Speed of convergence of Chernoff approximations for two model examples:
heat equation and transport equation
Pavel S. Prudnikov

<EMAIL_ADDRESS>National Research University Higher School of Economics

Nizhny Novgorod City, Russia

Abstract. Paul Chernoff in 1968 proposed his approach to approximations of
one-parameter operator semigroups while trying to give a rigorous mathematical
meaning to Feynman’s path integral formulation of quantum mechanics. In early
2000’s Oleg Smolyanov noticed that Chernoff’s theorem may be used to obtain
approximations to solutions of initial-value problems for linear partial
differential equations (LPDEs) of evolution type with variable coefficients,
including parabolic equations, Schrödinger equation, and some other. Chernoff
expressions are explicit formulas containing variable coefficients of LPDE and
the initial condition, hence they can be used as a numerical method for
solving LPDEs. However, the speed of convergence of such approximations at the
present time is understudied which makes it risky to employ this class of
numerical methods.

In the present paper we take two equations with known solutions (heat equation
and transport equation) and study both analytically and numerically the speed
of decay of the norm of the difference between Chernoff approximations and
exact solutions. We also provide graphical illustrations of convergence and
its rate. These model examples, being relatively simple, allow to demonstrate
general properties of Chernoff approximations. The observations obtained build
a base for the future employment of the approach based on Chernoff’s theorem
to the problem of construction of new numerical methods for solving initial-
value problem for parabolic LPDEs with variable coefficients.

Keywords: parabolic partial differential equation, Cauchy problem solution,
Chernoff approximations, convergence speed, estimates of convergence rate,
numerical experiment

MSC2010: 35K15, 47D06, 65M12

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 $C_{0}$-semigroups and Chernoff’s theorem
    2. 2.2 Conjectures on convergence speed for Chernoff approximations
  3. 3 Exact formulas and estimates for convergence speed
    1. 3.1 Transport equation
    2. 3.2 Heat equation
  4. 4 Numerical experiments on model examples
    1. 4.1 Transport equation
    2. 4.2 Heat equation
  5. A Code for numerical experiments

##  1 Introduction

The Chernoff theorem [3] is an effective tool of functional analysis which
allows to construct approximations to $C_{0}$-semigroups that, in turn,
provide solutions to some evolution equations (e.g., heat equation, parabolic
equation with variable coefficients, Schrödinger equation, etc.); standard
textbooks on the topic are [5, 12, 8]. If one finds so-called operator-valued
Chernoff function then Chernoff’s theorem generates a sequence of functions,
also referred to as Chernoff approximations, that converges to the solution of
a Cauchy problem for the evolution equation. In many cases constructing
Chernoff function is the only method that allows to express the solution in
terms of both variable coefficients of the equation and the chosen initial
condition.

At the time the subject area has a decent amount of data on different methods
to construct Chernoff functions. For instance, the members of Smolyanov’s
group implemented the use of integral operators to find the solutions to
parabolic equations in a plethora of cases during the last 15 years (refer to
pioneering papers [18, 17], overview [19, 2] and some other results [16, 4,
21]). The solutions obtained in the above mentioned studies were represented
in the form of Feynman formula, i.e., as a limit of multiple integral when
multiplicity goes to infinity. In turn, Schrödinger equation also belongs to
the class of evolution equations, and the use of the same technique allows to
represent the solution in the form of Feynman and quasi-Feynman integral
formulas as well, see [15, 13].

When dealing with expressions provided by the Chernoff theorem, one might be
interested in obtaining the estimate for the rate of decay of approximation
error as $n$ tends to infinity. Some results regarding this topic were already
proposed for the case of Schrödinger equation in [11]. Similarly, the question
of convergence speed arises for the Trotter product formula
$e^{A+B}=\lim_{n\to\infty}(e^{A/n}e^{B/n})^{n}$ which we do not exploit in
current paper but which is a particular exemplar of Chernoff’s theorem for
$G(t)=e^{tA}e^{tB}$ (see recent papers [9, 10, 22]). In a less specific
setting, other than within the framework of Chernoff’s theorem, one may refer
to the results presented in [6, 1, 7].

Nevertheless, this area is not well studied at the present time in general.
Starting from [20] members of I.D. Remizov’s group examine the problem
systematically. So, this paper is dedicated to the analysis of the convergence
speed for some set of initial conditions and Chernoff functions. We suggest
few analytic and computational methods in order to obtain the information on
the order of approximation subspaces to which the examined model examples
belong to as well as demonstrate the dependence of the order of the
approximation on the choice of the initial condition. It is also shown on the
model example that if the initial condition does not belong to the domain of
the generator of the corresponding $C_{0}$-semigroup then the convergence
speed can be slower than the one attained on the functions from the domain.

Conducting this study implies the use of basic methods of infinite-dimensional
functional analysis and operator semigroups, as well as the use of Matlab
software under student license in order to perform numerical experiments and
visualise the results.

##  2 Preliminaries

###  2.1 $C_{0}$-semigroups and Chernoff’s theorem

This section is solely dedicated to the introduction of basic concepts that
will be used throughout the rest of the paper. During the paper we will work
with the system of equations given by

######  Definition 1.

Let $X$ be an infinite set, and $\mathcal{F}$ be a Banach space of all number-
valued functions on $X$. Moreover, assume having $dom(L)\subset\mathcal{F}$
and a closed linear operator $L\colon dom(L)\to\mathcal{F}$. Then the system
of equations

$\begin{cases}u^{\prime}_{t}(t,x)=Lu(t,x),\\\ u(0,x)=u_{0}(x),\end{cases}$ (1)

where $x\in X$, $u_{0}\in\mathcal{F}$, and $u(t,\cdot)\in\mathcal{F}$ for all
$t\geq 0$, is called a Cauchy problem for the evolution equation.

######  Remark.

In case of existence of the $C_{0}$-semigroup $(G(t))_{t\geq 0}$ with the
generator $(L,dom(L))$ the solution of (1) exists and is given by the equality
$u(t,x)=(e^{tL}u_{0})(x)$ for all $t\geq 0$ and $x\in X$.

######  Definition 2.

If $u_{0}\in dom(L)$ then $u(t,\cdot)\in dom(L)$ for all $t\geq 0$ and $u$ is
called a classical solution.

######  Remark.

For an arbitrary $u_{0}\in\mathcal{F}$ the solution of (1) exists only as a
solution of the corresponding integral equation
$u(t,\cdot)=L\int_{0}^{t}u(s,\cdot)ds+u_{0}$.

After defining a Cauchy problem, it appears necessary to provide a common
definition of a $C_{0}$-semigroup [5].

######  Definition 3.

Let $\mathcal{F}$ be a Banach space, and $\mathscr{L}(\mathcal{F})$ be a space
of all linear bounded operators in $\mathcal{F}$. Let a mapping
$V\colon[0,+\infty)\to\mathscr{L}(\mathcal{F})$ also be given, i.e., for a
fixed $t\geq 0$ we have $V(t)$ being a linear bounded operator. Then $V$ is
called a $C_{0}$-semigroup, or the same as strongly continuous one-parametric
semigroup of linear bounded operators, if the following conditions hold:

  1. 1.

$V(0)=I$, i.e., for all $\varphi\in\mathcal{F}$ we have $V(0)\varphi=\varphi$;

  2. 2.

under the action of $V$ the addition in $[0,+\infty)$ is mapped to the
composition in $\mathscr{L}(\mathcal{F})$, i.e., for all $t,s\geq 0$ we have
$V(t+s)=V(t)\circ V(s)$, where $\circ$ denotes the composition of operators
such that $(A\circ B)(\varphi)=A(B(\varphi))$ for all $\varphi\in\mathcal{F}$;

  3. 3.

$V$ is continuous in $\mathscr{L}(\mathcal{F})$ endowed with the strong
operator topology, i.e., for all $\varphi\in\mathcal{F}$ a mapping $t\mapsto
V(t)\varphi$ is continuous on $[0,+\infty)$.

The concept of a $C_{0}$-semigroup is closely related to the notion of its
generator [5]. So, we propose the following

######  Definition 4.

Let $(V(t))_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $\mathcal{F}$.
Then the linear operator $L$ defined as

$L\varphi=\lim_{t\to+0}\frac{V(t)\varphi-\varphi}{t}$ (2)

on a linear subspace

$dom(L)=\left\\{\varphi\in\mathcal{F}\;|\>\exists\lim_{t\to+0}\frac{V(t)\varphi-\varphi}{t}\right\\}\subset\mathcal{F}$
(3)

is called an infinitesimal generator of a $C_{0}$-semigroup $(V(t))_{t\geq
0}$. We also say that operator $L\colon dom(L)\to\mathcal{F}$ generates a
$C_{0}$-semigroup and denote $V(t)=e^{tL}$.

######  Definition 5.

Let $dom(L)\subset\mathcal{F}$, and $L\colon dom(L)\to\mathcal{F}$ be a linear
operator. Then $L$ is called

  * •

closed if its graph
$\Gamma_{L}=\\{(x,Lx)\in\mathcal{F}\times\mathcal{F}\;|\;x\in dom(L)\\}$ is a
closed subset in space $\mathcal{F}\times\mathcal{F}$ endowed with the norm
$\|\cdot\|_{\mathcal{F}\times\mathcal{F}}$ given by
$\|(x,y)\|_{\mathcal{F}\times\mathcal{F}}=\|x\|_{\mathcal{F}}+\|y\|_{\mathcal{F}}$
for all $x,y\in\mathcal{F}$;

  * •

closable (or admitting closure) if the closure of its graph
$\overline{\Gamma_{L}}$ in space $\mathcal{F}\times\mathcal{F}$ with the norm
$\|\cdot\|_{\mathcal{F}\times\mathcal{F}}$ is a graph of some operator
$(\overline{L},dom(\overline{L}))$, meaning
$\Gamma_{\overline{L}}=\overline{\Gamma_{L}}$.

######  Remark.

If operator $\overline{L}$ exists then it is a linear closed operator
extending operator $L$, i.e., $dom(L)\subset dom(\overline{L})$, and
$L|_{dom(L)}=\overline{L}|_{dom(L)}$.

The next set of definitions is provided in the wording of I.D. Remizov [14].

######  Definition 6.

Let $\mathcal{F}$ be a Banach space, and $\mathscr{L}(\mathcal{F})$ be a space
of all linear bounded operators in $\mathcal{F}$. Let a mapping
$G\colon[0,+\infty)\to\mathscr{L}(\mathcal{F})$, or the same as a family of
linear bounded operators $(G(t))_{t\geq 0}$ in $\mathcal{F}$, also be given.
Moreover, assume $dom(L)$ is dense in $\mathcal{F}$ and a linear operator
$L\colon dom(L)\to\mathcal{F}$ is closed. Then function $G$ is called Chernoff
tangent to operator $L$ if the following conditions are satisfied:

  1. (CT1).

function $G$ is strongly continuous, or the same as continuous in
$\mathscr{L}(\mathcal{F})$ endowed with the strong operator topology, i.e.,
the mapping $t\mapsto G(t)f$ is continuous on $[0,+\infty)$ for all
$f\in\mathcal{F}$;

  2. (CT2).

$G(0)=I$, i.e., for all $f\in\mathcal{F}$ we have $G(0)f=f$;

  3. (CT3).

there exists a dense linear subspace $D\subset\mathcal{F}$ such that for all
$f\in D$ we have

$\lim_{t\to+0}(G(t)\varphi-\varphi)/t\overset{denote}{=}G^{\prime}(0)f\in\mathcal{F};$

  4. (CT4).

the closure of operator $(G^{\prime}(0),D)$ exists and is equal to
$(L,dom(L))$.

######  Remark.

Conditions (CT3) and (CT4) together mean that there exists a core $D$ for $L$.

