Journal of Function Spaces and Applications

VolumeΒ 2012, Article IDΒ 982360, 41 pages

http://dx.doi.org/10.1155/2012/982360

## Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145 Balikesir, Turkey

Received 9 July 2009; Accepted 13 December 2009

Academic Editor: V. M.Β Kokilashvili

Copyright Β© 2012 Ramazan AkgΓΌn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let and be, respectively, bounded and unbounded components of a plane curve satisfying Dini's smoothness condition. In addition to partial sum of Faber series of belonging to weighted Smirnov-Orlicz space (), we prove that interpolating polynomials and Poisson polynomials are near best approximant for . Also considering a weighted fractional moduli of smoothness, we obtain direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems, we prove direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces and .

#### 1. Introduction

Let and be, respectively, bounded and unbounded components of a closed rectifiable curve of complex plane . Without loss of generality we may suppose that . By Riemann conformal mapping theorem [1, page 26], if is connected Jordan curve that consists of more than one point, there exists a conformal mapping of complex unit disc onto . Let for a given . We denote by , , Smirnovβs classes of analytic functions satisfying where positive constant is independent of .

It is well known that for every and every function has a nontangential boundary values a.e. on , the boundary function belongs to Lebesgue space on . If , then is a Banach space with the norm Smirnov classes , , of analytic functions can be defined similarly and are fulfilling the same above properties to that of .

A smooth Jordan curve will be called *Dini-smooth*, if the function , the angle between the tangent line and the positive real axis expressed as a function of arclength , has modulus of continuity satisfying the Dini condition
A Jordan curve will be called *Radon curve*, if has bounded variation and it does not contain cusp point.

Main approximation problems in the spaces , , were dealt with by several mathematicians so far. Walsh and Russell gave [2] results in , , for algebraic polynomial approximation orders in case of analytic boundary. Alβper proved [3] direct and converse approximation theorems by algebraic polynomials in , , for Dini-smooth boundary. Kokilashvili improved [4] to Alβperβs direct and converse results of algebraic polynomial approximation, and then considering Regular curves that Cauchyβs Singular Integral Operator is bounded (corners are permitted), he obtained [5] improved direct and converse approximation theorems in Smirnov spaces , . Andersson proved [6] that Kokilashviliβs results also holds in . When the boundary is a regular curve, approximation of functions of , , by partial sum of Faber series was obtained by Israfilov in [7, 8]. These results are generalized to Muckenhoupt weighted Smirnovβs spaces in [9β12]. Approximation properties of Faber series in so-called weighted and unweighted Smirnov-Orlicz spaces are investigated in [13β20]. Most of the above results use the partial sums of Faber series as approximation tool. Interpolating polynomials [16] and Poisson polynomials [21] can be also considered as an approximating polynomial. In the present paper we obtain that in addition to partial sums of Faber series of belonging to weighted Smirnov-Orlicz space , interpolating polynomials and Poisson polynomials are near best approximant for . Also considering a weighted fractional moduli of smoothness, we obtain in Section 2 direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems we prove in Sectionβ3 direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces and .

Throughout the work, we will denote by , the constants that are different in different places.

#### 2. Approximation Theorems in Weighted Orlicz Space

A function is called *Young function* if is even, continuous, nonnegative in , increasing on such that
A Young function is said to satisfy ββ*condition* () if there is a constant such that
for all .

Two Young functions and are said to be *equivalent* if there are such that

A function is said to be *quasiconvex* if there exist a convex Young function and a constant such that
holds.

A nonnegative function defined on will be called weight if is measurable and a.e. positive. Let be a quasiconvex Young function. We denote by the class of Lebesgue measurable functions satisfying the condition
The linear span of the *weighted Orlicz class *, denoted by , becomes a normed space with the *Orlicz *norm
where , is the *complementary function* of .

