Simultaneous approximation by Gauss–Weierstrass–Wachnicki operators

Abstract

In this note, we spotlight a generalization of Weierstrass integral operators introduced by Eugeniusz Wachnicki. The construction involves modified Bessel functions. The operators are correlated with diffusion equation. Our main result consists in obtaining the asymptotic expansion of derivatives of any order of Wachnicki’s operators. All coefficients are explicitly calculated, and distinct expressions are provided for analytical functions

Authors

Ulrich Abel
Fachbereich MND, Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Strasse 13, 61169, Friedberg, Germany

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; Gamma function; hypergeometric function; asymptotic expansion

Paper coordinates

U. Abel, O. Agratini, Simultaneous approximation by Gauss-Weierstrass-Wachnicki operators, Mediterr. J. Math., 19 (2022), art. no. 267, https://doi.org/10.1007/s00009-022-02194-0

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Mediterranean Journal of Mathematics

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Springer

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1660-5446

Online ISSN

1660-5454

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Simultaneous approximation by Gauss–Weierstrass–Wachnicki operators

Simultaneous approximation by Gauss–Weierstrass–Wachnicki operators

Ulrich Abel Fachbereich MND,
Technische Hochschule Mittelhessen,
Wilhelm-Leuschner-Straße 13,
61169 Friedberg, Germany
ulrich.abel@mnd.thm.de
ORCID: 0000-0003-1889-4850
 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
ORCID: 0000-0002-2406-4274
Abstract.

In this note we spotlight a generalization of Weierstrass integral operators introduced by Eugeniusz Wachnicki in the year 2000. The operators are correlated with diffusion equation. Our main result consists in the obtaining of the asymptotic expansion of derivatives of any order of Wachnicki’s operators. All coefficients are explicitly calculated and distinct expressions are provided for analytical functions.

Mathematics Subject Classification: 41A36, 41A60, 26A24.

Keywords and phrases: Gauss–Weierstrass operator, Bessel function, Gamma function, hypergeometric function, asymptotic expansion.

1. Introduction

The starting point of this note is the following generalized Gauss-Weierstrass transform

W(f;x,t) =12πtexp((xy)24t)f(y)𝑑y
=12πtf(xy)exp(y24t)𝑑y, (1)

where t>0 is a parameter, x and f: is chosen such that the integral exists and is finite. Actually, (1) represents the convolution of f with the density of the normal distribution (also called Gaussian distribution) having the expectation null and the variance 2t. W(f;,t) is a smoothed out version of f and physically W(;,t)Wt is correlated with a heat or diffusion equation for t times units. It is additive

Wt1Wt2=Wt1+t2,(t1,t2)(0,)×(0,×),

this being read as follows: diffusion for t1 time units and then t2 time units, is equivalent to diffusion for t1+t2 times units.

Examining relation (1) shows that Wt is translation invariant, meaning that the transform of f(x+a) is W(f;x+a,t).

We mention that Wt transform one can extend to t=0 by setting W0 to be the convolution with the Dirac delta function. This case does not come to our attention.

The special case t=1 can be interpreted this way. Taking in view the following formula

eu2=12πeuyey2/4𝑑y,u,

if we replace u with the formal differential operator D=ddx and we utilize the Lagrange shift operator

eyDf(x)=f(xy),x,

then we get

eD2f(x)=12πeyDf(x)ey2/4𝑑y=W(f;x,1)

which allows us to get the following formal expression for this particular transform

W1=eD2.

The operator eD2 is to be understood as acting on the signal f as

eD2f(x)=k=0D2kf(x)k!,x.

For more documentation on Wt transform, see Zayed’s book [5, Chapter 18: The Weierstrass Transform].

In 2000 Eugeniusz Wachnicki [6] defined and studied an integral operator representing a generalization of Wt which involves modified Bessel function. Our paper focuses bringing to light new properties of Weierstrass–Wachnicki integral operators. In the next section we present these operators pointing out some already established properties based on which we will highlight noteworthy features of them. The main results are set out in Section 3.

2. The operators

At the beginning we recall the modified Bessel function of the first kind and fractional order α>1, see [2, Chapter 10] or [8]. Using the traditional notation Iα, it is described by the series

Iα(z)=k=01k!Γ(k+α+1)(z2)2k+α, (2)

where Γ is the Gamma function. Iα forms a class of particular solutions of the following ordinary linear differential equation

z2w′′(z)+zw(z)(α2+z2)w(z)=0.

In the motivation of our results we will also use the low-order differentiation with respect to z described as follows

ddz(zαIα(z))=zαIα+1(z), (3)

see [8, Formula 03.02.20.0009.01].