Classical wording of the next theorem can be found in [3, 5]. Based on the
notion of Chernoff tangency we provide

######  Theorem 1 (The Chernoff theorem).

Let $\mathcal{F}$ be a Banach space, and $\mathscr{L}(\mathcal{F})$ be a space
of all linear bounded operators in $\mathcal{F}$. Let a mapping
$G\colon[0,+\infty)\to\mathscr{L}(\mathcal{F})$, or the same as a family of
linear bounded operators $(G(t))_{t\geq 0}$ in $\mathcal{F}$, also be given.
Moreover, assume having a closed linear operator $L\colon
dom(L)\subset\mathcal{F}\to\mathcal{F}$. Suppose that the following conditions
are met:

  * (E).

there exists a $C_{0}$-semigroup $(e^{tL})_{t\geq 0}$ with generator
$(L,dom(L))$;

  * (CT).

function $G$ is Chernoff tangent to operator $L$;

  * (N).

there exists number $\omega\in\mathbb{R}$ such that $\|G(t)\|\leq e^{\omega
t}$ for all $t\geq 0$.

Then for each $f\in\mathcal{F}$ we have
$(G(t/n))^{n}f\xrightarrow{n\to\infty}e^{tL}f$ uniformly in $t\in[0,T]$ for
any fixed $T\geq 0$, i.e., for each $f\in\mathcal{F}$ and each $T\geq 0$ we
get

$\lim_{n\to\infty}\sup_{t\in[0,T]}\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}f-e^{tL}f\right\|=0.$
(4)

######  Definition 7.

If the above conditions hold then $G$ is called a Chernoff function for $L$.

###  2.2 Conjectures on convergence speed for Chernoff approximations

During the analysis of the rate of convergence both for the transport and heat
equations, our aim is to check the validity of the following conjectures for
several model examples of initial conditions.

######  Conjecture 1 (I. Remizov, 2018).

Let $(e^{tL})_{t\geq 0}$ be a $C_{0}$-semigroup in a Banach space
$\mathcal{F}$ with generator $(L,dom(L))$, and $G$ be a Chernoff function for
operator $L$. Moreover, assume $t_{0}\geq 0$ and $f\in\mathcal{F}$ are given,
and suppose that for all $t\in[0,t_{0}]$ we have $f\in dom(G^{\prime}(t))\cap
dom(G^{\prime\prime}(t))$ and functions $t\to G^{\prime}(t)f$ and $t\to
G^{\prime\prime}(t)f$ are continuous. Then there exists a number $C\geq 0$
such that for all $t\in[0,t_{0})$ and all $n\in\mathbb{N}$ the estimate
$\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}f-e^{tL}f\right\|\leq\frac{C}{n}$
holds.

######  Conjecture 2 (I. Remizov, 2018).

Let $(e^{tL})_{t\geq 0}$ be a $C_{0}$-semigroup in a Banach space
$\mathcal{F}$ with generator $(L,dom(L))$, and $G$ be a Chernoff function for
operator $L$. Moreover, assume number $t_{0}\geq 0$ and $f\in\mathcal{F}$ are
given, and suppose that for all $t\in[0,t_{0}]$ we have function $f$ being
from the intersection of the domains of $G^{\prime}(t)$,
$G^{\prime\prime}(t)$, $G^{\prime\prime\prime}(t)$,
$G^{\prime\prime\prime\prime}(t)$, $G^{\prime}(t)G^{\prime\prime}(t)$,
$G^{\prime}(t)^{2}G^{\prime\prime}(t)$, $G^{\prime\prime}(t)^{2}$, having that
each of these operators is continuous for all $t\in[0,t_{0}]$. Then there
exists a number $C\geq 0$ such that for all $t\in[0,t_{0})$ and all
$n\in\mathbb{N}$ the following inequality is valid:

$\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}f-e^{tL}f+\frac{t^{2}}{2n}e^{tL}(L^{2}-G^{\prime\prime}(0))f\right\|\leq\frac{C}{n^{2}}.$

For the further convenience of the interpretation of numerical studies we
provide the definition of an approximation subspace proposed in [20].

######  Definition 8.

Let $\tau\subset[0,+\infty)=\mathbb{R}^{+}$, and map
$\psi\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ be such that
$\lim_{x\to+\infty}\psi(x)=0$. Then the set
$A^{\tau}_{\psi}=\\{f\in\mathcal{F}\;|\;\sup_{t\in\tau}\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}f-e^{tL}f\right\|=O(\psi(n))\;\mbox{as}\;n\to\infty\\}$
is called an approximation subspace of order $\psi$.

######  Remark.

The approximation subspace $A^{\tau}_{\psi}$ is indeed a linear subspace of
$\mathcal{F}$.

Hence, we now state the previous Conjecture 2 in the new wording in

######  Conjecture 3.

Let $(e^{tL})_{t\geq 0}$ be a $C_{0}$-semigroup in a Banach space
$\mathcal{F}$ with generator $(L,dom(L))$, and $G$ be a Chernoff function for
operator $L$. More than that, assume number $t_{0}\geq 0$ is given, and denote
the intersection of the domains of $G^{\prime}(t)$, $G^{\prime\prime}(t)$,
$G^{\prime\prime\prime}(t)$, $G^{\prime\prime\prime\prime}(t)$,
$G^{\prime}(t)G^{\prime\prime}(t)$, $G^{\prime}(t)^{2}G^{\prime\prime}(t)$,
$G^{\prime\prime}(t)^{2}$ for all $t\in[0,t_{0}]$ as $D(t_{0})$. Assume that
$D(t_{0})$ is dense in $\mathcal{F}$ and each of these operators are
continuous in $t$ on all vectors from $D(t_{0})$ for each $t\in[0,t_{0}]$.
Then function $G$ has a dense approximation subspace $A^{[0,t_{0})}_{\zeta}$
of order $\zeta(n)=\frac{1}{n^{2}}$, and this subspace is a subset of
$D(t_{0})$.

Despite the fact that only fast converging approximations have a practical
value for the research as a whole, it is worth mentioning that Chernoff
functions can be likewise constructed in a way to provide an arbitrary slow
convergence rate. This result is presented in the next

######  Proposition 2 (I. Remizov et al., 2019).

The convergence speed in the Chernoff theorem can be arbitrary slow. That is,
if a function $w\colon[0,+\infty]\to[0,+\infty]$ such that
$\lim_{x\to+\infty}w(x)=0$ is given then there exist such
$X,\mathcal{F},L,e^{tL},G,f$ for which the equality
$w(n/t)=O(\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}f-e^{tL}f\right\|)$
holds for all $t\geq 0$ when $n\to\infty$.

##  3 Exact formulas and estimates for convergence speed

During this section we will propose the results on the approximation speed for
some model initial conditions and both one-dimensional transport and heat
equations.

###  3.1 Transport equation

As a preliminary we will commence with the notion of the translation semigroup
that gives the solution for one-dimensional transport equation. The examined
system of equations is stated in

######  Definition 9.

The system of equations defined as

$\begin{cases}u^{\prime}_{t}(t,x)=u^{\prime}_{x}(t,x),\\\
u(0,x)=u_{0}(x),\end{cases}$ (5)

where $x\in\mathbb{R}$, $u_{0}\in UC_{b}(\mathbb{R})$ for all $t\geq 0$, is
called a Cauchy problem for the first order partial differential equation on
the real line, also referred to as the transport equation.

######  Remark.

The solution of the system (5) and, hence, the corresponding $C_{0}$-semigroup
is given by the formula $u(t,x)=(e^{tL}u_{0})(x)=u_{0}(x+t)$. This
$C_{0}$-semigroup is called the translation semigroup.

First, let us verify that the generator of the translation semigroup is indeed
the differentiation operator.

######  Theorem 3.

Let $(G(t))_{t\geq 0}$ be the $C_{0}$-semigroup given by the formula
$(G(t)f)(x)=f(x+t)$ for $f\in UC_{b}(\mathbb{R})$ and $t\geq 0$. Then its
generator is differentiation operator $L=\frac{d}{dx}$, and its domain
$dom(L)$ is

$UC^{1}_{b}(\mathbb{R})\overset{denote}{=}\left\\{f\in
UC_{b}(\mathbb{R})\;|\;f^{\prime}\in UC_{b}(\mathbb{R})\right\\}.$

###### Proof.

Using Definition 4 of the generator of a $C_{0}$-semigroup we get

$dom(L)=\left\\{f\in
UC_{b}(\mathbb{R})\;|\;\lim_{t\to+0}\left\|\frac{G(t)f-f}{t}-Lf\right\|=0\right\\},$
(6)

where $\|\cdot\|$ denotes the standard supremum norm in $UC_{b}(\mathbb{R})$.
Applying the definition of the $C_{0}$-semigroup, we obtain

$\lim_{t\to+0}\sup_{x\in\mathbb{R}}\left|\frac{f(x+t)-f(x)}{t}-f^{\prime}(x)\right|=0.$
(7)

Now, let us show that $f^{\prime}$ is uniformly continuous. Take a sequence
$(t_{n})_{n\in\mathbb{N}}$ such that $\lim_{n\to\infty}t_{n}=0$ and $t_{n}\geq
0$ for all $n\in\mathbb{N}$. Then it follows that
$\frac{f(x+t_{n})-f(x)}{t_{n}}\xrightarrow{n\to\infty}f^{\prime}(x)$ uniformly
in $UC_{b}(\mathbb{R})$ norm for some $x\in\mathbb{R}$. Since functions given
by the formula $\frac{f(x+t_{n})-f(x)}{t_{n}}$ are uniformly continuous for
all $n\in\mathbb{N}$, by Weierstrass theorem $f^{\prime}$ is also uniformly
continuous.

Next, we will show that $f^{\prime}$ is bounded. By the definition of uniform
continuity we have that

$\forall\varepsilon>0\;\exists\,N=N(\varepsilon)>0\>\colon\forall n\geq
N\;\forall
x\in\mathbb{R}\implies\sup_{x\in\mathbb{R}}\left|\frac{f(x+t_{n})-f(x)}{t_{n}}-f^{\prime}(x)\right|<\varepsilon.$

So we get
$\left|f^{\prime}(x)\right|<\left|\frac{f(x+t_{n})-f(x)}{t_{n}}\right|+\varepsilon$.
Take $\varepsilon_{0}=1$ then $f^{\prime}$ is bounded.

Finally, let us show that if $f^{\prime}\in UC_{b}(\mathbb{R})$ then
$\lim_{t\to-0}\sup_{x\in\mathbb{R}}\left|\frac{f(x+t)-f(x)}{t}-f^{\prime}(x)\right|=0$.
For all $t<0$ we have

$\begin{split}\left|\frac{f(x+t)-f(x)}{t}-f^{\prime}(x)\right|&=\left|\frac{f(x-|t|)-f(x)}{-|t|}-f^{\prime}(x)\right|\\\
&=\left|\frac{f(x)-f(x-|t|)}{|t|}-f^{\prime}(x)\right|\\\
&\leq\left|\frac{f((x-|t|)+|t|)-f(x-|t|)}{|t|}-f^{\prime}(x-|t|)\right|+\left|f^{\prime}(x-|t|)-f^{\prime}(x)\right|.\end{split}$

For small $|t|$ the former term is small due to (7), and the latter one due to
$f^{\prime}$ being uniformly continuous. So, we get that

$\lim_{t\to+0}\frac{f(x+t)-f(x)}{t}=f^{\prime}(x)=\lim_{t\to-0}\frac{f(x+t)-f(x)}{t}.$

Hence, if the limit given by (6) exists, meaning $f\in dom(L)$, then $f\in
UC^{1}_{b}(\mathbb{R})$. On the other hand, if $f\in UC^{1}_{b}(\mathbb{R})$
then by mean value theorem

$\left|\frac{f(x+t_{n})-f(x)}{t_{n}}-f^{\prime}(x)\right|=\left|f^{\prime}(x+\theta)-f^{\prime}(x)\right|<\varepsilon\;\mbox{for}\;|t|<\delta\;\mbox{and}\;0<\theta<t_{n},$

i.e., $f\in dom(L)$. So, we proved that $L$ is generator of this semigroup and
$dom(L)=UC^{1}_{b}(\mathbb{R})$. ∎

Prior to the numerical studies we will find the composition degree of some
Chernoff functions $G$ for an arbitrary initial condition $u_{0}\in
UC_{b}(\mathbb{R})$.