If is quasiconvex and is its complementary function, then *Youngβs inequality* holds
For a quasiconvex function we define the indice of as
The indice was first defined and used by Gogatishvili and Kokilashvili in [22] to obtain weighted inequalities for maximal function. We note that the indice is much more convenient than Gustavsson and Peetreβs lower index and Boydβs upper index. If , then it can be easily seen that and becomes a Banach space with the Orlicz norm. The Banach space is called *weighted Orlicz space*.

We define the *Luxemburg *functional as
There exist [23, page 23] constants such that
For a weight we denote by the class of measurable functions on such that belongs to Lebesgue space on . We set for .

A 2-periodic weight function belongs to the Muckenhoupt class , , if with a finite constant independent of , where is any subinterval of and denotes the length of .

We will denote by a class of functions satisfying condition such that is quasiconvex for some .

In the present section we consider the trigonometric polynomial approximation problems for functions and its fractional derivatives in the spaces , , where . We prove a Jackson type direct theorem and a converse theorem of trigonometric approximation with respect to the fractional order moduli of smoothness in weighted Orlicz spaces with Muckenhoupt weights. In the particular case, we obtain a constructive characterization of Lipschitz class in these spaces.

In weighted Lebesgue and Orlicz spaces with Muckenhoupt weights, these results were investigated in [24β29]. For more general doubling weights, some of these problems were investigated in [30]. Jackson and converse inequalities were proved for Lebesgue spaces with Freud weight in [31]. For a general discussion of weighted polynomial approximation, we can refer to the books [32, 33].

Let , , , ,

be the *Fourier* and the *conjugate Fourier series* of , respectively. Putting in (2.12), we define for

For a given , assuming
we define th* fractional ** integral* of as [34, v.2, page 134]
where
as principal value.

Let be given. We define *fractional derivative* of a function , satisfying (2.15), as
provided the right hand side exists.

Setting , , , , and , we define where for and are Binomial coefficients, is Steklovβs mean operator, and is identity operator.

Theorem A (see [23, page 278, Theoremββ6.7.1]). *One suppose that is anyone of the operators , , and . If , , and , then there exists a constant such that
**
holds.*

Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from (2.20) that with a constant independent of .

By [35, page 14, (1.51)], there exists a constant depending only on such that we have and therefore provided , , where .

Let . For , we define the *fractional modulus of smoothness of index * for , as
where denotes the integer part of a real number .

Since the operator is bounded in , , where , we have by (2.24) that where the constant , dependent only on and .

*Remark 2.1. **The modulus of smoothness **, where **, **, *, βhas the following properties:(i)* is nonnegative, nondecreasing function of ** and subadditive,*(ii)*.*

For formulations of our results, we need several lemmas.

Lemma A (see [36]). *For , we suppose that*(i)*,
*(ii)*,
**
be two series in a Banach space . Let**
for . Then,
**
for some if and only if there exists a such that
**
where and are constants depending only on one another.*

If , , and , then from Theorem A(ii) and Abelβs transformation we get and therefore from (2.14) and(2.30) From the property it is known that for , .

Lemma 2.2. *Let , and . If , then there exists a constant independent of such that
**
holds.*

*Proof. *Without loss of generality one can assume that . Since
we have by (2.30) and Theorem A(iii) that
and from Lemma A
Hence from (2.33) and (2.31), we find
General case follows immediately from this.

Let . We denote by , , , the linear space of -periodic real valued functions such that .

Lemma 2.3. *Let . If with and , then for , there is a constant dependent only on and such that
**
holds.*

*Proof. *If , then from boundedness (see (2.21)) of the operator we get that
Let , . Since
we have
From (2.21) we get the boundedness of in and we have
From Lemma 2.2 we get
Now we have
Since
we get
Now we show that
For this we set
For given and , by Lemma 3 of [37], there exists a trigonometric polynomial such that
which by (2.7) this implies that
and hence we obtain
In this case from (2.40) we have
in norm. If , then
Hence,
Therefore,
Since
we obtain
Consequently,
and (2.48) holds. Now (2.47) and (2.48) imply the result.

