On Wachnicki’s Generalization of the Gauss–Weierstrass Integral

3 years ago

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Abstract

The paper aims at a generalization of the Gauss–Weierstrass integral introduced by Eugeniusz Wachnicki two decades ago. It is intimately connected to a generalization of the heat equation. The main result is an asymptotic expansion for the operators when applied to a function belonging to a rather large class. An essential auxiliary result is a localization theorem which is interesting in itself.

Authors

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

Gauss–Weierstrass operator; Bessel function; Kummer function; Asymptotic expansion; Degree of approximation

Paper coordinates

U. Abel and O. Agratini, On Wachnicki’s Generalization of the Gauss-Weierstrass Integral, In: Candela, A.M., Cappelletti Montano, M., Mangino, E. (eds) Recent advances in Mathematical Analysis. Trends in Mathematics, October 2022, Birkhäuser, Cham., pp 1–13, 2023.
https://doi.org/10.1007/978-3-031-20021-2_1

PDF

https://link.springer.com/chapter/10.1007/978-3-031-20021-2_1

About this paper

Journal

Recent Advances in Mathematical Analysis

Publisher Name

Birkhäuser, Cham

DOI

https://doi.org/10.1007/978-3-031-20021-2_1

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On Wachnicki’s generalization of the Gauss–Weierstrass integral

Ulrich Abel Fachbereich MND,
Technische Hochschule Mittelhessen,
Wilhelm-Leuschner-Straße 13,
61169 Friedberg, Germany ulrich.abel@mnd.thm.de and Octavian Agratini Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kogălniceanu, 1, 400084 Cluj-Napoca, Romania
and
Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy,
Str. Fântânele, 57, 400320 Cluj-Napoca, Romania agratini@math.ubbcluj.ro

Abstract.

Mathematics Subject Classification: 41A36, 41A60.

Keywords and phrases: Gauss–Weierstrass operator, Bessel function, Kummer function, asymptotic expansion, degree of approximation.

1. Introduction

It is acknowledged that linear positive operators are a useful tool in approximating signals from various spaces. Referring to operators of either discrete or continuous type, a natural challenge is to highlight their properties. Two decades ago Eugeniusz Wachnicki [16] defined an integral operator representing a generalization of the classical Gauss–Weierstrass operators. One of the genuine Gauss–Weierstrass operator’s write modes is as follows

W (f; x, t) = \frac{1}{2 \sqrt{π t}} \int_{ℝ} \exp (- \frac{{(x - y)}^{2}}{4 t}) f (y) 𝑑 y,

(1)

where $t > 0$ is a parameter, $x \in ℝ$ and $f : ℝ \to ℝ$ is chosen so that the integral exists and is finite. A concise but relevant presentation of these operators can be found, e.g., in the monograph [6, Section 5.2.9]. We mention that a modification of the operators (1) was presented by Altomare and Sabina, see [7].

Actually, the $W f$ transform is a smoothed out version of $f$ obtained by averaging the values of $f$ with a Gaussian signal centered at $x$ . Specifically, it is the function $G_{t}$ given by

G_{t} (x) = \frac{1}{2 \sqrt{π t}} \int_{ℝ} f (y) \exp (- \frac{{(x - y)}^{2}}{4 t}) 𝑑 y = \frac{1}{2 \sqrt{π t}} \int_{ℝ} f (x - y) \exp (- \frac{y^{2}}{4 t}) 𝑑 y

representing the convolution of $f$ with the Gaussian function

x \mapsto \frac{1}{2 \sqrt{π t}} \exp (- x^{2} / (4 t)) .

Weierstrass used a variant of this transform in his original proof of the famous Approximation Theorem which bears his name. Also, this transform is intimately related to the heat equation or, equivalently, the diffusion equation with constant diffusion coefficient. These are only a few reasons why it has been intensively studied over time. We note that an extension of this integral was performed even in q-Calculus [8]. Among the most recent papers published in 2021, without intending to bring up them all, we mention [9], [17].

