^{2}}{r^{2}}\Big{)}^{N}r^{|m|}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{p}r^{k}w\left(\frac{z}{r}\right)\frac{r^{2}}{(r^{2}-|z|^{2})^{n+1}}dv(z)$
$\displaystyle\leq
C\frac{r^{2+p|m|}}{r^{k+2pN}}\sum_{|m|=N}\int_{\mathbb{B}_{n}}\left|\big{(}1-|z|^{2}\big{)}^{N}\frac{\partial^{m}f}{\partial
z^{m}}(z)\right|^{p}w(z)\frac{dv(z)}{(r^{2}-|z|^{2})^{n+1}}.$

Again, this argument shows that

$\limsup_{r\to
1^{-}}\|f_{r}\|^{p}_{\mathcal{B}^{p}_{w}}\leq\|f\|^{p}_{\mathcal{B}^{p}_{w}},$

from which the result follows. $\square$

##  4 Besov spaces: alternative definition

Another approach in the study of Besov spaces in several complex variables is
to use the concept of radial derivative. Let $f$ be a holomorphic function in
the unit ball $\mathbb{B}_{n}$. The radial derivative of $f$ is defined by

$Rf(z)=\sum_{k=1}^{n}z_{k}\frac{\partial f}{\partial z_{k}}(z).$

According to [17], if

$f(z)=\sum_{k=0}^{\infty}f_{k}(z)$ (5)

is the homogenous expansion of $f$ (each $f_{k}$ is a polynomial of degree $k$
in $z_{1}$,…,$z_{n}$), then

$Rf(z)=\sum_{k=1}^{\infty}kf_{k}(z).$

The weighted analytic Besov space $\mathbf{B}_{w}^{p}(\mathbb{D}^{n})$
consists of analytic functions $f$ on a neighborhood of polydisk for which the
integral

$\int_{\mathbb{D}^{n}}\Big{(}\prod_{j=1}^{n}(1-|z_{j}|^{2})^{p-2}\Big{)}|Rf(z)|^{p}w(z)dV(z)$

is finite. The norm of a function in $\mathbf{B}_{w}^{p}(\mathbb{D}^{n})$ is
given by

$\|f\|^{p}_{\mathbf{B}^{p}_{w}(\mathbb{D}^{n})}=|f(0)|^{p}+\int_{\mathbb{D}^{n}}\Big{(}\prod_{j=1}^{n}(1-|z_{j}|^{2})^{p-2}\Big{)}|Rf(z)|^{p}w(z)dV(z).$

###### Theorem 4.1

Let $2\leq p<\infty$ and $d\mu(z)=w(z)dV(z)$ be a finite positive measure on
the polydisk such that $w$ satisfies the condition (1). Then the polynomials
are dense in the weighted analytic Besov space
$\mathbf{B}_{w}^{p}(\mathbb{D}^{n})$.

Proof. As we pointed out earlier, the constant term in the definition of norm
does not play any role in approximation, so that we just look at

$\|f_{r}\|^{p}_{\mathbf{B}^{p}_{w}(\mathbb{D}^{n})}=\int_{\mathbb{D}^{n}}\Big{(}\prod_{j=1}^{n}(1-|z_{j}|^{2})^{p-2}\Big{)}|Rf_{r}(z)|^{p}w(z)dV(z).$

We first note that by (5)

$f_{r}(z)=f(rz)=\sum_{m=0}^{\infty}f_{m}(rz),$

from which it follows that

$(Rf_{r})(z)=\sum_{m=0}^{\infty}mf_{m}(rz).$

Replacing $z$ by $z/r$ we obtain

$\displaystyle\|f_{r}\|^{p}_{\mathbf{B}^{p}_{w}(\mathbb{D}^{n})}$
$\displaystyle=\int_{\mathbb{D}^{n}}\Big{[}\prod_{j=1}^{n}(1-|z_{j}|^{2})^{p-2}\Big{]}|Rf_{r}(z)|^{p}w(z)dV(z)$
$\displaystyle=\frac{1}{r^{k+2n}}\int_{r\mathbb{D}\times\cdots\times
r\mathbb{D}}\Big{[}\prod_{j=1}^{n}\Big{(}\frac{r^{2}-|z_{j}|^{2}}{r^{2}}\Big{)}^{p-2}\Big{]}|Rf(z)|^{p}r^{k}w(\frac{z}{r})dV(z)$
$\displaystyle\leq\frac{C}{r^{k+2n}}\int_{r\mathbb{D}\times\cdots\times
r\mathbb{D}}\Big{[}\prod_{j=1}^{n}\Big{(}\frac{r^{2}-|z_{j}|^{2}}{r^{2}}\Big{)}^{p-2}\Big{]}|Rf(z)|^{p}w(z)dV(z).$