Lemma 2.4. *Let , , , and . If , then
**
hold, where the constant is dependent only on and .*

*Proof. *First we prove that if , then
It is easily seen that if , then (2.61) holds. Now, we assume . In this case, putting , we have
Then,
and hence (2.61) holds. We note that if taking , for the remaining cases or or , it can easily be obtained from the last inequality that the required inequality (2.61) holds. Now we will show that if , , then
Putting
we have
Therefore,
Since
we obtain that
Using (2.61), (2.64), and Lemma 2.2, we get
which is the required result (2.60) for . On the other hand in case of the inequality (2.60) can be obtained by Marcinkiewicz Multiplier Theorem for where and .

*Definition 2.5. *For , and , the Peetre*-*functional is defined as

Proposition 2.6. *Let , , and . Then the functional in (2.71) and the modulus , are equivalent.*

*Proof. *If , then we have

Putting
we have
and hence
On the other hand, we find
Now, let . Then and
Since
we get
Taking into account
by a recursive procedure, we obtain

Now we can formulate the results.

Theorem 2.7. *Let and . If with , then there is a constant dependent only on and such that for **
holds.*

*Proof. *We put , . From Remark 2.1(i), (2.64), (2.71), Proposition 2.6, and (2.61), we get for every and

Theorem 2.8. *Let and . If with , then there is a constant dependent only on and such that for **
holds.*

*Proof. *Let be the best approximating polynomial of and let . Then,
By Lemma 2.4 we have
Since
we get
Fractional Bernstein inequality of Lemma 2.2 gives
Hence,
It is easily seen that
where
Therefore,
If we choose , then
Last two inequalities complete the proof.

From Theorems 2.7 and 2.8 we have the following corollaries.

Corollary 2.9. *Let and . If with and
**
then
**
hold.*

*Definition 2.10. *Let and *. If * and then for we set *. *

Corollary 2.11. *Let and . If , , and , then .*

Corollary 2.12. *Let and let , , where . Then the following conditions are equivalent:
*

Theorem 2.13. *Let , , where . If and
**
then
**
hold where the constant is dependent only on and .*

*Proof of Theorem 2.13. *The condition (2.98) and Lemma 2.3 implies that exist and . Since
we have for
On the other hand, we find
and Theorem 2.13 is proved.

As a corollary of Theorems 2.7, 2.8, and 2.13 we have the following.

Corollary 2.14. *Let , , , and
**
for some . In this case for , there exists a constant dependent only on , , and such that
**
hold.*

#### 3. Near Best Approximants in Weighted Smirnov-Orlicz Space

Let and be the conformal mappings of and onto the complement of , normalized by the conditions
respectively. We denote by and the inverse mappings of and , respectively, and . These mappings and have in some deleted neighborhood of the representations
Therefore, the functions
are analytic in and have, respectively, simple zero and zero of order 2 at . Hence they have expansions
where and are, respectively, Faber Polynomials of degree for continuums and , with the integral representations [38, pp. 35, 255]
We put
and correspond the series
with the function , that is,
This series is called the *Faber-Laurent* series of the function and the coefficients and are said to be the *Faber-Laurent coefficients* of . For further information about the Faber polynomials and Faber Laurent series, we refer to monographs [39, Chapter I, Section 6], [40, Chapter II], and [38].

It is well known that, using the Faber polynomials, approximating polynomials can be constructed [3]. The interpolating polynomials can also be used for this aim. In their work [41] under the assumption , , Shen and Zhong obtain a series of interpolation nodes in and show that interpolating polynomials and best approximating polynomial in , , have the same order of convergence. In [42] considering and choosing the interpolation nodes as the zeros of the Faber polynomials, Zhu obtain similar result.

In the above-cited works, does not admit corners, whereas many domains in the complex plain may have corners. When is a piecewise Vanishing Rotation curve [43] Zhong and Zhu show that the interpolating polynomials based on the zeros of the Faber polynomials converge to