######  Proposition 4.

Let $G$ be the Chernoff function such that for all $f\in UC_{b}(\mathbb{R})$
we have

$(G(t)f)(x)=f(x+t+at^{k+1})$ (8)

for some fixed $a,k>0$. Then its $n$-th composition degree is given by the
formula

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=u_{0}\left(x+t+\frac{at^{k+1}}{n^{k}}\right).$
(9)

###### Proof.

By the definition of Chernoff function we get

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=u_{0}\left(x+n\left(\frac{t}{n}+a\left(\frac{t}{n}\right)^{k+1}\right)\right)=u_{0}\left(x+t+\frac{at^{k+1}}{n^{k}}\right)$

for all $x\in\mathbb{R}$ and $t\geq 0$. ∎

#### Analysis of convergence speed for $u_{0}=[x\to\sin(x)]$

Now, let us check the convergence speed for $u_{0}=[x\mapsto\sin(x)]$ and
Chernoff function (8), the prototype of which was proposed in [20]. The main
result of this paragraph is presented in the following

######  Theorem 5.

Let $(e^{tL})_{t\geq 0}$ be the translation semigroup on the real line, and
$G$ be the Chernoff function (8) for some fixed $a,k>0$. Then function
$u_{0}=[x\mapsto\sin(x)]$ belongs to the approximation subspace of order
$\frac{1}{n^{k}}$, and the error is given by the formula

$\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}-e^{tL}u_{0}\right\|=2\left|\sin(at^{k+1}/2n^{k})\right|.$

###### Proof.

Denote
$\xi(n)=\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}-e^{tL}u_{0}\right\|$
and $\tau=at^{k+1}/n^{k}$. Applying the definition of a supremum norm in
$UC_{b}(\mathbb{R})$ and using (4), we get

$\begin{split}\xi(n)&=\sup_{x\in\mathbb{R}}\left|\sin(x+t+\tau)-\sin(x+t)\right|\\\
&=\sup_{x\in\mathbb{R}}\left|(\sin(x)\cos(t+\tau)+\cos(x)\sin(t+\tau))-(\sin(x)\cos(t)+\cos(x)\sin(t))\right|\\\
&=\sup_{x\in\mathbb{R}}\left|(\cos(t+\tau)-\cos(t))\sin(x)+(\sin(t+\tau)-\sin(t))\cos(x)\right|.\\\
\end{split}$

The above supremum can be found based on the following assumption: as $\sin$
and $\cos$ are bounded functions then the value of the supremum is reached in
an extremum of the internal function. Let us now find an extreme point. We
have that

$(\cos(t+\tau)-\cos(t))\cos(x)-(\sin(t+\tau)-\sin(t))\sin(x)=0,$

and

$\tan(x)=\frac{\cos(t+\tau)-\cos(t)}{\sin(t+\tau)-\sin(t)}=\frac{-2\sin(t+\tau)\sin(\tau)}{2\cos(t+\tau)\sin(\tau)}=\tan(-(t+\tau)),$

thus meaning that the extremum is reached at the point $x=-(t+\tau)$
regardless of the period of $\tan$. So, we get

$\begin{split}\xi(n)&=\left|(\sin(t+\tau)-\sin(t))\cos(t+\tau)-(\cos(t+\tau)-\cos(t))\sin(t+\tau)\right|\\\
&=\left|2\cos(t+\tau)\cos(t+\tau)\sin(\tau)+2\sin(t+\tau)\sin(t+\tau)\sin(\tau)\right|\\\
&=2\left|(\cos^{2}(t+\tau)+\sin^{2}(t+\tau))\sin(\tau)\right|\\\
&=2\left|\sin(\tau)\right|.\\\ \end{split}$

Now, using Taylor expansion $\sin(z)=z-\frac{z^{3}}{3!}+O(z^{5})$ when $z\to
0$, we finally get

$\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}-e^{tL}u_{0}\right\|=2\sin(\tau)=\frac{at^{k+1}}{n^{k}}+O\left(\frac{1}{n^{k+1}}\right)\;\mbox{when}\;n\to\infty,$

which proves that $u_{0}=[x\mapsto\sin(x)]$ belongs to an approximation
subspace of order $\frac{1}{n^{k}}$. ∎

#### Analysis of slow converging approximations for $u_{0}=[x\to\sin(x)]$

At last, we will generalize Theorem 5 for an arbitrary function
$w\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$.

######  Theorem 6.

Let $(e^{tL})_{t\geq 0}$ be the translation semigroup on the real line, and
$G$ be the Chernoff function such that for all $f\in UC_{b}(\mathbb{R})$ we
have

$(G(t)f)(x)=f(x+t+t\cdot w(1/t)),$ (10)

where $w\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
$\lim_{n\to\infty}w(n)=0$. Then function $u_{0}=[x\mapsto\sin(x)]$ belongs to
the approximation subspace of order $w$, and the error is given by the formula

$\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}-e^{tL}u_{0}\right\|=2\left|\sin(t/2\cdot
w(n/t))\right|.$

###### Proof.

Similarly to Proposition 4 we first obtain $n$-th composition degree for
Chernoff function $G$. We get

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=u_{0}\left(x+n\left(\frac{t}{n}+\frac{t}{n}\cdot
w(n/t)\right)\right)=u_{0}\left(x+t+w(n/t)t\right).$ (11)

It now suffices to set $\tau=t\cdot w(n/t)$ and use the same technique as of
Theorem 5. Hence, since $w(n)\xrightarrow{n\to\infty}0$, we have

$\xi(n)=2\left|\sin(t/2\cdot w(n/t))\right|=\left|t\cdot
w(n/t)+o\left(w^{2}(n)\right)\right|\;\mbox{as}\;n\to\infty,$

which proves that $u_{0}=[x\mapsto\sin(x)]$ belongs to an approximation
subspace of order $w$. ∎

Later in Section 4 we will check the convergence speed attained for the slow
converging Chernoff function $G$ given by (10) with
$w(t)=\frac{1}{t^{\gamma}}$ for different values of $0<\gamma<1$ applied to
approximate the solution for $u_{0}=[x\mapsto\sin(x)]$.

###  3.2 Heat equation

The next differential equation we are going to analyse is the heat equation.
Now, we will state the corresponding Cauchy problem and give an explicit
formula for the solution in a form of a $C_{0}$-semigroup.

######  Definition 10.

The system of equations defined the following way

$\begin{cases}u^{\prime}_{t}(t,x)=Lu(t,x)=a^{2}u^{\prime\prime}_{xx}(t,x),\\\
u(0,x)=u_{0}(x),\end{cases}$ (12)

where $a>0$, $x\in\mathbb{R}$, and $u_{0}\in UC_{b}(\mathbb{R})$ for all
$t\geq 0$, is called a Cauchy problem for the heat equation on the real line
with constant coefficient of thermal conductivity.

######  Remark.

The solution of the system (12), and, hence, the corresponding
$C_{0}$-semigroup, is given by the Poisson integral

$u(t,x)=(e^{tL}u_{0})(x)=\int_{\mathbb{R}}\Phi(t,x-y)u_{0}(y)dy,$ (13)

where

$\Phi(t,x)=\frac{1}{2a\sqrt{\pi t}}\exp\left(-\frac{x^{2}}{4a^{2}t}\right).$
(14)

We will analyse the performance of approximations for the following three
Chernoff functions:

$\left(G(t)u_{0}\right)(x)=\frac{1}{4}u_{0}(x+2a\sqrt{t})+\frac{1}{4}u_{0}(x-2a\sqrt{t})+\frac{1}{2}u_{0}(x)$
(15)

proposed by I. Remizov in [14],

$(G(t)u_{0})(x)=\frac{1}{6}u_{0}(x+a\sqrt{6t})+\frac{1}{6}u_{0}(x-a\sqrt{6t})+\frac{2}{3}u_{0}(x)$
(16)

proposed by A. Vedenin in [20], and

$(G(t)u_{0})(x)=\frac{1}{30}u_{0}(x+a\sqrt{12t})+\frac{1}{30}u_{0}(x-a\sqrt{12t})+\frac{3}{10}u_{0}(x+a\sqrt{2t})+\frac{3}{10}u_{0}(x-a\sqrt{2t})+\frac{1}{3}u_{0}(x)$
(17)

also proposed by A. Vedenin in oral communication to the author of the present
paper in 2020. To simplify the complexity of computational algorithms one
might need a general formula for the $n$-th composition degree for the above
Chernoff functions. So we present the following

######  Proposition 7.

Let $G$ be the Chernoff function given by (15) for all $f\in
UC_{b}(\mathbb{R})$ and fixed $a>0$. Then its $n$-th composition degree is
given by the formula

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=\frac{1}{4^{n}}\sum^{n}_{p=-n}\alpha_{p,n}u_{0}(x+2ap\sqrt{t/n}),$
(18)

where for each $n\in\mathbb{N}$ and $p\in\\{-n,\dots,n\\}$ coefficients
$\alpha_{p,n}$ are defined as

$\alpha_{p,n}=\sum^{\left[\frac{n-|p|}{2}\right]}_{k=0}C^{n}_{k}C^{k+|p|}_{n-k}2^{n-|p|-2k}.$

###### Proof.

Consider $G(t)=\frac{1}{4}(S_{2a\sqrt{t}}+S_{-2a\sqrt{t}}+2I)$ for all $t>0$,
where $I$ is an identity operator and $S_{\tau}$ is a shift operator given by
$(S_{\tau}f)(x)=f(x+\tau)$ for some $\tau\in\mathbb{R}$. It then yields that

$\begin{split}G(t)^{n}_{t/n}&=\frac{1}{4^{n}}(S_{2a\sqrt{t/n}}+S_{-2a\sqrt{t/n}}+2I)^{n}\\\
&=\frac{1}{4^{n}}\sum_{0\leq k+m\leq
n}S^{k}_{2a\sqrt{t/n}}S^{m}_{-2a\sqrt{t/n}}2^{n-k-m}\frac{n!}{k!m!(n-k-m)!}\\\
&=\frac{1}{4^{n}}\sum_{0\leq k+m\leq
n}S_{2a(k-m)\sqrt{t/n}}2^{n-k-m}\frac{n!}{k!m!(n-k-m)!}.\end{split}$

Now, set $k-m=p$ with $-n\leq p\leq n$. We obtain

$\begin{split}G(t)^{n}_{t/n}&=\frac{1}{4^{n}}\sum^{n}_{p=-n}\sum^{\left[\frac{n-|p|}{2}\right]}_{k=0}C^{k}_{n}C^{k+|p|}_{n-k}2^{n-|p|-2k}S_{2ap\sqrt{t/n}}\\\
&=\frac{1}{4^{n}}\sum^{n}_{p=-n}\alpha_{p,n}S_{2ap\sqrt{t/n}},\end{split}$

where
$\alpha_{p,n}=\sum^{\left[\frac{n-|p|}{2}\right]}_{k=0}C^{n}_{k}C^{k+|p|}_{n-k}2^{n-|p|-2k}$.
∎

Similarly, one may derive the next

######  Proposition 8.

Let $G$ be the Chernoff function given by (16) for all $f\in
UC_{b}(\mathbb{R})$ and fixed $a>0$. Then its $n$-th composition degree is
given by the formula

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=\frac{1}{6^{n}}\sum^{n}_{p=-n}\beta_{p,n}u_{0}(x+ap\sqrt{6t/n}),$
(19)

where for each $n\in\mathbb{N}$ and $p\in\\{-n,\dots,n\\}$ coefficients
$\beta_{p,n}$ are defined as

$\beta_{p,n}=\sum^{\left[\frac{n-|p|}{2}\right]}_{k=0}C^{n}_{k}C^{k+|p|}_{n-k}4^{n-|p|-2k}.$

#### Analysis of convergence rate for the initial condition
$u_{0}(x)=[x\to\sin(x)]$

Let us now find the solution of the above Cauchy problem (12) for
$u_{0}=[x\mapsto\sin(x)]$.

######  Theorem 9.

Suppose $u_{0}=[x\mapsto\sin(x)]$. Then the solution of the system (12) with
initial condition $u_{0}$ is given by the formula
$u(t,x)=(e^{tL}u_{0})(x)=e^{-a^{2}t}\sin(x)$.