Set +=[0,). For a fixed constant K0, we consider the space

EK={ f:+f is locally integrable and exists Mf0,
|f(s)|Mfexp(Ks2),s>0}.

The space can be endowed with the norm K as such

fK=sups+|f(s)|exp(Ks2).

We consider the operator Wα defined on EK by the following relation

Wα(f;r,t)=12t0rαsα+1exp(r2+s24t)Iα(rs2t)f(s)𝑑s, (4)

where α1/2, (r,t)(0,)×(0,×) and Iα is given at (2).

This operator was introduced in [6, Eq. (1)] with a minor modification of the domain EK in which the author inserted fC(+), the space of all real-valued continuous functions defined on +. Since Wαf, fEK, is well defined for any K>0, we can consider the domain of Wα as

E=K>0EK.

Wachnicki [6, Theorem 4] showed the convergence

limt0+Wα(f;r,t)=f(r),fEC(+),

uniformly on compact subintervals of (0,).

Also, in [6] the author specified that for α=1/2 the operator defined by (4) turns out to be the authentic Weierstrass operators. Because this statement was not accompanied by a proof, we insert it as a detail in our paper. More precisely we prove

W1/2f=Wf^,fEK, (5)

see (1), where f^(s)=f((sgns)s), s.

By using the hyperbolic cosine, the identity

I1/2(z)=2πzcosh(z)

takes place, see [2, page 443]. Further, for any (r,t)(0,)×(0,) we can write successively

W1/2(f;r,t)
=12t4tπ0exp(r2+s24t)cosh(rs2t)f(s)𝑑s
=12πt0exp(r2+s24t)(exp(rs2t)+exp(rs2t))f(s)𝑑s
=12πt0(exp((rs)24t)+exp((r+s)24t))f(s)𝑑s
=12πt(0exp((rs)24t)f(s)𝑑s+0exp((rs)24t))f(s)ds
=W(f^;r,t)

and the statement (5) is completed.

Wαf is intimately connected to a generalized heat equation having the following expression, see [3, Eq. (1.3)]

u(r,t)t=Δμu(r,t) (6)

where μ=2(α+1), α>1/2, and the operator

Δμ=2r2+μ1rr

is the Laplacian in radial coordinates when μ=n. It is usual to refer to t as time. If fEKC(), then Wαf with α=n/21 is a caloric function, this means it is a solution of equation (6) on a certain domain D,

D={(r,t):r>0, 0<t<14K}.

For the detailed proof see [3, pages 254-255].

Recently, these operators have come back to the attention of some authors. For example in [4] an extension of Wαf was achieved for continuous functions defined on the domain (0,)× and bounded by certain two-dimensional exponential functions. In [1], the authors obtained the asymptotic expansion of the operator Wα(f;r,t) as t0+, for functions fE being sufficiently smooth at a point r>0.

In order to obtain an autonomous exposure, in this preliminary section we recall some notions which will be used in establishing our results.

The factorial powers (falling and rising factorial, respectively) are marked as follows

uj¯=l=0j1(ul),uj¯=l=0j1(u+l),j.

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

For |z|<1 and generic parameters a,b,c, the Gauss hypergeometric function F12 is defined by

2F1(a,b;c;z)=j0aj¯bj¯cj¯zjj!, (7)

this series being convergent, see [2, Chapter 15]. Outside the disk with unit radius the function is defined as the analytic continuation with respect to z of this sum, with the parameters a,b,c held fixed. For particular case z=1, the following identity

2F1(a,b;c;1)=Γ(c)Γ(cab)Γ(ca)Γ(cb),Re(cab)>0, (8)

c0,1,2,, takes place.

3. Results

In the first stage we establish some technical formulas gathered in a few lemmas. Set 0={0}.

The first result is an explicit representation of (r)mWα(f;r,t) in terms of Wα+j(f;r,t), j0. We mention the identity that we will state contains a finite sum.

Lemma 1.

Let f belong to EK and let Wαf be defined by (1). For any m0,

(r)mWα(f;r,t)=j0(2j)!2jj!(m2j)rm2j(12tΔ)mjWα(f;r,t) (9)

holds, where Δ denoted the forward difference of step one with respect to α.

Proof.

If fEK, for s>0 we have

rWα(f;r,t)=12t0sα+1r(exp(r2+s24t)rαIα(rs2t))f(s)𝑑s.