Since $p-2\geq 0$, each of the functions

$r\mapsto\Big{(}\frac{r^{2}-|z_{j}|^{2}}{r^{2}}\Big{)}^{p-2}$

is increasing in $r$, so that we can apply the generalized version of
dominated convergence theorem to get

$\lim_{r\to
1^{-1}}\|f_{r}\|^{p}_{\mathbf{B}^{p}_{w}(\mathbb{D}^{n})}=\|f\|^{p}_{\mathbf{B}^{p}_{w}(\mathbb{D}^{n})}.$

This latter implies that the dilations $f_{r}$ tend to $f$ in norm;

$\|f-f_{r}\|^{p}_{\mathbf{B}^{p}_{w}(\mathbb{D}^{n})}=0,$

which leads to the desired result. $\square$

##  5 Angular weights on polydisk

In contrast to radial weights defined on the unit disk, we may consider
weights that depend only on the argument of $z$; that is,
$w(re^{i\theta})=w(\theta)$. We may call such weights angular weights. For
example, given $\alpha>0$,

$w(z)=w(re^{i\theta})=(4\pi^{2}-\theta^{2})^{\alpha},\quad 0\leq\theta<2\pi,$

is an angular weight on the unit disk. Similarly, a positive continuous
function $w$ on the polydisk $\mathbb{D}^{n}$ is said to be angular if

$w(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})=w(\theta_{1},...,\theta_{n}),$

meaning that $w$ just depends on the arguments of the components of
$z\in\mathbb{D}^{n}$.

Some notations
For simplicity, given an $n$-tuple $\mathbf{r}=(r_{1},...,r_{n})$, we write
$d\mathbf{r}=dr_{1}\cdots dr_{n}$, and for a given weight function $w$, we
write $w(\mathbf{r})$ instead of $\omega(r_{1},...,r_{n})$. In the same
manner, $w(\mathbf{\theta})$ is a short form for
$w(\theta_{1},...,\theta_{n})$, and $d\mathbf{\theta}$ stands for
$d\theta_{1}\cdots d\theta_{n}$. The distinguished boundary of
$\mathbb{D}^{n}$ is denoted by $\mathbb{T}^{n}$, and $|m|=m_{1}+\cdots+m_{n}$
where $m=(m_{1},...,m_{n})$ is a multi-index consisting of non-negative
integers.

###### Proposition 5.1

Let $w$ be an angular weight function on the polydisk that satisfies

$\int_{\mathbb{T}^{n}}w(\mathbf{\theta})d\mathbf{\theta}<\infty.$

Then the Taylor polynomials of each function in
$A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})$ tend to the same function in norm. In
particular, the polynomials are dense in
$A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})$.

Proof. Let $m=(m_{1},...,m_{n})$ be a multi-index, and

$f(z)=\sum_{|m|\geq 0}a_{m}z^{m}=\sum_{|m|\geq
0}a_{m_{1},...,m_{n}}z_{1}^{m_{1}}\cdots z_{n}^{m_{n}}$

be a function in $A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})$. Let $I^{n}$ stand
for the $n$ times Cartesian product of the unit interval $[0,1]$. Then using
polar coordinates we have

$\displaystyle\|f\|^{2}_{A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})}$
$\displaystyle=\int_{I^{n}}\int_{\mathbb{T}^{n}}\sum_{|m|\geq
0}|a_{m}|^{2}\prod_{k=1}^{n}\left(r_{k}^{2m_{k}}(1-r_{k}^{2})^{\alpha}r_{k}dr_{k}\right)\,w(\mathbf{\theta})d\mathbf{\theta}$
$\displaystyle=\Big{(}\int_{\mathbb{T}^{n}}w(\mathbf{\theta})d\mathbf{\theta}\Big{)}\sum_{|m|\geq
0}\frac{|a_{m}|^{2}\,\Gamma(m_{1}+1)\cdots\Gamma(m_{n}+1)(\Gamma(\alpha+1))^{n}}{2^{n}\Gamma(m_{1}+\alpha+1)\cdots\Gamma(m_{n}+\alpha+1)}.$