###### Proof.

Using (13) and substituting $u_{0}=[x\mapsto\sin(x)]$, we get

$u(t,x)=(e^{tL}u_{0})(x)=\int_{\mathbb{R}}\frac{1}{2a\sqrt{\pi
t}}\exp\left(-\frac{(x-y)^{2}}{4a^{2}t}\right)\sin(y)dy.$

Now, let us apply the following identity $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$.
So, we obtain

$u(t,x)=\frac{1}{2i}\frac{1}{2a\sqrt{\pi
t}}\left(\int_{\mathbb{R}}\exp\left(-\frac{(x-y)^{2}}{4a^{2}t}\right)\exp(iy)dy-\int_{\mathbb{R}}\exp\left(-\frac{(x-y)^{2}}{4a^{2}t}\right)\exp(-iy)dy\right).\\\
$

The former integral now takes the form

$\begin{split}\int_{\mathbb{R}}\frac{1}{2a\sqrt{\pi
t}}\exp\left(-\frac{(y-(x+2a^{2}ti))^{2}-(x+2a^{2}ti)^{2}+x^{2}}{4a^{2}t}\right)dy&=\exp\left(-\frac{-(x+2a^{2}ti)^{2}+x^{2}}{4a^{2}t}\right)\\\
&=\exp\left(ix-a^{2}t\right).\end{split}$

Similarly, the latter integral is equal to $\exp\left(-ix-a^{2}t\right)$,
i.e., the solution can now be expressed as

$u(t,x)=(e^{tL}u_{0})(x)=e^{-a^{2}t}\left(\frac{e^{ix}-e^{-ix}}{2i}\right)=e^{-a^{2}t}\sin(x).$
(20)

So, we found the solution of the Cauchy problem given by (12) for
$u_{0}=[x\mapsto\sin(x)]$. ∎

Prior to analysing the convergence rate for Chernoff function (15) and
$u_{0}=[x\to\sin(x)]$ let us compute its $n$-th composition degree.

######  Proposition 10.

Let $G$ be the Chernoff function given by (15) for all $f\in
UC_{b}(\mathbb{R})$ and fixed $a>0$. Then its $n$-th composition degree for
$u_{0}=[x\mapsto\sin(x)]$ has the form

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=\left(\cos\left(a\sqrt{\frac{t}{n}}\right)\right)^{2n}\sin(x).$
(21)

###### Proof.

Substituting the initial condition into (15), we get

$\begin{split}\left(G(t)u_{0}\right)(x)&=\frac{1}{4}\sin(x+2a\sqrt{t})+\frac{1}{4}\sin(x-2a\sqrt{t})+\frac{1}{2}\sin(x)\\\
&=\left(\frac{1+\cos(2a\sqrt{t})}{2}\right)\sin(x)\\\
&=\cos^{2}(a\sqrt{t})\sin(x).\end{split}$

So, for all $t\geq 0$ operator $G(t)$ is a function multiplication operator,
meaning its $n$-th composition degree has the following form

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=\left(\cos\left(a\sqrt{\frac{t}{n}}\right)\right)^{2n}u_{0}(x)=\left(\cos\left(a\sqrt{\frac{t}{n}}\right)\right)^{2n}\sin(x)$

for all $t\geq 0$ and $x\in\mathbb{R}$. ∎

Finally, let us find the convergence speed for $u_{0}=[x\mapsto\sin(x)]$ and
the above derived Chernoff function (21). The main result of this subsection
is presented in the next

######  Theorem 11.

Let $(e^{tL})_{t\geq 0}$ be the heat $C_{0}$-semigroup, and $G$ be the
Chernoff function given by (15) for all $f\in UC_{b}(\mathbb{R})$ and fixed
$a>0$. Then function $u_{0}=[x\mapsto\sin(x)]$ belongs to the approximation
subspace of order $\frac{1}{n}$.

###### Proof.

Applying the definition of a standard norm in $UC_{b}(\mathbb{R})$, we get

$\begin{split}\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}-e^{tL}u_{0}\right\|&=\sup_{x\in\mathbb{R}}\left|\left(\cos\left(a\sqrt{\frac{t}{n}}\right)\right)^{2n}\sin(x)-e^{-a^{2}t}\sin(x)\right|\\\
&=\left|\left(\cos\left(a\sqrt{\frac{t}{n}}\right)\right)^{2n}-e^{-a^{2}t}\right|\sup_{x\in\mathbb{R}}\left|\sin(x)\right|.\end{split}$

Now, let us exploit the following Taylor expansions:

$\cos(z)=1-\frac{z^{2}}{2!}+O(z^{4}),$

and

$\ln(1+z)^{\alpha}=z-\frac{z^{2}}{2!}+O(z^{4}),$

and

$\exp(z)=1+z+\frac{z^{2}}{2!}+O(z^{3})$

when $z\to 0$. So, we have

$\begin{split}\left\|\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}-e^{tL}u_{0}\right\|&=\left|e^{2n\ln\left(\cos\left(a\sqrt{\frac{t}{n}}\right)\right)}-e^{-a^{2}t}\right|\\\
&=\left|-\frac{a^{4}t^{2}}{6n}+O\left(\frac{1}{n}\right)\right|,\;\mbox{when}\;n\to\infty.\end{split}$

This proves that $u_{0}=[x\mapsto\sin(x)]$ belongs to the approximation
subspace of order $\frac{1}{n}$. ∎

Similarly, applying the same technique as of Theorem 11 one may propose the
next series of results containing the estimates of the convergence rate for
Chernoff functions (16) and (17), starting with

######  Proposition 12.

Let $G$ be the Chernoff function given by (16) (respectively (17)) for all
$f\in UC_{b}(\mathbb{R})$ and fixed $a>0$. Then its $n$-th composition degree
for $u_{0}=[x\mapsto\sin(x)]$ is given by

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=\frac{1}{3^{n}}\left(2+\cos\left(a\sqrt{\frac{6t}{n}}\right)\right)^{n}\sin(x)$
(22)

respectively

$\left(\left(G\left(\frac{t}{n}\right)\right)^{n}u_{0}\right)(x)=\frac{1}{15^{n}}\left(5+\cos\left(a\sqrt{\frac{12t}{n}}\right)+9\cos\left(a\sqrt{\frac{2t}{n}}\right)\right)^{n}\sin(x).$
(23)

######  Theorem 13.

Let $(e^{tL})_{t\geq 0}$ be the heat $C_{0}$-semigroup, and $G$ be the
Chernoff function given by (16) (respectively (17)) for all $f\in
UC_{b}(\mathbb{R})$ and fixed $a>0$. Then function $u_{0}=[x\mapsto\sin(x)]$
belongs to the approximation subspace of order $\frac{1}{n^{2}}$ and
$\frac{1}{n^{3}}$ respectively.

#### Finding the exact solution for the initial condition
$u_{0}=[x\to\exp(-|x|)]$

Here we will find the solution of the Cauchy problem given by (12) for
$u_{0}=[x\mapsto e^{-|x|}]$.

######  Theorem 14.

Suppose $u_{0}=[x\mapsto e^{-|x|}]$. Then the solution of the system (12) with
initial condition $u_{0}$ is given by the formula

$u(t,x)=(e^{tL}u_{0})(x)=e^{t-x}\left(1-\frac{1}{2}\operatorname{erfc}\left(\frac{x}{2\sqrt{t}}-\sqrt{t}\right)\right)+e^{t+x}\frac{1}{2}\operatorname{erfc}\left(\frac{x}{2\sqrt{t}}+\sqrt{t}\right),$
(24)

where

$\operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}}\int^{+\infty}_{x}e^{-y^{2}}dy.$
(25)

###### Proof.

First, let us plug the formula giving the initial condition into the solution
defined in (13) and (14). We have

$\begin{split}u(t,x)&=\frac{1}{2\sqrt{\pi
t}}\int^{+\infty}_{-\infty}e^{-\frac{(x-y)^{2}}{4t}}u_{0}(y)dy\\\
&=\frac{1}{2\sqrt{\pi
t}}\int^{+\infty}_{-\infty}e^{-\frac{(x-y)^{2}}{4t}}e^{-|y|}dy.\end{split}$

Now, change the variables within the integral by denoting $x-y=2z\sqrt{t}$. It
then yields that $y=x-2z\sqrt{t}$ and $dy=2\sqrt{t}dz$. So,

$\begin{split}u(t,x)&=\frac{1}{2\sqrt{\pi
t}}\int^{+\infty}_{-\infty}e^{-z^{2}}e^{-|x-2z\sqrt{t}|}2\sqrt{t}dz\\\
&=\frac{1}{\sqrt{\pi}}\int^{\frac{x}{2\sqrt{t}}}_{-\infty}e^{-(z^{2}-2z\sqrt{t}+x)}dz+\frac{1}{\sqrt{\pi}}\int^{+\infty}_{\frac{x}{2\sqrt{t}}}e^{-(z^{2}+2z\sqrt{t}-x)}dz\\\
&=\frac{1}{\sqrt{\pi}}\int^{\frac{x}{2\sqrt{t}}}_{-\infty}e^{-(z-\sqrt{t})^{2}+t-x}dz+\frac{1}{\sqrt{\pi}}\int^{+\infty}_{\frac{x}{2\sqrt{t}}}e^{-(z+\sqrt{t})^{2}+t+x}dz.\end{split}$

Now, set $z-t=w$ and $z+t=w$ for the former respectively latter integral.
Hence, we obtain

$\begin{split}u(t,x)&=\frac{1}{\sqrt{\pi}}e^{t-x}\int^{\frac{x}{2\sqrt{t}}-\sqrt{t}}_{-\infty}e^{-w^{2}}dw+\frac{1}{\sqrt{\pi}}e^{t+x}\int^{+\infty}_{\frac{x}{2\sqrt{t}}+\sqrt{t}}e^{-w^{2}}dw\\\
&=e^{t-x}\left(1-\frac{1}{2}\frac{2}{\sqrt{\pi}}\int^{+\infty}_{\frac{x}{2\sqrt{t}}-\sqrt{t}}e^{-w^{2}}dw\right)+\frac{1}{2}\frac{2}{\sqrt{\pi}}e^{t+x}\int^{+\infty}_{\frac{x}{2\sqrt{t}}+\sqrt{t}}e^{-w^{2}}dw\\\
&=e^{t-x}\left(1-\frac{1}{2}\operatorname{erfc}\left(\frac{x}{2\sqrt{t}}-\sqrt{t}\right)\right)+e^{t+x}\frac{1}{2}\operatorname{erfc}\left(\frac{x}{2\sqrt{t}}+\sqrt{t}\right).\end{split}$

Besides, as $\operatorname{erfc}(x)\xrightarrow{x\rightarrow+\infty}0$, then
for each $t>0$ one has $u(t,x)\sim e^{t-x}$ as $x\to+\infty$ and, moreover,
$u(t,x)\sim e^{t-|x|}$ as $x\to\infty$. ∎

As analysing the convergence speed for the obtained solution is technically a
sophisticated task, we will present only the numerical estimates of the norm
decay later on. Even though it is not a rigorous proof, it gives hope that one
can obtain a formal reasoning.

##  4 Numerical experiments on model examples

This chapter is devoted to the results of numerical experiments performed for
the previously analysed initial conditions $u_{0}=[x\to\sin(x)]$ and
$u_{0}=[x\to\exp(-|x|)$, and different Cauchy problems (5) and (10).

###  4.1 Transport equation

#### Convergence speed for $u_{0}=[x\to\sin(x)]$

First, we will analyse the convergence speed for $u_{0}=[x\mapsto\sin(x)]$.

Figure 1: Graphs of initial condition and solution of the transport equation
($t=2$) Figure 2: Graphs of approximations to the solution of the transport
equation ($t=2$, $n=1$)

As the initial condition as well as the Chernoff approximations have period
equal to $2\pi$, the uniform norm in $UC_{b}(\mathbb{R})$ is reached on the
interval of this period. Note that the domain of the graphs presented below
varies significantly and is chosen in favor of visual clarity.

So, Figure 1 containes the graphs of the initial condition and the
corresponding solution for $t=2$. Whereas, Figures 2 and 3 represent some
examples of Chernoff approximations with respect to the initial condition
$u_{0}=[x\mapsto\sin(x)]$ and $t=2$ with composition degree $n=1$ and $n=5$
respectively.