2. The operators

First of all we recall the modified Bessel function of the first kind and fractional order $α > - 1$ , see, e.g., [5, Chapter 10]. Denoting by $I_{α}$ , it is described by the series

I_{α} (z) = {(\frac{z}{2})}^{α} \sum_{k = 0}^{\infty} \frac{{(\frac{1}{4} z^{2})}^{k}}{k! Γ (α + k + 1)},

(2)

where $Γ$ is the Gamma function. An integral formula for this function (for $Re (z) > 0$ ) will be read as follows

I_{α} (z) = \frac{1}{π} \int_{0}^{π} e^{z \cos θ} \cos (α θ) 𝑑 θ - \frac{\sin (α π)}{π} \int_{0}^{\infty} e^{- z \cosh (t) - α t} 𝑑 t .

We indicate two particular cases useful throughout the paper

I_{- 1 / 2} (z) = \sqrt{\frac{2}{π z}} \cosh (z), I_{1 / 2} (z) = \sqrt{\frac{2}{π z}} \sinh (z),

(3)

see, e.g., [5, p. 433]. In the above, $\cosh$ , $\sinh$ represent the usual notation of hyperbolic functions.

Set $ℝ_{+} = [0, \infty)$ . For a fixed constant $K \geq 0$ , we consider the space

	$E_{K} = {$	$f : ℝ_{+} \to ℝ ∣ f is locally integrable and (\exists) M_{f} \geq 0,$
		$\| f (s) \| \leq M_{f} e^{K s^{2}}, s > 0} .$

The space is endowed with the norm $∥ \cdot ∥_{K}$ ,

{‖ f ‖}_{K} = sup_{s \in ℝ_{+}} | f (s) | e^{- K s^{2}} .

We consider the operator $W_{α}$ defined on $E_{K}$ by the following relation

W_{α} (f; r, t) = \frac{1}{2 t} \int_{0}^{\infty} r^{- α} s^{α + 1} \exp (- \frac{r^{2} + s^{2}}{4 t}) I_{α} (\frac{r s}{2 t}) f (s) 𝑑 s,

(4)

where $α \geq - 1 / 2$ , $(r, t) \in (0, \infty) \times (0, \infty)$ and $I_{α}$ is given at (2).

$W_{α}$ was introduced in [16, Eq. (1)] with a minor modification of the domain $E_{K}$ in which the author inserted $f \in C (ℝ_{+})$ .

Since $W_{α} f$ , $f \in E_{K}$ , is well defined for any $K > 0$ , we can consider the domain of $W_{α}$ as $E := ⋃_{K > 0} E_{K}$ .

Wachnicki [16, Theorem 4] showed the convergence

lim_{t \to 0^{+}} W_{α} (f; r, t) = f (r),

for $f \in E \cap C (ℝ_{+})$ , uniformly on compact subintervals of $(0, + \infty)$ .

For $α \geq - 1 / 2$ , the function $W_{α} (f; r, t)$ is an example of the solution of the generalized heat equation

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial r^{2}} + \frac{2 α + 1}{r} \frac{\partial u}{\partial r} .

(5)

If $α = n / 2 - 1$ , $n \in ℕ$ , Eq. $(5)$ is the heat equation in $ℝ^{n + 1}$ in radial coordinates.

Recently, in [12] an extension of this operator was achieved for continuous functions defined on the domain $(0, \infty) \times ℝ$ and bounded by certain two-dimensional exponential functions.

Remark. For $α = - 1 / 2$ the operator defined by (4) becomes the authentic Gauss–Weierstrass operator, a specification that can be found in [16]. Because the statement was not accompanied by a proof, we insert it as a detail in our paper. More precisely we prove

W_{- 1 / 2} f = W \hat{f}, f \in E_{K},

(6)

see (1), where $\hat{f} (s) = f ((sgn s) s)$ , $s \in ℝ$ .