This implies that for $p_{k}(z)=\sum_{0\leq|m|\leq k}a_{m}z^{m}$ we have

$\|f-p_{k}\|^{2}_{A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})}=\Big{(}\int_{\mathbb{T}^{n}}w(\mathbf{\theta})d\mathbf{\theta}\Big{)}\sum_{|m|\geq
k+1}\frac{|a_{m}|^{2}\,\Gamma(m_{1}+1)\cdots\Gamma(m_{n}+1)(\Gamma(\alpha+1))^{n}}{2^{n}\Gamma(m_{1}+\alpha+1)\cdots\Gamma(m_{n}+\alpha+1)}.$

Now, we let $k$ tend to infinity, so that $p_{k}\to f$ in
$A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})$. $\square$

This result can be generalized in the following way.

###### Theorem 5.2

Let $w$ be an angular weight function on the polydisk that satisfies

$\int_{\mathbb{T}^{n}}w(\mathbf{\theta})d\mathbf{\theta}<\infty.$

Then the polynomials are dense in
$A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha}),\,0<p<\infty$.

Proof. Since $w$ is angular, we see that $w(z)=w(\frac{z}{r})$, so that

$\displaystyle\|f_{r}\|^{p}_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}$
$\displaystyle=\int_{\mathbb{D}^{n}}|f_{r}(z)|^{p}w(z)\prod_{k=1}^{n}(1-|z_{k}|^{2})^{\alpha}dV(z)$
$\displaystyle=\frac{1}{r^{2n}}\int_{r\mathbb{D}^{n}}|f(z)|^{p}w(\frac{z}{r})\prod_{k=1}^{n}\left(\frac{r^{2}-|z_{k}|^{2}}{r^{2}}\right)^{\alpha}dV(z)$
$\displaystyle=\frac{1}{r^{2n}}\int_{r\mathbb{D}^{n}}|f(z)|^{p}w(z)\prod_{k=1}^{n}\left(\frac{r^{2}-|z_{k}|^{2}}{r^{2}}\right)^{\alpha}dV(z).$

Again, we use $\alpha\geq 0$ and the fact that the function

$r\mapsto\Big{(}\frac{r^{2}-|z_{k}|^{2}}{r^{2}}\Big{)}^{\alpha},\quad|z_{k}|\leq
r,$

is increasing in $r$ to obtain

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}^{n}}|f_{r}(z)|^{p}w(z)dV_{\alpha}(z)=\int_{\mathbb{D}^{n}}|f(z)|^{p}w(z)dV_{\alpha}(z),$

As in the proof of Theorem 2.1, this equality leads to

$\|f_{r}-f\|_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}=0,\quad r\to 1^{-},$

from which the result follows. $\square$

We now consider weights that are products of a radial and an angular weight.

###### Proposition 5.3

Let
$w(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})=\omega(r_{1},...,r_{n})\nu(\theta_{1},...,\theta_{n}),$
where $\omega$ and $\nu$ are two weights on $\mathbb{D}^{n}$ satisfying

$\int_{I^{n}}\Big{[}\prod_{k=1}^{n}(1-r_{k}^{2})^{\alpha}r_{k}\Big{]}\omega(\mathbf{r})d\mathbf{r}<\infty,\quad\int_{\mathbb{T}^{n}}\nu(\mathbf{\theta})d\mathbf{\theta}<\infty.$

Then the Taylor polynomials of each function in
$A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})$ tend to the same function in norm. In
particular, the polynomials are dense in
$A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})$.

