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

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

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

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

Another way of construction would be to replace x by z directly in the expression forThen,some standard reasonings lead,for all|z|≤M,to

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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 follows Now,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.

It is well-known by the Fubini type result that a sufficient condition for the commutativity is thatApplied to our case,we get

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

Then for all s≥1 and|z|≤r,is analytic,

and the following estimate hold:

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Then for all s≥1 and|z|≤r 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

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

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

for all s≥1 and|z|≤r with rA<1.

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

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which proves the theorem.

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

which also is equivalent with i.e.,equivalent with valid for all k≥2.Thus,(3.2)is proved.

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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

Proof.Byand since A|z|

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

and if we impose to have

Now,denoting firstly we will deduce the following recurrence formula

Indeed,we can write

is studied.Note that simple calculations lead us to

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

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this is equivalent with

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

Finally,we get

which proves the theorem.

Denoting =sup{|f(z)|;|z|≤r},in what follows we obtain the exact order of approximation by the complex Post-Widder operators

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 disk we getIndeed,supposing the contrary,it follows that z2 f (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

where

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

which implies that

for all s≥s0.

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For 1≤s≤s0−1,we evidently getwithf r>0(since for 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

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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]).

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