Indeed, for any $(r, t) \in (0, \infty) \times (0, \infty)$ , taking in view (3), we can write

	$W_{- 1 / 2} (f; r, t)$
	$= \frac{1}{2 t} \sqrt{\frac{4 t}{π}} \int_{0}^{\infty} \exp (- \frac{r^{2} + s^{2}}{4 t}) \cosh (\frac{r s}{2 t}) f (s) 𝑑 s$
	$= \frac{1}{\sqrt{π t}} \int_{0}^{\infty} \exp (- \frac{r^{2} + s^{2}}{4 t}) \frac{1}{2} (\exp (\frac{r s}{2 t}) + \exp (- \frac{r s}{2 t})) f (s) 𝑑 s$
	$= \frac{1}{2 \sqrt{π t}} \int_{0}^{\infty} (\exp (- \frac{{(r - s)}^{2}}{4 t}) + \exp (- \frac{{(r + s)}^{2}}{4 t})) f (s) 𝑑 s$
	$= \frac{1}{2 \sqrt{π t}} (\int_{0}^{\infty} \exp (- \frac{{(r - s)}^{2}}{4 t}) f (s) 𝑑 s + \int_{- \infty}^{0} \exp (- \frac{{(r - s)}^{2}}{4 t})) f (- s) d s$
	$= W (\hat{f}; r, t) .$

3. The asymptotic expansion

To achieve our goal and to obtain a self contained exposure, we recall the following notions.

For factorial powers (falling respective rising factorial) we use the notations

u^{\underline{k}} = \prod_{ℓ = 0}^{k - 1} (u - ℓ), u^{\bar{k}} = \prod_{ℓ = 0}^{k - 1} (u + ℓ), k \in ℕ .

An empty product $(k = 0)$ is taken to be 1.

Kummer’s function of the first kind, also known as the confluent hypergeometric function of the first kind, is commonly denoted $M (a, b, z)$ or ${}_{1}F_{1} (a, b, z)$ and defined as follows

	$M (a, b, z)$	$\equiv_{1} F_{1} (a, b, z) = 1 + \frac{a}{b} z + \frac{a (a + 1)}{b (b + 1) 2!} z^{2} + \dots$
		$= \sum_{k = 0}^{\infty} \frac{a^{\bar{k}}}{b^{\bar{k}} k!} z^{k}, z \in ℂ,$		(7)

see [5, Chapter 13]. If $b \in ℤ ∖ ℕ$ , then $M$ is undefined. Otherwise the series is convergent for all $z \in ℂ$ .

Let us denote $e_{i}$ the monomials $e_{i} (x) = x^{i}$ , $i \in ℕ_{0} = ℕ \cup {0}$ . Clearly, $e_{i} \in E_{K}$ , for each $K$ . We determine all moments of the operator $W_{α}$ .

Proposition 1.

Let $W_{α}$ be defined by (4). For each $i \in ℕ_{0}$ the identity

W_{α} (e_{i}; r, t) = {(4 t)}^{i / 2} \exp (- \frac{r^{2}}{4 t}) \frac{Γ (α + 1 + i / 2)}{Γ (α + 1)} M (α + 1 + \frac{i}{2}, α + 1, \frac{r^{2}}{4 t})

(8)

takes place, where $M$ stands for Kummer function (3). Moreover

W_{α} (e_{i}; r, t) \sim r^{i} \sum_{n = 0}^{\infty} \frac{1}{n!} {(\frac{i}{2})}^{\underline{n}} {(α + \frac{i}{2})}^{\underline{n}} {(\frac{4 t}{r^{2}})}^{n} (t \to 0^{+}) .

(9)

Proof.