Figure 3: Graphs of approximations to the solution of the transport equation
($t=2$, $n=5$) Figure 4: Graphs of estimates of the convergence speed ($t=1$)

Figure 4 represents the graphs of convergence rate, i.e., the decay in the
norm of the difference of the Chernoff approximations and the corresponding
solution depending on the growth of composition degree.

Figure 5: Graphs of estimates of the convergence speed in log-log scale
($t=1$)

Apart from that, the above series of figures illustrates the results of a non-
linear regression applied to analyse the order of approximation subspaces to
which the examined initial condition belongs to. So, Figures 4 and 5 using
both standard and log-log scale represent the convergence rate and the
corresponding obtained regression results for Chernoff functions from Theorem
5.

#### Convergence speed for $u_{0}=[x\to\exp(-|x|)]$

Figure 6: Graphs of initial condition and solution of the transport equation
($t=1$)

Second, we will analyse the convergence rate for $u_{0}=[x\mapsto e^{-|x|}]$.
The graphs of the initial condition and examined solution for $t=1$ are shown
in Figure 6.

Figure 7: Graphs of approximations to the solution of the transport equation
($t=2$, $n=1$) Figure 8: Graphs of approximations to the solution of the
transport equation ($t=2$, $n=5$)

Figures 7 and 8 represent some model examples of Chernoff approximations based
on the initial condition $u_{0}=[x\mapsto e^{-|x|}]$ for $t=2$ with
composition degree $n=1$ and $n=5$ respectively.

As the initial condition and the Chernoff approximations are decaying outside
of some segment, we assume that the value of the uniform norm of the
difference is reached on the interval of correctly selected length (e.g.,
$[-5,5]$). So, Figure 9 provides a graph of convergence rate, i.e., the decay
of the norm of the difference depending on the growth of composition degree.

Figure 9: Graphs of estimates of the convergence speed ($t=1$) Figure 10:
Graphs of estimates of the convergence speed in log-log scale ($t=1$)

More than that, Figures 9 and 10, using both standard and log-log scale,
illustrate the results of a non-linear regression applied to analyse the
approximation subspaces to which the examined initial condition belongs to for
Chernoff functions from Theorem 5.

#### Numerical demonstration of slow convergence for $u_{0}=[x\to\sin(x)]$

Third, we will analyse the convergence rate for $u_{0}=[x\mapsto\sin(x)]$ and
Chernoff function $G$ (10) given in Theorem 6 with $w(t)=\frac{1}{t^{\gamma}}$
for $\gamma=\frac{1}{2}$, $\gamma=\frac{1}{3}$, and $\gamma=\frac{1}{6}$.

Figure 11: Graphs of approximations to the solution of the transport equation
($t=2$, $n=1$) Figure 12: Graphs of approximations to the solution of the
transport equation ($t=2$, $n=5$)

Figures 11 and 12 demonstrate some common examples of Chernoff approximations
for the initial condition $u_{0}=[x\mapsto\sin(x)]$ and $t=2$ with composition
degree $n=1$ and $n=5$ respectively.

Here, we will again exploit the periodic nature of the solution and the
Chernoff approximations and suggest that the uniform norm is reached on the
properly selected interval. So, Figure 13 provides a graph of convergence rate
when $t=2$.

Apart from that, the below series of figures also illustrates the order of
approximation subspaces to which the examined initial condition belongs to.
Thus, Figures 13 and 14, using both standard and log-log scale, represent the
convergence rate and the corresponding obtained regression results for
Chernoff functions from Theorem 6 with $\gamma=\frac{1}{2}$,
$\gamma=\frac{1}{3}$, and $\gamma=\frac{1}{6}$.

Figure 13: Graphs of estimates of the convergence speed ($t=1$) Figure 14:
Graphs of estimates of the convergence speed with log-log scale ($t=1$)

One can clearly notice that, independent of the considered initial value of
the equation, all of the approximations performed as if the examined model
function belonged to the domain of generator, i.e., even
$u_{0}=[x\to\exp(-|x|)]\notin UC^{1}_{b}(\mathbb{R})$ belongs to the higher
order approximation subspaces. We will examine later in this chapter that for
the heat equation this is not the case already.

###  4.2 Heat equation

#### Convergence speed for $u_{0}=[x\to\sin(x)]$

Figure 15: Graphs of initial condition and solution of the heat equation
($t=1$) Figure 16: Graphs of approximations to the solution of the heat
equation ($t=2$, $n=1$)

Here, we will analyse the convergence speed of Chernoff approximations defined
by (15), (16), and (17) for $u_{0}=[x\mapsto\sin(x)]$ to the exact solution
given by (20).

The graphs of the initial condition and investigated solution for $t=1$ are
shown in Figure 15.

Figure 17: Graphs of approximations to the solution of the heat equation
($t=2$, $n=2$)

Whereas, Figures 16 and 17 represent some model examples of Chernoff
approximations for the initial condition $u_{0}=[x\mapsto\sin(x)]$ and $t=2$
with composition degree $n=1$ and $n=2$ respectively.

Exploiting one more time the periodic nature of the initial condition and the
Chernoff approximations, we state that the uniform norm in
$UC_{b}(\mathbb{R})$ is reached on the interval of the proper period, e.g.,
$[0,2\pi]$.

Figure 18: Graphs of estimates of the convergence speed ($t=2$) Figure 19:
Graphs of estimates of the convergence speed with log-log scale ($t=2$)

Figures 18 and 19, using both standard and logarithmic scale, represent the
convergence rate and the corresponding obtained regression results for
Chernoff functions from Theorem 11 and Proposition 13 for $t=2$. Hence, one we
can state that the results of numerical experiments confirmed the estimates
obtained in Section 3.

#### Convergence speed for $u_{0}=[x\to\exp(-|x|)]$

Figure 20: Graphs of initial condition and solution of the heat equation
($t=1$)

Here, we will analyse the convergence rate for the initial condition
$u_{0}=[x\mapsto e^{-|x|}]$. The graphs of the initial condition and solution
given by the $C_{0}$-semigroup for $t=1$ are shown in Figure 20.

Figure 21: Graphs of approximations to the solution of the heat equation
($t=1$, $n=1$) Figure 22: Graphs of approximations to the solution of the heat
equation ($t=1$, $n=5$)

Figures 21 and 22 demonstrate two model examples of Chernoff approximations
for the initial condition $u_{0}=[x\mapsto e^{-|x|}]$ and $t=1$ with
composition degree $n=1$ and $n=5$ respectively.

Figure 23: Graphs of estimates of the convergence speed ($t=1$)

Let us once again refer to the decaying property of the initial condition and
the Chernoff approximations. We propose that the standard norm in
$UC_{b}(\mathbb{R})$ is reached on the appropriately defined interval (e.g.,
$[-5,5]$). So, Figure 23 contains two graphs of convergence rate for $t=1$.

Figure 24: Graphs of estimates of the convergence speed with log-log scale
($t=1$)

The above series of figures illustrates the results of a non-linear regression
applied to analyse the approximation subspaces to which the examined initial
condition belongs to. So, Figures 23 and 24, using both standard and
logarithmic scale, represent the convergence rate and the corresponding
obtained regression results for the heat equation.

One can easily derive now that the behaviour of the Chernoff approximations
depends on the choice of initial condition, e.g., for $u_{0}=[x\to\exp(-|x|)]$
we have it belonging to the approximation subspaces of the same order
$\frac{1}{n}$ when applying expressions via different Chernoff functions (15),
(16), and (17).

Acknowledgements. The author would like to express deep gratitude to his
scientific supervisor I.D. Remizov for the problem setting and attention to
the research, and to O.E. Galkin, and other members of research group
’Evolution semigroups and their applications’ for useful critiques. The
publication was prepared within the framework of the Academic Fund Program at
HSE University in 2020–2021 (grant No.20-04-022, project Evolution semigroups
and their applications) and within the framework of the Russian Academic
Excellence Project ’5-100’.

## References

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##  Appendix A Code for numerical experiments

https://gitlab.com/tervenar/matlab


# Random Sampling of Mellin Band-limited Signals

Shivam Bajpeyi Dhiraj Patel S. Sivananthan
Department of Mathematics, Indian Institute of Technology Delhi,
New Delhi-110016, India Email id<EMAIL_ADDRESS>id:
<EMAIL_ADDRESS>id<EMAIL_ADDRESS>
###### Abstract

In this paper, we address the random sampling problem for the class of Mellin
band-limited functions $\mathcal{B}_{T}$ which is concentrated on a bounded
cube. It is established that any function in $\mathcal{B}_{T}$ can be
approximated by an element in a finite-dimensional subspace of
$\mathcal{B}_{T}.$ Utilizing the notion of covering number and Bernstein’s
inequality to the sum of independent random variables, we prove that the
random sampling inequality holds with an overwhelming probability provided the
sampling size is large enough.

Keywords: Sampling Inequality; Random sampling; Mellin Transform; Mellin band-
limited functions; Reproducing kernel space
Mathematics Subject Classification(2020): 94A20, 41A05,42A61,42C15

##  1 Introduction

The theory of sampling and reconstruction has been receiving significant
attention from the broad areas of approximation theory, signal processing, and
digital communication. The problem of sampling for the function space $V$ is
mainly concerned with the stable recovery of a function $f\in V$ from its
sample values $\\{f(x_{j})\\}$ on some countable set $X=\\{x_{j}:j\in J\\}$,
without losing any information. The sampling set $X$ is known as a stable
sampling set. For the space of Mellin band-limited functions $\mathcal{B}_{T}$
(defined in Section 2), the sampling problem is equivalent to finding a
sampling set $\\{x_{j}:j\in J\\}$ such that the sampling inequality

$A\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2}\leq\sum_{j\in
J}|f(x_{j})|^{2}x_{j}^{2c}\leq B\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2},$
(1.1)

holds for all $f$ in $\mathcal{B}_{T}$, for some $A,B>0$. Here
$\mathbb{R}_{+}$ denotes the set of positive real numbers. The Banach space
$X_{c}^{2}(\mathbb{R}^{n}_{+})$ is a collection of all measurable function $f$
with

$\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2}=\int_{\mathbb{R}_{+}^{n}}|x|^{2c-1}|f(x)|^{2}\,dx<\infty.$

An important result in this direction was collectively given by Bertero and
Pike [14], and Gori [25] which asserts that any function $f$ in
$\mathcal{B}_{T}$ can be stably recovered from its sample at exponentially
spaced sample points $\\{e^{\frac{k}{T}}:k\in\mathbb{Z}\\}$ and the
reconstruction formula is given by

$f(x)=\sum_{k\in\mathbb{Z}}f(e^{\frac{k}{T}})\,\text{lin}_{\frac{c}{T}}(e^{-k}x^{T}),\qquad
x\in\mathbb{R}_{+},$ (1.2)

where $\text{lin}_{c}(x)=x^{-c}\text{sinc}\,(\log x)$ and
$\text{sinc}\,(u)=\frac{\sin\pi u}{\pi u}.$ In this case, the sampling set
$\\{e^{k/T}:k\in\mathbb{Z}\\}$ is a stable sample set for $\mathcal{B}_{T}$
and $A=B=T.$ The exponentially spaced sampling set is useful in the
application where the signal information is accumulated near zero. The
exponential sampling method is also useful to deal with various real-world
problems arising in science and engineering (see [19, 30]). Butzer and Jansche
[16] pioneered the mathematical study of the exponential sampling formula
(1.2) employing the tools of the theory of Mellin transform. Some recent
developments related to the exponential sampling problem can be observed in
[1, 3, 8, 9].

The signals are used to recover from their random measurements in the field of
image processing [20], compressed sensing [18, 23], and machine learning [21,
22]. The random sampling problem deals with the following question. What is
the probability that the uniformly distributed random sample points satisfy
the sampling inequality (1.1) for some class of functions in
$\mathcal{B}_{T}$? Due to the randomness of the samples, the sampling
inequality (1.1) may not hold surely for $\mathcal{B}_{T}$. So, the random
sampling problem estimates the following probability

$P\Big{(}A\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2}\leq\sum_{j\in
J}|f(x_{j})|^{2}x_{j}^{2c}\leq
B\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2}\Big{)}\geq 1-\epsilon,$

for some class of Mellin band-limited function, where $\epsilon>0$ is
arbitrary small.