Proof. A computation shows that

$\displaystyle\|f\|^{2}_{A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})}$
$\displaystyle=\int_{I^{n}}\int_{\mathbb{T}^{n}}\sum_{|m|\geq
0}|a_{m}|^{2}\prod_{k=1}^{n}\left(r_{k}^{2m_{k}}(1-r_{k}^{2})^{\alpha}r_{k}dr_{k}\right)\omega(\mathbf{r})\nu(\mathbf{\theta})d\mathbf{r}d\mathbf{\theta}$
$\displaystyle=\Big{(}\int_{\mathbb{T}^{n}}\nu(\mathbf{\theta})d\mathbf{\theta}\Big{)}\sum_{|m|\geq
0}|a_{m}|^{2}\int_{I^{n}}\Big{[}\prod_{k=1}^{n}r_{k}^{2m_{k}}(1-r_{k}^{2})^{\alpha}r_{k}\Big{]}\omega(\mathbf{r})d\mathbf{r}$
$\displaystyle=\Big{(}\int_{\mathbb{T}^{n}}\nu(\mathbf{\theta})d\mathbf{\theta}\Big{)}\sum_{|m|\geq
0}|a_{m}|^{2}\gamma_{m},$

where

$\displaystyle\gamma_{m}$
$\displaystyle=\int_{I^{n}}\Big{[}\prod_{k=1}^{n}r_{k}^{2m_{k}}(1-r_{k}^{2})^{\alpha}r_{k}\Big{]}\omega(\mathbf{r})d\mathbf{r}$
$\displaystyle\leq\int_{I^{n}}\Big{[}\prod_{k=1}^{n}(1-r_{k}^{2})^{\alpha}r_{k}\Big{]}\omega(\mathbf{r})d\mathbf{r}<\infty.$

Now, for $p_{k}(z)=\sum_{0\leq|m|\leq k}a_{m}z^{m}$ we have

$\|f-p_{k}\|^{2}_{A^{2}_{w}(\mathbb{D}^{n},dV_{\alpha})}=\Big{(}\int_{\mathbb{T}^{n}}\nu(\mathbf{\theta})d\mathbf{\theta}\Big{)}\sum_{|m|\geq
k+1}^{\infty}|a_{m}|^{2}\gamma_{m},$

which tends to zero as $k\to\infty$. $\square$

###### Theorem 5.4

Let
$w(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})=\omega(r_{1},...,r_{n})\nu(\theta_{1},...,\theta_{n}),$
where $\omega$ and $\nu$ are two weights on $\mathbb{D}^{n}$ satisfying

$\int_{I^{n}}\Big{[}\prod_{k=1}^{n}(1-r_{k}^{2})^{\alpha}r_{k}\Big{]}\omega(\mathbf{r})d\mathbf{r}<\infty,\quad\int_{\mathbb{T}^{n}}\nu(\mathbf{\theta})d\mathbf{\theta}<\infty,$

and $s^{k}\omega(r/s)\leq C\omega(r)$ for some integer $k\geq 0$. Then the
polynomials are dense in the weighted Bergman space
$A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha}),\,0<p<\infty$.

Proof. Again, we see that for
$z=(r_{1}e^{i\theta_{1}},...,r_{n}e^{i\theta_{n}})$,

$\displaystyle\|f_{s}\|^{p}_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}$
$\displaystyle=\int_{\mathbb{D}^{n}}|f_{s}(z)|^{p}\omega(r)\nu(\theta)\prod_{j=1}^{n}(1-|z_{j}|^{2})^{\alpha}dv(z)$
$\displaystyle=\frac{1}{s^{2n+k}}\int_{s\mathbb{D}^{n}}|f(z)|^{p}s^{k}\omega(r/s)\prod_{j=1}^{n}\left(\frac{s^{2}-|z_{j}|^{2}}{s^{2}}\right)^{\alpha}\nu(\theta)dv(z)$
$\displaystyle\leq\frac{C}{s^{2n+k}}\int_{s\mathbb{D}^{n}}|f(z)|^{p}\omega(r)\prod_{j=1}^{n}\left(\frac{s^{2}-|z_{j}|^{2}}{s^{2}}\right)^{\alpha}\nu(\theta)dv(z).$

Using the fact that for $\alpha\geq 0$ and $j$ fixed,

$s\mapsto\Big{(}\frac{s^{2}-|z_{j}|^{2}}{s^{2}}\Big{)}^{\alpha},\quad|z_{j}|\leq
s,$

is increasing in $s$ we obtain

$\limsup_{s\to
1^{-}}\|f_{s}\|^{p}_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}\leq\|f\|^{p}_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}.$

We now use the method introduced in the proof of Theorem 2.1 to get

$\|f_{s}-f\|_{A^{p}_{w}(\mathbb{D}^{n},dV_{\alpha})}=0,\quad s\to 1^{-},$

which in turn leads to the desired result. $\square$

##  6 Declarations

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The author declares no competing interests.

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