By a straightforward calculation we obtain

	$W_{α} (e_{i}; r, t)$
	$= \frac{1}{2 t} {(\frac{1}{4 t})}^{α} \exp (- \frac{r^{2}}{4 t}) \sum_{k = 0}^{\infty} \frac{Γ (k + α + 1 + i / 2)}{k! Γ (k + α + 1)} {(\frac{r}{4 t})}^{2 k} 2^{i + 2 α + 1 + 2 k} t^{α + 1 + k + i / 2}$
	$= {(4 t)}^{i / 2} \exp (- \frac{r^{2}}{4 t}) \sum_{k = 0}^{\infty} \frac{Γ (k + α + 1 + i / 2)}{k! Γ (k + α + 1)} {(\frac{r^{2}}{4 t})}^{k} .$

By using Kummer function defined by (3) and the well-known relation

Γ (u + k) = Γ (u) u^{\bar{k}}, u > 0, k \in ℕ_{0},

we arrive at (8).

Further, we use the asymptotic expansion of Kummer function for large argument. To accomplish this, we use for example Digital Library of Mathematical Functions [18, Formulas 13.2.4, 13.7.1].

M (a, b, x) \sim \frac{Γ (b)}{Γ (a)} e^{x} x^{a - b} \sum_{n = 0}^{\infty} \frac{{(b - a)}^{\bar{n}} {(1 - a)}^{\bar{n}}}{n!} x^{- n} (x \to \infty), a \notin ℤ ∖ ℕ_{0} .

Returning at (8) with $x = \frac{r^{2}}{4 t}$ , we get

W_{α} (e_{i}; r, t) \sim r^{i} \sum_{n = 0}^{\infty} \frac{1}{n!} {(- \frac{i}{2})}^{\bar{n}} {(- α - \frac{i}{2})}^{\bar{n}} (\frac{4 t}{r^{2}}) (t \to 0^{+}) .

Knowing that ${(- u)}^{\bar{n}} = {(- 1)}^{n} u^{\underline{n}}$ , we obtain (9) and the proof is ended. ∎

Remark. i) Formula (9) means that

W_{α} (e_{i}; r, t) = e_{i} (r) + \sum_{n = 1}^{q} \frac{1}{n!} {(\frac{i}{2})}^{\underline{n}} {(α + \frac{i}{2})}^{\underline{n}} {(\frac{4 t}{r^{2}})}^{n} + o (t^{q}) (t \to 0^{+})

for all $q \in ℕ$ .

ii) If $i = 2 k$ , $k \in ℕ$ , is an even integer, $W_{α} (e_{2 k}; r, t)$ is a polynomial as a function of $r$ , since $M (α + 1 + k, α + 1, \frac{r^{2}}{4 t})$ is a finite sum. The polynomials determined in this way are called radial heat polynomials (see [10]). For $k = 0, 1, 2$ formula (8) easily leads us to the explicit formulas [16, Eq. (13)]

$W_{α} (e_{0}; r, t)$	$=$	$1,$
$W_{α} (e_{2}; r, t)$	$=$	$r^{2} + 4 (α + 1) t,$
$W_{α} (e_{4}; r, t)$	$=$	$r^{4} + 8 (α + 2) r^{2} t + 16 (α + 1) (α + 2) t^{2} .$

The first formula implies the fact that the operator reproduces the constants. In the special case $α = - 1 / 2$ we have the representation (see [14, Eq. (1.2)])

W_{- 1 / 2} (e_{2 k}; r, t) = k! \sum_{j = 0}^{⌊ k / 2 ⌋} \frac{x^{k - 2 j}}{(k - 2 j)!} \frac{t^{j}}{j!} .

Further we introduce the $j$ -th central moments of $W_{α}$ operator, $j \in ℕ_{0}$ , i.e., $W_{α} ψ_{r}^{j}$ , where $ψ_{r} (s) = s - r$ , $s > 0$ , $r > 0$ .

Lemma 2.

For each $j \in ℕ$ , the operator $W_{α}$ defined by (4) satisfies the relation

W_{α} (ψ_{r}^{j}; r, t) \sim C_{α} (n, j) r^{j} \sum_{n = 0}^{\infty} \frac{1}{n!} {(\frac{4 t}{r^{2}})}^{n} (t \to 0^{+}),

(10)

where

C_{α} (n, j) = \sum_{i = 0}^{j} {(- 1)}^{j - i} (\binom{j}{i}) {(\frac{i}{2})}^{\underline{n}} {(α + \frac{i}{2})}^{\underline{n}} .