In the case of finite dimensional space, the random sampling problem was
studied for the space of multivariate trigonometric polynomials [10, 27]. For
the infinite-dimensional space of Fourier band-limited functions, the stable
recovery from their random samples scattered throughout $\mathbb{R}^{n}$ was
shown to be impossible [11]. This motivated to consider the random samples
distributed on a bounded cube $C\subset\mathbb{R}^{n}$ and the class of
functions concentrated on $C$. It was established in [11, 12] that the class
of Fourier band-limited functions concentrated on $C$ can be stably recovered
from its samples at uniformly distributed random points on $C$. The random
sampling problem has been investigated for the shift-invariant spaces in [2,
24, 35]. In recent years, the related random sampling problem has been studied
on the sphere in [33], for the image space of an idempotent integral operator
[26, 31, 32] which generalizes the space of shift-invariant space and the
signal space of finite rate of innovation, see [29].

A Mellin band-limited function can not be Fourier band-limited at the same
time, and it is analytically extensible to the Riemann surface of the
(complex) logarithm (see [7]). Since one must appropriately extend the idea of
the Bernstein spaces, involving Riemann surfaces, the theory of Mellin band-
limited functions differs greatly from the theory of Fourier band-limited
functions. The main aim of the paper is to study the stability of the random
sampling set for the space Mellin bandlimited functions, using the Mellin
transform and its inversion theory. Motivated from [11], we consider the
problem of random sampling for the class of $\delta$-concentrated Mellin band-
limited functions on $C_{R}:=[1/R,R]^{n}\subset\mathbb{R}_{+}^{n}$, i.e.,

$\mathcal{B}_{T,\delta}:=\Big{\\{}f\in\mathcal{B}_{T}:\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx\geq(1-\delta)\|f\|^{2}_{X_{c}^{2}(\mathbb{R}_{+}^{n})}\Big{\\}},$

where $\delta\in(0,1),$ $R>1,$ and $c\in\mathbb{R}^{n}.$

The proposed plan of the paper goes as follows. Section 2 provides some
preliminaries required to derive the desired results. In Section 3, we first
establish that any element in $\mathcal{B}_{T}$ can be approximated by an
element in a finite-dimensional subspace of $\mathcal{B}_{T}$ with the help of
the representation formula (1.2). Then we show that any bounded subset of
$\mathcal{B}_{T,\delta}$ is totally bounded with respect to
$\|\cdot\|_{X_{c}^{\infty}(C_{R})}$. In the end, by using the notion of
covering number and the well-known Bernstein’s inequality, we prove that the
random sampling inequality holds with overwhelming probability for the
functions concentrated on the compact set.

##  2 Preliminaries

In this section, we define some preliminary results, and basic notations used
in the rest of the paper are given in Table 1.

For $1\leq p<\infty$, let $L^{p}(\mathbb{R}^{n}_{+})$ be the space of
$p-$integrable functions in the Lebesgue sense on $\mathbb{R}^{n}_{+}$ with
usual $p-$norm. Moreover, $L^{\infty}(\mathbb{R}^{n}_{+})$ denotes the class
of bounded measurable functions defined on $\mathbb{R}^{n}_{+}$ with norm
$\|f\|_{L^{\infty}(\mathbb{R}^{n}_{+})}:=ess\sup_{x\in\mathbb{R}^{n}_{+}}|f(x)|.$
Let $c\in\mathbb{R}^{n}$ be fixed and for $1\leq p\leq\infty,$ we define the
space

$X_{c}^{p}(\mathbb{R}^{n}_{+}):=\big{\\{}f:\mathbb{R}^{n}_{+}\rightarrow\mathbb{C}:(\cdot)^{c-1/p}f(\cdot)\in
L^{p}(\mathbb{R}^{n}_{+})\big{\\}}$

equipped with norm

$\|f\|_{X_{c}^{p}(\mathbb{R}^{n}_{+})}=\|(\cdot)^{c-1/p}f(\cdot)\|_{L^{p}(\mathbb{R}^{n}_{+})}.$

It is important to mention that $X_{c}^{p}(\mathbb{R}^{n}_{+})$ is Banach
space (see [17]). Moreover, for $p=2$, $X_{c}^{2}(\mathbb{R}^{n}_{+})$ is a
Hilbert space with the inner product

$\langle
f,g\rangle_{X_{c}^{2}(\mathbb{R}^{n}_{+})}:=\int_{\mathbb{R}_{+}^{n}}f(x)\overline{g(x)}x^{2c-1}dx.$

Let $c+it=:s\in\mathbb{C}^{n}.$ Then the multi-dimensional Mellin transform of
any function $f\in X_{c}^{1}(\mathbb{R}^{n}_{+})$ is given by (see [34])

$\hat{M}[f](s):=\int_{\mathbb{R}^{n}_{+}}f(x)\ x^{s-1}\ dx.$

For a fixed $c\in\mathbb{R}^{n},$ the corresponding inverse Mellin transform
is defined as

$\hat{M}^{-1}[f](x):=\frac{1}{(2\pi
i)^{n}}\int_{c+i\mathbb{R}^{n}}\hat{M}[f](s)\ x^{-s}\ ds.$

Riemann first employed Mellin transform in his renowned memoir on the Riemann
zeta function. Later, Mamedov studied the Mellin transform and its properties
in [28]. The article [15] presents various theoretical aspects of Mellin
transform along with its applications to different areas of mathematical
analysis. For some significant development of the Mellin theory, we also refer
to [4, 6, 5, 7].

######  Definition 2.1.

For $c,t\in\mathbb{R}^{n}$ and $T>0,$ any function $f\in
X_{c}^{2}(\mathbb{R}^{n}_{+})$ is said to be Mellin band-limited to
$[-T,T]^{n}$ if $\hat{M}[f](c+it)=0$ for all $\|t\|_{\infty}>T.$ We denote
$\mathcal{B}_{T}$ the space of all Mellin band-limited functions, i.e.,

$\mathcal{B}_{T}=\big{\\{}f\in
X_{c}^{2}(\mathbb{R}^{n}_{+}):\hat{M}[f](c+it)=0\ \mbox{ for \ all}\
\|t\|_{\infty}>T\big{\\}}.$

Notation | Remark
---|---
$x:=\big{(}x(1),x(2),\dots,x(n)\big{)}\in\mathbb{R}^{n}$ | $x(i)\in\mathbb{R}$
$\frac{x}{y}:=\Big{(}\frac{x(1)}{y(1)},\frac{x(2)}{y(2)},\dots,\frac{x(n)}{y(n)}\Big{)}$ | $x,y\in\mathbb{R}_{+}^{n}$
$\log x:=\big{(}\log x(1),\log x(2),\dots,\log x(n)\big{)}$ | $x\in\mathbb{R}_{+}^{n}$
$x^{c}:=x(1)^{c(1)}\cdot x(2)^{c(2)}\cdots x(n)^{c(n)}$ |  $x\in\mathbb{R}_{+}^{n}$, $c\in\mathbb{R}^{n}$
$\alpha^{x}:=\big{(}\alpha^{x(1)},\alpha^{x(2)},\dots,\alpha^{x(n)}\big{)}$ |  $\alpha\in\mathbb{R}_{+}$, $x\in\mathbb{R}^{n}$
$\text{sinc}\,(x):=\frac{\sin\pi x(1)}{\pi x(1)}\cdot\frac{\sin\pi x(2)}{\pi x(2)}\cdots\frac{\sin\pi x(n)}{\pi x(n)}$ | $x\in\mathbb{R}^{n}$
$\text{lin}_{c}(x):=x^{-c}\text{sinc}\,(\log x)$ | $x\in\mathbb{R}_{+}^{n}$
Table 1: Some notations used for the rest of the paper

A Hilbert space $H$ of functions defined on $\Lambda$ is a reproducing kernel
Hilbert space if for each $x\in\Lambda$, the point evaluation functional
$f\mapsto f(x)$ is continuous, i.e., for each $x\in\Lambda,$ there exists
positive constant $C_{x}$ such that

$|f(x)|\leq C_{x}\|f\|\qquad\forall\,f\in H.$

######  Lemma 2.2.

The space $\mathcal{B}_{T}$ is a reproducing kernel space. Moreover, for
$x\in\mathbb{R}^{n}_{+},$ we have

$|f(x)|\leq x^{-c}\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}\ ,\ \ \forall
f\in\mathcal{B}_{T}.$

###### Proof.

Let $f$ be Mellin band-limited to $[-T,T]$, and $T>0.$ Then from [4, Theorem
4], we have

$f(x)=T\int_{\mathbb{R}_{+}}f(y)\,\text{lin}_{\frac{c}{T}}\big{(}(x/y)^{T}\big{)}\frac{dy}{y},\qquad
x\in\mathbb{R}_{+}.$ (2.1)

Using $n-$dimensional Mellin transform and inverse Mellin transform, we extend
(2.1) for $n$-dimensional Mellin band-limited function. In particular, we have

$f(x)=T^{n}\int_{\mathbb{R}_{+}^{n}}f(y)\,\text{lin}_{\frac{c}{T}}\Big{(}\frac{x}{y}\Big{)}^{T}y^{-1}dy,$
(2.2)

where $\text{lin}_{\frac{c}{T}}\big{(}\frac{x}{y}\big{)}^{T}:=x^{-c}y^{c}\
\text{sinc}\,(T\log x-T\log y).$ In view of (2.2) and Hölder’s inequality, we
have

$\displaystyle|f(x)|$ $\displaystyle\leq$ $\displaystyle
T^{n}x^{-c}\left(\int_{\mathbb{R}^{n}_{+}}|f(y)|^{2}y^{2c-1}dy\right)^{1/2}\left(\int_{\mathbb{R}^{n}_{+}}\text{sinc}\,^{2}\left(T\log(x/y)\right)y^{-1}dy\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
x^{-c}\|f\|_{X_{c}^{2}(\mathbb{R}_{+}^{n})}\|\text{sinc}\,\|_{L^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq$ $\displaystyle
x^{-c}\|f\|_{X_{c}^{2}(\mathbb{R}_{+}^{n})}.$

This completes the proof. ∎

Now we see an example of Mellin band-limited function which is important due
to its major application in approximation theory. For instance, it plays a
vital role in the study of the convergence of generalized exponential sampling
series, see [8].

