Sorin G.Galand Vijay Gupta

1 Department of Mathematics and Computer Science,University of Oradea,Str.Universitatii No.1,410087 Oradea,Romania

2 Department of Mathematics,Netaji Subhas Institute of Technology,Sector 3 Dwarka,New Delhi-110078,India

Abstract.In the present article,we deal with the so-called overconvergence phenomenon in C of a slightly modified Post-Widder operator of real variable,that is with the extension of its approximation properties from the real axis in the complex plane.In this sense,error estimates in approximation and a quantitative Voronovskaya-type asymptotic formula are established.

Key Words:Real and complex Post-Widder type operator,overconvergence phenomenon,approximation estimate,Voronovskaya-type result,exact error estimation.

1 Introduction

In the case of real functions,in e.g.,[2],Chapter 9,the slightly modified Post-Widder operator given by

is considered,where f:[0,+∞)→R,x>0.

It is clear that and passing here from the discrete parameter n to a continuous parameter s ≥1,we can consider the form(after the change of variable w=t/x)

Denoting ei(x)=xi,i=0,1,2,according to[2],Chapter 9(see,also[8])we have

and

Remark 1.1.In the paper[1](see also[9],pp.287),the original Post-Widder operator given by the formula

is studied.Note that simple calculations lead us to

In any case,the original Post-Widder operators Ln(f;x),do not reproduce the linear functions as the modified onesdo.

The overconvergence phenomenon,that is the extension of approximation properties of the positive and linear operators from the real axis in the complex plane,is an intensively studied topic in approximation theory.Thus,for example,the first author estimated the approximation properties of many complex operators in the book[3],while some other complex operators of Durrmeyer type have been discussed in,e.g.,[4,6,7]and[5],to mention only a few.

In the present paper,we study the approximation properties of a complex operatorof Post-Widder type.

2 The complex case

A way to construct a complex type Post-Widder approximation operator would be that in[1],namely to define for|z|≤1,the complex operators of convolution type

Similar reasonings with those in[1]immediately lead to the estimates

which unfortunately is not a good quantitative estimate because for s→+∞,s·ω1(f;1/s)does not converge to 0.Here ω1(f;δ)denotes the modulus of continuity of f on C.

However,by using a different method,in what follows we will prove a quantitative estimate forunder the hypothesis that f is an entire function. Also,a quantitative Voronovskaya-type asymptotic formula is obtained.

3 Main results

Firstly,we need the following auxiliary result.

Lemma 3.1.For any k∈N and s≥1,we have

Proof.We proceed by mathematical induction after k∈N,with s≥1 fixed,arbitrary.For k=1,it followsNow,supposing that the inequality is valid for k,we will prove that it is valid for k+1 too.Indeed,we get

But

Indeed,the above inequality is equivalent with

which after simple calculation reduces to the valid inequality

We complete the proof.

Now,we are in position to prove the following error estimate in approximation by

Theorem 3.1.Suppose that f is an entire function,i.e.,for all z∈C such that there exist M>0 and A∈(0,1),with the propertyfor all k=0,1,···,(which impliesfor all z∈C).Consider

and the following estimate hold:

for all|z|≤r,which shows that Ps(f;z)is analytic in|z|≤r.

Now,since we can write

if above the integral would commute with the infinite sum,then we would obtain

for all s≥1 and|z|≤r with rA<1.

Finally,taking into account the inequality(1.1)too,we obtain

which proves the theorem.

Suggested by the Voronovskaya-type result for Ps(f;x)in[2]given by

in what follows we will deduce the following quantitative estimate in the Voronovskayatype result for the complex Post-Widder operator.

Theorem 3.2.Suppose that f is an entire function,i.e.,for all z∈C such that there exist M>0 and A∈(0,1),with the propertyfor all k=0,1,···,(which impliesfor all z∈C).Consider

Then for all s≥1 and|z|≤r and the following estimate hold:

Proof.It is immediate that for all|z|≤r and s≥1,we can write

Indeed,we can write

Now,based on the relation(3.1)we will prove that

We use the mathematical induction.For k=1,we obtainSuppose now that(3.2)is valid for k and we will prove that it is valid for k+1 too.

By using(3.1),we get

and if we impose to have

this is equivalent with

which also is equivalent withi.e.,equivalent withvalid for all k≥2.Thus,(3.2)is proved.

Finally,we get

which proves the theorem.

Theorem 3.3.In the hypothesis of Theorem 3.2,if f is not a polynomial of degree ≤1 then we have

where the constant C in the equivalence depends only on f and r.

Proof.For all|z|≤r we can write the identity

Using the inequality

we get

Since f is not a polynomial of degree ≤1 in any diskwe getIndeed,supposing the contrary,it follows that z2f(z)=0,for all|z|≤r.

The last equality is equivalent to f(z)=0,for all,a contradiction with the hypothesis.Now by Theorem 3.2,for all s≥1,we have

But evidently that there exists s0>2 such that for all s≥s0we have

which implies that

for all s≥s0.

For 1≤s≤s0?1,we evidently getwithfr>0(sincefor a certain s is valid only for f a polynomial of degree ≤1,contradicting the hypothesis on f).

Therefore,finally we have

for all s≥1,where

which combined with Theorem 3.1,proves the desired conclusion.

Remark 3.1.We mention in passing that if we denote f(z)=U(x,y)+iV(x,y),x+iy,thencan be written under the form

fact which suggests to attach to any multivariate real valued function φ(x1,···,xp),the multivariate Post-Widder type operators

to study their convergence properties and possible applications to inverse Laplace transform(analogous with Chapter VII in[9]).