(11)

Proof.

Since $ψ_{r}^{j} (s) = \sum_{i = 0}^{j} (\binom{j}{i}) {(- r)}^{j - i} e_{i} (s)$ , based on relation (9) we can write

W_{α} (ψ_{r}^{j}; r, t) \sim \sum_{i = 0}^{j} (\binom{j}{i}) {(- r)}^{j - i} r^{i} \sum_{n = 0}^{\infty} \frac{1}{n!} {(\frac{i}{2})}^{\underline{n}} {(α + \frac{i}{2})}^{\underline{n}} {(\frac{4 t}{r^{2}})}^{n} (t \to 0^{+}) .

Using the coefficients defined in (11) we arrive at (10). ∎

Remark. Examining the coefficients $C_{α} (n, j)$ , $j \in ℕ_{0}$ , we notice that they can be written as follows

	$C_{α} (n, j)$	$= {\sum_{i = 0}^{j} {(- 1)}^{j - i} (\binom{j}{i}) {(\frac{\partial}{\partial w})}^{n} [w^{n + α} {(\frac{\partial}{\partial w})}^{n} w^{i / 2}] \|}_{w = 1}$
		$= {{(\frac{\partial}{\partial w})}^{n} [w^{n + α} {(\frac{\partial}{\partial w})}^{n} {(\sqrt{w} - 1)}^{j}] \|}_{w = 1} .$		(12)

The following theorem is our main result. It presents a complete asymptotic expansion of the operators $W_{α} (f; r, t)$ as $t \to 0^{+}$ , for functions $f \in E$ being sufficiently smooth at a point $r > 0$ .

Theorem 3.

Let $W_{α}$ be defined by (4) and $r > 0, q \in ℕ$ be given. If $f \in E$ is $q$ times differentiable at $r$ , then

W_{α} (f; r, t) = \sum_{n = 0}^{q} \frac{1}{n!} {(\frac{4 t}{r^{2}})}^{n} \sum_{j = 0}^{2 q} \frac{f^{(j)} (r)}{j!} r^{j} C_{α} (n, j) + o (t^{q}) (t \to 0^{+}),

(13)

where the coefficients $C_{α} (n, j)$ , $j = 0, \dots, 2 q$ , are defined at (11).

If $f$ is a real analytic function, then

W_{α} (f; r, t) \sim \sum_{n = 0}^{\infty} c_{n} (α, f, r) t^{n} (t \to 0^{+}),

(14)

where

c_{n} (α, f, r) = {\frac{4^{n}}{n! r^{2 n}} {(\frac{\partial}{\partial w})}^{n} [w^{n + α} {(\frac{\partial}{\partial w})}^{n} f (r \sqrt{w})] |}_{w = 1} .

(15)

Proof.

Let $r > 0$ and put $U_{δ} (r) = (r - δ, r + δ) \cap [0, + \infty)$ , for $δ > 0$ . Let $δ > 0$ be given. Choose a function $φ \in C^{\infty} ([0, + \infty))$ with $φ (x) = 1$ on $U_{δ} (r)$ and $φ (x) = 0$ on $[0, + \infty) ∖ U_{2 δ} (r)$ . Put $\tilde{f} = φ f$ . Then we have $\tilde{f} \equiv f$ on $U_{δ} (r)$ and $\tilde{f} \equiv 0$ on $[0, + \infty) ∖ U_{3 δ} (r)$ . In particular $f^{(2 q)} (r) = {\tilde{f}}^{(2 q)} (r)$ . By the localization theorem (Theorem 5), $W_{α} (f - \tilde{f}; r, t)$ deceases exponentially fast as $t \to 0^{+}$ . Consequently, $\tilde{f}$ and $f$ possess the same asymptotic expansion of the form $(13)$ . Therefore, without loss of generality, we can assume that $f \equiv 0$ on $[0, + \infty) ∖ U_{3 δ} (r)$ . By Lemma 2, we have $W_{α} (ψ_{r}^{2 s}; r, t) = O (t^{s})$ as $t \to 0^{+}$ . Under these conditions, a general approximation theorem due to Sikkema [15, Theorem 3] implies that

W_{α} (f; r, t) = \sum_{j = 0}^{2 q} \frac{f^{(j)} (r)}{j!} W_{α} (ψ_{r}^{j}; r, t) + o (t^{q}),

see [15, Eq. (15)]. Taking in view (10), identity (13) is proved.