######  Example 2.3.

[8] Let $c\in\mathbb{R},\alpha\geq 1,$ and $k\in\mathbb{N}$. Then the Mellin
Jackson kernel is defined as

$J_{\alpha,k}(x):=C_{\alpha,k}\ x^{-c}\text{sinc}\,^{2k}\left(\frac{\log
x}{2\alpha k\pi}\right),$

where $x\in\mathbb{R}_{+}$ and $\displaystyle
C^{-1}_{\alpha,k}:=\int_{0}^{\infty}\text{sinc}\,^{2k}\left(\frac{\log
x}{2\alpha k\pi}\right)\frac{dx}{x}.$ Since $\hat{M}[J_{\alpha,k}](c+it)=0$
for $|t|\geq\frac{1}{\alpha},$ the function $J_{\alpha,\eta}$ is Mellin band-
limited. In particular, for $k=1,$ this is well-known as Mellin Fejer kernel,
is given by

$F_{\rho}^{c}(x)=\frac{\rho}{2\pi}x^{-c}\text{sinc}\,^{2}\left(\frac{\rho\log\sqrt{x}}{\pi}\right).$

##  3 Main Results

In this section, we discuss the proposed results of the paper. We see that for
each $f\in\mathcal{B}_{T},$ the following representation holds:

$f(x)=\sum_{k\in\mathbb{Z}^{n}}f(e^{k/T})\,\text{lin}_{\frac{c}{T}}(e^{-k}x^{T}),\qquad
x\in\mathbb{R}_{+}^{n}.$

Now we aim to show that any $f\in\mathcal{B}_{T}$ can be approximated on
$C_{R}$ by an element from a finite-dimensional subspace $\mathcal{B}^{N}_{T}$
of $\mathcal{B}_{T}$ where $\mathcal{B}^{N}_{T}$ is defined by

$\mathcal{B}_{T}^{N}=\Big{\\{}\sum_{k\in\mathbb{Z}^{n}\cap[-\frac{N}{2},\frac{N}{2}]^{n}}c_{k}\,\text{lin}_{\frac{c}{T}}\big{(}e^{-k}(\cdot)\big{)}:c_{k}\in\mathbb{R}\Big{\\}}.$

######  Theorem 3.1.

Let $\epsilon>0.$ Then for each $f\in\mathcal{B}_{T},$ there exists
$f_{N}\in\mathcal{B}_{T}^{N}$ such that

$\|f-f_{N}\|_{X_{c}^{\infty}(C_{R})}<\epsilon\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})},$

whenever $N>4T\pi^{-2}\epsilon^{-2/n}+2T\log R.$

###### Proof.

For any $f\in\mathcal{B}_{T}$ and $x\in C_{R},$ we have
$f_{N}\in\mathcal{B}^{N}_{T}$ as

$f_{N}(x)=\sum_{k\in\mathbb{Z}^{n}\cap[-\frac{N}{2},\frac{N}{2}]^{n}}f(e^{k/T})\,\text{lin}_{\frac{c}{T}}(e^{-k}x^{T}).$

In view of Cauchy-Schwarz inequality, we obtain

$\displaystyle x^{c}\big{|}f(x)-f_{N}(x)\big{|}$ $\displaystyle=$
$\displaystyle\bigg{|}x^{c}\sum_{k\in\mathbb{Z}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}f(e^{k/T})\,\text{lin}_{\frac{c}{T}}(e^{-k}x^{T})\bigg{|}$
$\displaystyle=$
$\displaystyle\bigg{|}\sum_{k\in\mathbb{Z}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}f(e^{k/T})\
e^{\frac{\langle c,k\rangle_{2}}{T}}\,\text{sinc}\,(T\log x-k)\bigg{|}$
$\displaystyle\leq$
$\displaystyle\bigg{(}\sum_{k\in\mathbb{Z}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}|f(e^{k/T})|^{2}e^{\frac{2\langle
c,k\rangle_{2}}{T}}\bigg{)}^{1/2}\bigg{(}\sum_{k\in\mathbb{Z}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}\text{sinc}\,^{2}(T\log
x-k)\bigg{)}^{1/2}.$ (3.1)

In view of [4, Theorem 4], for $j,k\in\mathbb{Z}^{n},$ we have

$T^{n}e^{\frac{-2\langle
c,k\rangle_{2}}{T}}\big{\langle}\text{lin}_{\frac{c}{T}}(e^{-k}x^{T}),\text{lin}_{\frac{c}{T}}(e^{-j}x^{T})\big{\rangle}_{X_{c}^{2}(\mathbb{R}^{n}_{+})}=\text{lin}_{\frac{c}{T}}(e^{j-k})=\delta_{j,k}\
,$

where $\delta_{j,k}=$ $\begin{cases}1,&\text{if }j=k,\\\ 0,&\text{if }j\neq
k.\end{cases}$

This gives

$\displaystyle\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}^{n}}\sum_{j\in\mathbb{Z}^{n}}f(e^{k/T})\overline{f(e^{j/T})}\,\big{\langle}\text{lin}_{\frac{c}{T}}(e^{-k}x^{T}),\text{lin}_{\frac{c}{T}}(e^{-j}x^{T})\big{\rangle}_{X_{c}^{2}(\mathbb{R}^{n}_{+})}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{n}}\sum_{k\in\mathbb{Z}^{n}}|f(e^{k/T})|^{2}e^{\frac{2\langle
c,k\rangle_{2}}{T}}.$

Now for $\displaystyle x\in C_{R},$ we have

$\displaystyle\sum_{k\in\mathbb{Z}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}\text{sinc}\,^{2}\pi(T\log
x-k)\leq$
$\displaystyle\pi^{-2n}\sum_{k\in\mathbb{Z}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}\left(\prod_{i=1}^{n}\sin^{2}(T\log
x_{i}-k_{i})\,(T\log x_{i}-k_{i})^{-2}\right)$ $\displaystyle\leq$
$\displaystyle\pi^{-2n}\int_{\mathbb{R}^{n}\setminus[-\frac{N}{2},\frac{N}{2}]^{n}}\left(\prod_{i=1}^{n}(T\log
x_{i}-y_{i})^{-2}\right)dy$ $\displaystyle\leq$
$\displaystyle\pi^{-2n}\int_{\mathbb{R}^{n}\setminus[-\frac{N}{2}+T\log
R,\frac{N}{2}-T\log R]^{n}}y^{-2}\,dy$ $\displaystyle=$
$\displaystyle\frac{{4^{n}}}{\pi^{2n}(N-2T\log R)^{n}}.$

On combining these estimates, we obtain from (3.1) that

$\|f-f_{N}\|_{X_{c}^{\infty}(C_{R})}\leq\left(T^{n/2}\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}\right)\left(\frac{4}{\pi^{2}(N-2T\log
R)}\right)^{n/2}.$

This implies that if $N>4T\pi^{-2}\epsilon^{-2/n}+2T\log R,$ then

$\|f-f_{N}\|_{X_{c}^{\infty}(C_{R})}<\epsilon\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}.$

This completes the proof. ∎

In order to derive the desired probability estimate for the sampling
inequality (1.1) to hold, we first prove that the bounded subset of
$\delta$-concentrated functions is totally bounded.

######  Lemma 3.2.

The set
$\mathcal{B}_{T,\delta}^{*}:=\big{\\{}f\in\mathcal{B}_{T,\delta}:\|f\|_{X_{c}^{2}(\mathbb{R}_{+}^{n})}=1\big{\\}}$
is totally bounded with respect to $\|\cdot\|_{X_{c}^{\infty}(C_{R})}.$

###### Proof.

It follows similar arguments as in [31, Lemma 2.4]. By Theorem 3.1, for given
$\epsilon>0$ and $f\in\mathcal{B}^{*}_{T,\delta},$ there exists
$f_{N}\in\mathcal{B}^{N}_{T}$ such that
$\|f-f_{N}\|_{X_{c}^{\infty}(C_{R})}<\frac{\epsilon}{2}.$ Let
$\overline{B\left(0;\epsilon\right)}$ represents the closed ball in
$\mathcal{B}^{N}_{T}$ having radius $\epsilon$ and centred at origin. Now from
Lemma 2.2, we have $\|f\|_{X_{c}^{\infty}(C_{R})}\leq 1$ which implies that
$f_{N}\in\overline{B\left(0;1+\frac{\epsilon}{2}\right)}.$ Since
$\overline{B\left(0;1+\frac{\epsilon}{2}\right)}$ is totally bounded, the
desired result follows. ∎

######  Definition 3.3.

The covering number of any bounded subset $Y$ of a Banach space $X$ with
respect to $\beta$ is defined by

$\mathcal{N}(Y,\beta):=\min\Big{\\{}\ell\in\mathbb{N}:\exists\
a_{1},a_{2},...,a_{\ell}\in X\ \mbox{such that}\
Y\subset\bigcup_{m=1}^{\ell}B(a_{m};\beta)\Big{\\}},$

where $B(a_{m};\beta)$ represents open ball of radius $\beta$ and centered at
$a_{m}$ in $X.$

The following lemma provides a bound for the covering number of any closed
ball in a finite-dimensional Banach space.

######  Lemma 3.4.

[22] Let X be a Banach space of dimension $d$ and $\overline{B(0;s)}$
represents the closed ball of radius $s$ centered at the origin. Then the
minimum number of open balls of radius $r$ required to cover
$\overline{B(0;s)}$ is bounded by
$\displaystyle\Big{(}\frac{4s}{r}\Big{)}^{d}.$

######  Remark 3.5.

From Theorem 3.1 and Lemma 3.2, for $f\in\mathcal{B}^{*}_{T,\delta},$ it can
be seen that for $\|f-f_{N}\|_{X_{c}^{\infty}(C_{R})}<\frac{\epsilon}{2},$ we
need to choose $N$ in such a way that

$N>\left[4T\pi^{-2}2^{\frac{n}{2}}\epsilon^{-2/n}\right]+2T\log R.$

Particularly, if we set $\displaystyle
N=4T\pi^{-2}2^{\frac{n}{2}}\epsilon^{-2/n}+2T\log R+1,$ then the dimension of
$B_{N}^{T}$ will be bounded by

$\displaystyle N^{n}$ $\displaystyle=$
$\displaystyle\left[T2^{\frac{n}{2}+2}\pi^{-2}\epsilon^{-2/n}+2T\log
R+1\right]^{n}$ $\displaystyle\leq$ $\displaystyle
2^{n}\left[\frac{T^{n}2^{n\left(\frac{n}{2}+2\right)}\pi^{-2n}}{\epsilon^{2}}+(2T\log
R+1)^{n}\right]:=d_{\epsilon}.$

Now from Lemma 3.2 and 3.4, the bound for the covering number of
$\mathcal{B}_{T,\delta}^{*}$ is given by

$\mathcal{N}\left(\mathcal{B}_{T,\delta}^{*},\epsilon\right)\leq
M\bigg{(}\overline{B\Big{(}0;1+\frac{\epsilon}{2}\Big{)}},\frac{\epsilon}{2}\bigg{)}\leq\exp\Big{(}d_{\epsilon}\
\log\Big{(}\frac{16}{\epsilon}\Big{)}\Big{)}.$ (3.2)

###  3.1 Random Sampling

In this section, we deduce the probability estimate concerning the random
sampling inequality (3.5). We first define independent random variables on
$\mathcal{B}_{T,\delta}$ by using random samples. The main tool to derive the
probability estimate is the Bernstein inequality which utilizes the upper
bounds of the random variables and their variance.

######  Lemma 3.6 (Bernstein’s Inequality [13]).

Let $Y_{j},$ $j=1,2,\dots,r$ be a sequence of bounded, independent random
variables with $E[Y_{j}]=0,$ $Var[Y_{j}]\leq\sigma^{2},$ and
$\|Y_{j}\|_{\infty}\leq M$ for $j=1,2,\dots,r.$ Then

$P\bigg{(}\Big{|}\sum_{j=1}^{r}Y_{j}\Big{|}\geq\lambda\bigg{)}\leq
2\exp\Big{(}-\frac{\lambda^{2}}{2r\sigma^{2}+\frac{2}{3}M\lambda}\Big{)}.$
(3.3)

Let $S:=\big{\\{}x_{j}:j\in\mathbb{N}\big{\\}}$ be i.i.d. random variables
distributed uniformly on cube $C_{R}.$ For $f\in\mathcal{B}_{T,\delta},$ we
define random variable as follows:

$Z_{j}(f)=|f(x_{j})|^{2}x_{j}^{2c-1}-\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx.$
(3.4)

The set $\big{\\{}Z_{j}(f)\big{\\}}_{j\in\mathbb{N}}$ is a sequence of
independent random variables. We also see that

$E\big{[}Z_{j}(f)\big{]}=\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}\left[|f(y)|^{2}y^{2c-1}-\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx\right]dy=0$

######  Lemma 3.7.

Let $f,g\in\mathcal{B}_{T}$ with
$\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}=\|g\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}=1.$
Then the following estimates hold:

  1. (i)

$Var\big{[}Z_{j}(f)\big{]}\leq\frac{R^{2n}}{(R^{2}-1)^{n}},$

  2. (ii)

$\|Z_{j}(f)\|_{\infty}\leq R^{n},$

  3. (iii)