Formula (3) yields the relation

rWα(f;r,t)=r2t(Wα+1(f;r,t)Wα(f;r,t))=r2tΔWα(f;r,t). (10)

We will realize the proof of identity (10) by mathematical induction. Obviously, the assertion is valid for m=0. Assuming that it is true for an arbitrary m, we show that it takes place for m+1. Relations (9) and (10) imply

(r)m+1Wα(f;r,t)=j0(2j)!2jj!(m2j)
×[(m2j)rm2j1(12tΔ)mj+rm2j+1(12tΔ)mj+1]Wα(f;r,t)
=rm+1(12tΔ)m+1Wα(f;r,t)
+j1[(2j2)!2j1(j1)!(m2j2)(m2j+2)+(2j)!2jj!(m2j)]
×rm2j+1(12tΔ)m+1jWα(f;r,t)
=rm+1(12tΔ)m+1Wα(f;r,t)
+j1(2j)!2jj![(m2j1)+(m2j)]rm+12j(12tΔ)m+1jWα(f;r,t).

Considering the elementary identity

(m2j1)+(m2j)=(m+12j),

we obtain that (9) is true for m+1 and the induction is completed. ∎

From [1, Theorem 3] we extract the following relation.

If f belonging to E is a real analytic function, then

Wα(f;r,t)n=0cn(α,f,r)tn(t0+),

where

cn(α,f,r)=4nn!r2n(w)n[wn+α(w)nf(rw)]|w=1. (11)

In the following Δ[f] stands for forward difference with the step h=1 of a function f.

Lemma 2.

Let f belong to EK and let Wαf be defined by (1). For any m0,

ΔmWα(f;r,t)n=mΔmcn(α,f,r)tn(t0+),

where the quantities cn(α,f,r) are described by (11). For mn,

Δmcn(α,f,r)=4n(nm)!r2n(w)nm[wn+α(w)nf(rw)]|w=1 (12)

and for m<n, Δncn(α,f,r)=0.

Proof.

Based on the m-th order forward difference with the step h=1, we get

Δmwm+α=k=0m(1)mk(mk)wn+α+k=wn+α(w1)m.

Consequently, taking into accounr (11), we can write

Δmcn(α,f,r)=4nn!r2n(w)n[wn+α(w1)m(w)nf(fw)]|w=1.

Since

(w)k(w1)m|w=1=0

for km, we deduce that Δmcn(α,f,r) is null for m>n. Otherwise, if mn, application of the Leibniz rule for differentiation yields

Δmcn(α,f,r)=4nn!r2n(nm)m!(w)nm[wn+α(w)nf(rw)]|w=1

which leads us at (12) and the proof of our lemma is completed. ∎

Our first main result can be read as follows.

Theorem 3.

Let f belong to EK and let Wαf be defined by (1). For any m0 the following relation

(r)mWα(f;r,t)n=0cn[m](α,f,r)tn(t0+), (13)

holds, where

cn[m](α,f,r) =1n!(2r)2n+mj0(2j)!4jj!(m2j)(w)n
×[wn+mj+α(w)n+mjf(rw)]|w=1. (14)
Proof.

Let m0 be fixed. Concatenating the conclusions of Lemmas 1 and 2, we can write

(r)mWα(f;r,t)
12mj0(2j)!j!(m2j)rm2jn=mjΔmjcn(α,f,r)tnm+j
=12mj0(2j)!j!(m2j)rm2jn=0Δmjcn+mj(α,f,r)tn(t0+). (15)

Further, for each n0 we process the coefficient of tn with the help of the relation (12) applied for Δmjcn+mj(α,f,r).

12mj0(2j)!j!(m2j)rm2jΔmjcn+mj(α,f,r)
=12mj0(2j)!j!(m2j)rm2j4n+mjn!r2(n+mj)
×(w)n[wn+mj+α(w)n+mjf(rw)]|w=1
:=cn[m](α,f,r),

see (3) Returning at (3), the statement (13) is proven. ∎

Remark. Choosing in (3) m=0, the sum is reduced to a single term (j=0) and cn[0](α,f,r) coincides with cn(α,f,r) defined at (11).

Next we focus on a special case concerning the function f, namely we consider that it is an analytic function on an open set D[0,). Consequently, for any x0D one can write

f(x)=n=0an(xx0)n

for x in a neighborhood of x0.

We mention that the above sum can be finite, see the case where f is a polynomial.

Since f is a linear combination of the monomials ei, ei(r)=ri (i0), and Wα(f;,) is a linear operator with respect to f, it is sufficient to consider the behavior of the operator on the functions ei.

To present our result, we first need to establish some identities that involve hypergeometric functions defined by (7) and (8).

Lemma 4.

Let m0 and x>m1, x, arbitrarily chosen.

(i) For any z, |z|<1/4, the identity

j0(2jj)(xmj)((m2j)/(mj))zj=(xm)2F1(m2,m+12;xm+1;4z), (16)

holds, where F12 is given by (7).