If $f$ is a real analytic function we have the possibility to write

f (r \sqrt{w}) = \sum_{j \geq 0} \frac{f^{(j)} (r)}{j!} {(r \sqrt{w} - r)}^{j}, w > 0 .

Further, with the help of (3), formula (13) can be rewritten in the form

	$W_{α} (f; r, t)$
	$= {\sum_{n = 0}^{q} \frac{1}{n!} {(\frac{4 t}{r^{2}})}^{n} {(\frac{\partial}{\partial w})}^{n} [w^{n + α} {(\frac{\partial}{\partial w})}^{n} f (r \sqrt{w})] \|}_{w = 1} + o (t^{q}) (t \to 0^{+}) .$

Consequently we get

W_{α} (f; r, t) \sim {\sum_{n = 0}^{\infty} \frac{1}{n!} {(\frac{4 t}{r^{2}})}^{n} {(\frac{\partial}{\partial w})}^{n} [w^{n + α} {(\frac{\partial}{\partial w})}^{n} f (r \sqrt{w})] |}_{w = 1} (t \to 0^{+})

and (14) is substantiated. ∎

Applying formula (15) to find the first four coefficients of asymptotic expansion, we obtain the following values:

	$c_{0} (α, f, r)$	$= f (r),$
	$c_{1} (α, f, r)$	$= \frac{2 α + 1}{r} f^{'} (r) + f^{''} (r),$
	$c_{2} (α, f, r)$	$= \frac{1 - 4 α^{2}}{2 r^{3}} f^{'} (r) + \frac{4 α^{2} - 1}{2 r^{2}} f^{''} (r) + \frac{2 α + 1}{r} f^{'''} (r) + \frac{1}{2} f^{(4)} (r),$
	$c_{3} (α, f, r)$	$= (4 α^{2} - 1) (\frac{2 α - 3}{2 r^{5}} f^{'} (r) - \frac{2 α - 3}{2 r^{4}} f^{''} (r) + \frac{α - 3}{3 r^{3}} f^{'''} (r) + \frac{1}{2 r^{2}} f^{(4)} (r))$
		$+ \frac{2 α + 1}{2 r} f^{(5)} (r) + \frac{1}{6} f^{(6)} (r) .$

For the special case $α = - 1 / 2$ (nearly Gauss–Weierstrass operator, see (6)) the respective coefficients become

\begin{matrix} c_{0} (- 1 / 2, f, r) = f (r), & c_{1} (- 1 / 2, f, r) = f^{''} (r), \\ c_{2} (- 1 / 2, f, r) = \frac{1}{2} f^{(4)} (r), & c_{3} (- 1 / 2, f, r) = \frac{1}{6} f^{(6)} (r) . \end{matrix}

We have the following representation of the coefficients in the asymptotic expansion (cf. [3, Eq. (1.5)]).

Proposition 4.

For $α = - 1 / 2$ it holds

c_{n} (- 1 / 2, f, r) = \frac{1}{n!} f^{(2 n)} (r)

Proof.