$\|Z_{j}(f)-Z_{j}(g)\|_{\infty}\leq 2R^{n}\|f-g\|_{X_{c}^{\infty}(C_{R})}.$

###### Proof.

Using the definition of variance and (3.4), we obtain

$\displaystyle Var\big{[}Z_{j}(f)\big{]}$ $\displaystyle=$ $\displaystyle
E\big{[}|f(x_{j})|^{4}x_{j}^{4c-2}\big{]}-\big{[}E\big{(}|f(x_{j})|^{2}x_{j}^{2c-1}\big{)}\big{]}^{2}$
$\displaystyle\leq$ $\displaystyle E\big{[}|f(x_{j})|^{4}x_{j}^{4c-2}\big{]}$
$\displaystyle=$
$\displaystyle\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{4}x^{4c-2}dx$
$\displaystyle\leq$
$\displaystyle\frac{R^{2n}}{(R^{2}-1)^{n}}\|f\|^{2}_{X_{c}^{\infty}(C_{R})}\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx$
$\displaystyle=$
$\displaystyle\frac{R^{2n}}{(R^{2}-1)^{n}}\|f\|^{2}_{X_{c}^{\infty}(C_{R})}\|f\|^{2}_{X_{c}^{2}(C_{R})}$

Since
$\|f\|^{2}_{X_{c}^{\infty}(C_{R})}\leq\|f\|^{2}_{X_{c}^{2}(\mathbb{R}^{n}_{+})},$
and $\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}=1,$ we have

$Var\big{[}Z_{j}(f)\big{]}\leq\frac{R^{2n}}{(R^{2}-1)^{n}}.$

Now since $f\in\mathcal{B}_{T}$ and $\|f\|_{X^{2}_{c}(\mathbb{R}^{n}_{+})}=1,$
we can write

$\displaystyle\|Z_{j}(f)\|_{\infty}$ $\displaystyle=$
$\displaystyle\sup_{x_{j}\in
S}\left||f(x_{j})|^{2}x_{j}^{2c-1}-\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx\right|$
$\displaystyle\leq$
$\displaystyle\max\left\\{R^{n}\|f\|_{X_{c}^{\infty}(C_{R})}^{2},\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx\right\\}$
$\displaystyle=$ $\displaystyle R^{n}\|f\|_{X_{c}^{\infty}(C_{R})}^{2}\leq
R^{n}.$

Similarly for $(iv),$ we have

$\displaystyle\|Z_{j}(f)-Z_{j}(g)\|_{\infty}$ $\displaystyle=$
$\displaystyle\sup_{x_{j}\in
S}\Big{|}\left(|f(x_{j})|^{2}-|g(x_{j})|^{2}\right)x_{j}^{2c-1}-\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}(|f(x)|^{2}-|g(x)|^{2})\
x^{2c-1}dx\Big{|}$ $\displaystyle\leq$
$\displaystyle\max\left\\{(|f(x)|-|g(x)|)(\ |f(x)|+|g(x)|)x^{2c-1},\
\frac{R^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}(|f(x)|^{2}-|g(x)|^{2})x^{2c-1}dx\right\\}$
$\displaystyle\leq$ $\displaystyle 2R^{n}\|f-g\|_{X_{c}^{\infty}(C_{R})}.$

This completes the proof. ∎

Using these estimates, we now proceed toward probability estimation. The
following result will be helpful in this direction.

######  Theorem 3.8.

Let $\\{x_{j}:j\in\mathbb{N}\\}$ be the i.i.d. random variables drawn
uniformly from the cube $C_{R}.$ Then the following holds

$P\bigg{(}\sup_{f\in\mathcal{B}^{*}_{T,\delta}}\Big{|}\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f)\Big{|}\geq\epsilon\bigg{)}\leq
2\exp\bigg{[}2^{n}\bigg{(}\frac{16T\pi^{-2}R^{2n}2^{n\left(\frac{n}{2}+2\right)}}{\epsilon^{2}}+(2T\log
R+1)^{n}\bigg{)}\\\
\log\left(\frac{64R^{n}}{\epsilon}\right)-\frac{3r\epsilon^{2}(R^{2}-1)^{n}}{4R^{n}(6R^{n}+\epsilon(R^{2}-1)^{n})}\bigg{]}.$

###### Proof.

Let $\displaystyle
M:=\mathcal{N}\left(\mathcal{B}_{T,\delta}^{*},\frac{\epsilon}{4R^{n}}\right)$
be the covering number for the set $\mathcal{B}^{*}_{T,\delta}$ and
$f_{1},f_{2},...,f_{M}$ be the elements of $\mathcal{B}^{*}_{T,\delta}.$ Then
for any $f\in\mathcal{B}^{*}_{T,\delta},$ there exists $f_{m},\ 1\leq m\leq M$
such that
$\displaystyle\|f-f_{m}\|_{X^{\infty}_{c}(C_{R})}<\frac{\epsilon}{4R^{n}}.$
From (iii) of Lemma 3.7, we have

$\left|\frac{1}{r}\sum_{j=1}^{r}\left(Z_{j}(f)-Z_{j}(f_{m})\right)\right|\leq
2R^{n}\|f-f_{m}\|_{X^{\infty}_{c}(C_{R})}\leq\frac{\epsilon}{2}.$

In view of (3.3), for given $f_{m},\ 1\leq m\leq M,$ we have

$P\bigg{(}\bigg{|}\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f_{m})\bigg{|}\geq\frac{\epsilon}{2}\bigg{)}\leq
2\exp\left(-\frac{3r\epsilon^{2}(R^{2}-1)^{n}}{4R^{n}(6R^{n}+\epsilon(R^{2}-1)^{n})}\right).$

For fixed $m,$ we have

$\displaystyle
P\bigg{(}\sup_{\left\\{f:\|f-f_{m}\|_{X^{\infty}_{c}(C_{R})}\leq\frac{\epsilon}{4R^{n}}\right\\}}\bigg{|}\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f)\bigg{|}\geq\epsilon\bigg{)}$
$\displaystyle\leq$ $\displaystyle
P\bigg{(}\bigg{|}\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f_{m})\bigg{|}\geq\frac{\epsilon}{2}\bigg{)}$
$\displaystyle\leq$ $\displaystyle
2\exp\left(-\frac{3r\epsilon^{2}(R^{2}-1)^{n}}{4R^{n}(6R^{n}+\epsilon(R^{2}-1)^{n})}\right).$

Since $M$ is the covering number for $\mathcal{B}^{*}_{T,\delta},$ i.e.,

$\mathcal{B}^{*}_{T,\delta}\subset\bigcup_{m=1}^{M}\left\\{f:\|f-f_{m}\|_{X^{\infty}_{c}(C_{R})}\leq\frac{\epsilon}{4R^{n}}\right\\},$

we obtain

$\displaystyle
P\bigg{(}\sup_{f\in\mathcal{B}^{*}_{T,\delta}}\Big{|}\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f)\Big{|}\geq\epsilon\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sum_{m=1}^{M}P\bigg{(}\sup_{\left\\{f:\|f-f_{m}\|_{X^{\infty}_{c}(C_{R})}\leq\frac{\epsilon}{4R^{n}}\right\\}}\left|\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f)\right|\geq\epsilon\bigg{)}$
$\displaystyle\leq$ $\displaystyle
2M\exp\bigg{(}-\frac{3r\epsilon^{2}(R^{2}-1)^{n}}{4R^{n}(6R^{n}+\epsilon(R^{2}-1)^{n})}\bigg{)}.$

Using (3.2), we finally get

$P\bigg{(}\sup_{f\in\mathcal{B}^{*}_{T,\delta}}\Big{|}\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f)\Big{|}\geq\epsilon\bigg{)}\leq
2\exp\bigg{[}2^{n}\bigg{(}\frac{16T^{n}\pi^{-2n}R^{2n}2^{n\left(\frac{n}{2}+2\right)}}{\epsilon^{2}}+(2T\log
R+1)^{n}\bigg{)}\\\
\log\left(\frac{64R^{n}}{\epsilon}\right)-\frac{3r\epsilon^{2}(R^{2}-1)^{n}}{4R^{n}(6R^{n}+\epsilon(R^{2}-1)^{n})}\bigg{]}.$

∎

We now prove the following main result of the paper.

######  Theorem 3.9.

Let $\\{x_{j}:j\in\mathbb{N}\\}$ be the independent and identically
distributed random variables that are uniformly distributed over $C_{R}.$ For
$0<\mu<1-\delta,$ the following inequality

$\frac{R^{n-1}(1-\delta-\mu)}{(R^{2}-1)^{n}}\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2}\leq\frac{1}{r}\sum_{j=1}^{r}|f(x_{j})|^{2}x_{j}^{2c}\leq\frac{R^{n+1}(1+\mu)}{(R^{2}-1)^{n}}\|f\|_{X_{c}^{2}(\mathbb{R}^{n}_{+})}^{2},$
(3.5)

holds for every $f\in\mathcal{B}_{T,\delta}$ with probability at least
$\displaystyle
1-2\beta\exp\Big{(}-\frac{3r\mu^{2}}{4(R^{2}-1)^{n}(6+\mu)}\Big{)},$ where
$\displaystyle\beta:=\exp\bigg{(}2^{n}\Big{(}\frac{16T^{n}\pi^{-2n}(R^{2}-1)^{2n}\
2^{n\left(\frac{n}{2}+2\right)}}{\mu^{2}}+(2T\log
R+1)^{n}\Big{)}\log\Big{(}\frac{64(R^{2}-1)^{n}}{\mu}\Big{)}\bigg{)}.$

###### Proof.

Let $\\{x_{j}:j\in J\\}$ be the uniformly distributed samples taken from
$C_{R}$ and define the following event

$\mathcal{E}=\left\\{\sup_{f\in\mathcal{B}^{*}_{T,\delta}}\left|\frac{1}{r}\sum_{j=1}^{r}Z_{j}(f)\right|\leq\frac{\mu
R^{n}}{(R^{2}-1)^{n}}\right\\}.$

The event $\mathcal{E}$ is equivalent to the event

$\bigg{|}\sum_{j=1}^{r}|f(x_{j})|^{2}x_{j}^{2c-1}-\frac{rR^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}dx\bigg{|}\leq\frac{r\mu
R^{n}}{(R^{2}-1)^{n}},\qquad f\in\mathcal{B}_{T,\delta}^{*}.$

This implies that for all $f\in\mathcal{B}_{T,\delta}^{*}$

$\displaystyle\frac{rR^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}\,dx-\frac{r\mu
R^{n}}{(R^{2}-1)^{n}}$
$\displaystyle\leq\sum_{j=1}^{r}|f(x_{j})|^{2}x_{j}^{2c-1}$
$\displaystyle\leq\frac{rR^{n}}{(R^{2}-1)^{n}}\int_{C_{R}}|f(x)|^{2}x^{2c-1}\,dx+\frac{r\mu
R^{n}}{(R^{2}-1)^{n}}$
$\displaystyle\frac{R^{n-1}(1-\delta-\mu)}{(R^{2}-1)^{n}}\leq\frac{1}{r}\sum_{j=1}^{r}|f(x_{j})|^{2}x_{j}^{2c}$
$\displaystyle\leq\frac{R^{n+1}}{(R^{2}-1)^{n}}(1+\mu).$

It can be observed that the event $\mathcal{E}$ leads to the desired sampling
inequality (3.5). From Theorem 3.8, the probability that the random samples
$\\{x_{j}:j=1,2,\dots,r\\}$ satisfy the above sampling inequality, is given by

$\displaystyle P(\mathcal{E})$ $\displaystyle=$ $\displaystyle
1-P(\mathcal{E}^{c})$ $\displaystyle\geq$ $\displaystyle
1-2\exp\left(-r\alpha+\beta\right)$

where $\displaystyle\alpha:=\frac{3\mu^{2}}{4(R^{2}-1)^{n}(6+\mu)},\ $ and
$\displaystyle\beta:=2^{n}\bigg{(}\frac{16\pi^{-2n}T^{n}(R^{2}-1)^{2n}\
2^{n\left(\frac{n}{2}+2\right)}}{\mu^{2}}+(2T\log
R+1)^{n}\bigg{)}\log\bigg{(}\frac{64(R^{2}-1)^{n}}{\mu}\bigg{)}.$ ∎

######  Remark 3.10.

From the above estimate of $P(\mathcal{E}),$ it is easy to see that the
probability approaches to $1$ for sufficiently large sample size $r.$
Moreover, the sampling inequality (3.5) is true for large value of $r$ such
that $r>\frac{\beta}{\alpha}.$

## Acknowledgment

S. Bajpeyi and S. Sivananthan gratefully thank financial assistance from the
Department of Science and Technology, Government of India, via project no.
CRG/2019/002412.

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