Let

Let

It is well known that

A smooth Jordan curve

Main approximation problems in the spaces

Throughout the work, we will denote by

A function

Two Young functions

A function

A nonnegative function

If

We define the

A 2

We will denote by

In the present section we consider the trigonometric polynomial approximation problems for functions and its fractional derivatives in the spaces

In weighted Lebesgue and Orlicz spaces with Muckenhoupt weights, these results were investigated in [

Let

be the

For a given

Let

Setting

One suppose that

Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from (

By [

Let

Since the operator

For formulations of our results, we need several lemmas.

For

be two series in a Banach space

If

Let

Without loss of generality one can assume that

Let

Let

If

Let

First we prove that if

For

Let

If

Putting

Now we can formulate the results.

Let

We put

Let

Let

From Theorems

Let

Let

Let

Let

Let

The condition (

As a corollary of Theorems

Let

Let

It is well known that, using the Faber polynomials, approximating polynomials can be constructed [

In the above-cited works,

A function

The space

The Banach space

For

We denote by

Let

In this chapter, we prove that the convergence rate of the interpolating polynomials based on the zeros of the

In the case that all of the zeros of the

Let

We denote by

Let

If one of the functions

f

Assertion (

If

First of all we know [

If

We define Poisson polynomial for function

If

From (

Theorem

For

Let

It can easily be verified that the function

Let

We set

Now we can give several applications of approximation theorems of Section

Let

Let

Let

Let

Under the conditions of Corollary

Let

Let

By Corollaries

Let

The inverse theorem for unbounded domains has the following form.

Let

By the similar way to that of

Under the conditions of Corollary

Under the conditions of Theorem

By Corollaries

Let

Before the proofs, we need some auxiliary lemmas.

Let

Using

Let

Using Theorem

Let

Then it is readily seen that

If

a.e. on

Similarly taking from outside of

a.e. on

Since

Let

The set of trigonometric polynomials is dense in

Let

There are numbers

On the other hand, since the Poisson integral of a continuous function converges to it uniformly on

Let

The proof we give, only for the operator

Let

Therefore, the boundedness of the operator

Using here the relation (

Now we take a function

We prove that the rational function

First we prove (

Let

Using (

The proof of relation (

Let

Theorem