Since the formula is of algebraic form it is sufficient to prove it for polynomial functions $f$ . For $f = e_{i}$ , we have

$c_{n} (- 1 / 2, e_{i}, r)$	$=$	${\frac{4^{n}}{n! r^{2 n}} {(\frac{\partial}{\partial w})}^{n} [w^{n - 1 / 2} {(\frac{\partial}{\partial w})}^{n} {(r \sqrt{w})}^{i}] \|}_{w = 1}$
	$=$	${\frac{4^{n} r^{i}}{n! r^{2 n}} {(\frac{i}{2})}^{\underline{n}} {(\frac{\partial}{\partial w})}^{n} w^{i / 2 - 1 / 2} \|}_{w = 1}$
	$=$	$\frac{4^{n} r^{i - 2 n}}{n!} {(\frac{i}{2})}^{\underline{n}} {(\frac{i - 1}{2})}^{\underline{n}}$
	$=$	$\frac{4^{n} r^{i - 2 n}}{n!} \frac{i}{2} \frac{i - 1}{2} \frac{i - 2}{2} \frac{i - 3}{2} \dots \frac{i - 2 n + 2}{2} \frac{i - 2 n + 1}{2}$
	$=$	$\frac{1}{n!} i^{\underline{2 n}} r^{i - 2 n} = \frac{1}{n!} e_{i}^{(2 n)} (r) .$

∎

At the end of this section it is worth mentioning that the first author obtained asymptotic expansions for various classes of approximation linear positive operators, see, e.g., [1], [2], as well as [3], [4].

4. Localization result

The purpose of this paragraph is to characterize the function $W_{α} f$ according to its growth rate, where $f \in E_{K}$ has a specific property which means that it vanishes in a neighborhood of a fixed value $r > 0$ . For the modified Bessel function defined by (2), Luke [13, Eq. (6.25)] proved the estimate

1 < Γ (α + 1) {(\frac{2}{z})}^{α} I_{α} (z) < \cosh (z), z > 0, α > - \frac{1}{2} .

(16)

We emphasize that Ifantis and Siafarikas [11, Eq. (2.1)] gave more general bounds

{(\frac{x}{y})}^{α} \frac{\cosh (x)}{\cosh (y)} < \frac{I_{α} (x)}{I_{α} (y)} < {(\frac{\cosh (x)}{\cosh (y)})}^{1 / (2 α + 2)}, 0 < x < y, α > - \frac{1}{2} .

(17)

From (2) is immediately deduced

lim_{x \to 0^{+}} x^{- α} I_{α} (x) = \frac{2^{- α}}{Γ (α + 1)}

and in this way, out of (17), inequalities (16) can be reobtained.

Knowing the particular value of $I_{- 1 / 2} (z)$ , see (3), from the previous relations we can finally write

I_{α} (z) \leq L (α) z^{α} e^{z}, z > 0, α \geq - 1 / 2,

(18)

where $L (α) = 2^{- α} / Γ (α + 1)$ .

The following localization result has already been applied in the proof of Theorem 3. It is interesting in itself.

Theorem 5.

Let $W_{α}$ be defined by (4). Let $r > 0$ , $δ > 0$ be fixed. We consider the function $f \in E_{K}$ satisfying the condition

f (s) = 0, s \in (r - δ, r + δ) \cap [0, \infty) .

Then, it holds

W_{α} (f; r, t) = 𝒪 (\exp (- \frac{δ^{2}}{4 t})) (t \to 0^{+}) .

(19)

Proof.

By (18), we have

	$0$	$\leq \exp (- \frac{r^{2} + s^{2}}{4 t}) I_{α} (\frac{r s}{2 t})$
		$\leq L (α) \exp (- \frac{r^{2} + s^{2}}{4 t}) {(\frac{r s}{2 t})}^{α} \exp (\frac{r s}{2 t})$
		$= L (α) {(\frac{r s}{2 t})}^{α} \exp (- \frac{{(s - t)}^{2}}{4 t}) .$

Considering the conditions verified by the function $f$ , we obtain

| W_{α} (f; r, t) | \leq \frac{M L (α)}{{(2 t)}^{α + 1}} (I_{1} + I_{2}),

(20)

where

	$I_{1}$	$:= \int_{0}^{\max {0, r - δ}} s^{2 α + 1} \exp (- \frac{{(s - r)}^{2}}{4 t}) e^{K s^{2}} 𝑑 s,$
	$I_{2}$	$:= \int_{r + δ}^{\infty} s^{2 α + 1} \exp (- \frac{{(s - r)}^{2}}{4 t}) e^{K s^{2}} 𝑑 s .$

At first we estimate $I_{2}$ . A change of variable leads to

I_{2} = \int_{δ}^{\infty} {(s + r)}^{2 α + 1} \exp (- \frac{s^{2}}{4 t} + K {(s + r)}^{2}) 𝑑 s .