(ii) For z=1/4, the identity

j0(2j)!4jj!(m2j)xmj¯=2m(2x)m¯, (17)

holds.

Proof.

(i) The sum of the left-hand side of (16) is finite and has terms only for integer values of j satisfying 0jm/2, where stands for the floor function. Using well-known basic formulas

wp¯=Γ(w+1)Γ(wp+1)=w!(wp)!,wp¯=(1)p(w)p¯=(w+p1)p¯,

valid for any real or complex w, we can write the next set of identities

(2jj)(xmj)((m2j)/(mj))
=m2j¯j!m!xmj¯=(m)2j¯j!m!xm¯(xm+j)j¯
=xm¯m!2j(m2)j¯2j(m12)j¯(xm+1)j¯j!.

Since

xm¯m!=(xm),

from the above relations we obtain

j0(2jj)(xmj)((m2j)/(mj))zj=(xm)j0(m2j¯)(m12)j¯(xm+1)j¯(4z)jj!.

Considering (7), the relation (16) is proved.

(ii) Choosing z=1/4 in (16) and using (8), this identity can thus be rewritten

j0(2j)!4jj!(m2j)xmj¯=xm¯Γ(xm+1)Γ(x+12)Γ(xm2+1)Γ(xm2+12). (18)

The Legendre duplication formula for the Gamma function

Γ(2w)=1π 22w1Γ(w)Γ(w+12), 2w0,1,2,,

see, e.g., [7, Eq. 5.5.5], allows us to write

Γ(xm+1)Γ(x+12)Γ(xm2+1)Γ(xm2+12) =Γ(xm+1)Γ(2x+1)22xΓ(x+1)Γ(2xm+1)22x+m
=2m(2x)m¯xm¯.

Returning at (18), identity (17) is obtained and our lemma is completely motivated. ∎

In the light of those discussed before, exploring the case when f is an analytic function, we present

Theorem 5.

For any m0 and n0, the following identity

cn[m](α,ei,r)=(r)mcn(α,ei,r) (19)

holds, where cn[m](α,,r) and cn(α,,r) are respectively defined by (3) and (11).

Proof.

For m=0 we have highlighted this identity, see Remark. Further we consider m.

cn[m](α,ei,r)
=1n!(2r)2n+mj0(2j)!4jj!(m2j)(w)n[wn+mj+α(w)n+mjriwi/2]|w=1
=rin!(2r)2n+mj0(2j)!4jj!(m2j)(w)n[(i2)n+mj¯wi/2+α]|w=1
=rin!(2r)2n+mj0(2j)!4jj!(m2j)(i2)n+mj¯(i2+α)n¯. (20)

On the other hand,

(r)mcn(α,ei,r)
=4nn![(r)mrir2n](w)n[wn+α(w)nwi/2]|w=1
=4nn!(i2n)m¯ri2nm(w)n[wn+α(i2)n¯wi/2n]|w=1
=rin!22ni2n+m(i2n)m¯(i2)n¯(i2+α)n¯. (21)

To match the expressions (3) and (3) it remains to be shown that

2mj0(2j)!4jj!(m2j)(i2n)mj¯=(i2n)m¯. (22)

We have taken into account that

(i2)n+mj¯/(i2)n¯=(i2n)mj¯.

Now we turn to Lemma 4. By choosing in the identity (17) x:=i/2n, we obtain exactly (22). Thus relation (19) is proven. ∎

References

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[1] Abel, U., Agratini, O.: On Wachnicki’s generalization of the Gauss-Weierstrass integral, In: recent advances. In: Analysis, Mathematical (ed.) Anna Maria Candela. Mirella Cappelletti Montano, Elisabeta Mangino), Springer (2022) Google Scholar

[2] Abramowitz, M., Stegun, I.A. (Eds.): Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, Issued June 1964, Tenth Printing, with corrections (December 1972)

[3] Bragg, L.R.: The radial heat polynomials and related functions. Trans. Amer. Math. Soc. 119, 270–290 (1965) Article MathSciNet Google Scholar

[4] Krech, G., Krech, I.: On some bivariate Gauss-Weierstrass operators. Constr. Math. Anal. 2(2), 57–63 (2019) MathSciNet MATH Google Scholar

[5] Wachnicki, E.: On a Gauss-Weierstrass generalized integral. Rocznik Naukowo-Dydaktyczny Akademii Pedagogicznej W Krakowie, Prace Matematyczne 17, 251–263 (2000) MathSciNet MATH Google Scholar

[6] Zayed, A.I.: Handbook of function and generalized function transformations, 1st 1st Edition. CRC Press, London (1996) Google Scholar

[7] Digital Library of Mathematical Functions, https://dlmf.nist.gov Download references

2022

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