Next we consider $t > 0$ small enough, for example $t < {(8 K)}^{- 1}$ .

For each $m > 1$ , the derivative

		$\frac{\partial}{\partial s} [{(s + r)}^{m} \exp (- \frac{s^{2}}{4 t} + K {(s + r)}^{2})]$
		$= [m + (s + r) (- \frac{2 s}{4 t} + 2 K (s + r))] {(s + r)}^{m - 1} \exp (- \frac{s^{2}}{4 t} + K {(s + r)}^{2})$		(21)

has a unique positive zero, say $s (t)$ , where

s (t) = \frac{1}{2} {(4 K t - 1)}^{- 1} (r - 8 K r t - {({(8 K r t - r)}^{2} - 8 (4 K t - 1) (2 K r^{2} + m) t)}^{1 / 2}) .

As a function of $t$ , $s (t)$ tends to 0 when $t \to 0^{+}$ . Therefore, we can assume that $0 < s (t) < δ$ . Relation (4) and the above statements imply that the function $h_{r}$ ,

h_{r} (s) = {(s + r)}^{2 α + 1} \exp (- \frac{s^{2}}{4 t} + K {(s + r)}^{2})

is monotonically decreasing. Consequently, for $s \geq δ$ we have

	$I_{2}$	$\leq {(δ + r)}^{2 α + 3} \exp (- \frac{δ^{2}}{4 t} + K {(δ + r)}^{2}) \int_{δ}^{\infty} {(s + r)}^{- 2} 𝑑 s$
		$= {(δ + r)}^{2 α + 2} \exp (K {(δ + r)}^{2}) \exp (- \frac{δ^{2}}{4 t})$
		$= 𝒪 (\exp (- \frac{δ^{2}}{4 t})) (t \to 0^{+}) .$

We turn to the integral $I_{1}$ . In the case $r \leq δ$ there is nothing to prove. If $r > δ$ , we have

I_{1} \leq (r - δ) {(r - δ)}^{2 α + 1} \max_{0 \leq s \leq r - δ} \exp (- \frac{{(s - r)}^{2}}{4 t} + K s^{2}) .

Because $- {(s - r)}^{2} \leq - δ^{2}$ , for $0 \leq s \leq r - δ$ , we infer that

I_{1} \leq r^{2 α + 2} e^{K r^{2}} \exp (- \frac{δ^{2}}{4 t}) .

Combining the estimates of the integrals $I_{1}$ and $I_{2}$ , from (20) the statement (19) is proved. ∎

We note that Wachnicki [16, Lemma 2 and Lemma 3] showed that

lim_{t \to 0^{+}} \int_{0}^{r - δ} K_{α} (r, s, t) e^{K s^{2}} 𝑑 s = 0

uniformly on $r \in [α, β]$ , for $0 < α < δ < β$ , and

lim_{t \to 0^{+}} \int_{r + δ}^{+ \infty} K_{α} (r, s, t) e^{K s^{2}} 𝑑 s = 0

uniformly on $r \in (0, β]$ , for $β > 0$ , which implies the weaker estimate

W_{α} (f; r, t) = o (1) (t \to 0^{+}) .

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On Wachnicki’s Generalization of the Gauss–Weierstrass Integral

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On Wachnicki’s generalization of the Gauss–Weierstrass integral

Abstract.

1. Introduction

2. The operators

3. The asymptotic expansion

Proposition 1.

Proof.

Lemma 2.

Proof.

Theorem 3.

Proof.

Proposition 4.

Proof.

4. Localization result

Theorem 5.

Proof.

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