`Journal of Function Spaces and ApplicationsVolumeΒ 2012, Article IDΒ 982360, 41 pageshttp://dx.doi.org/10.1155/2012/982360`
Research Article

## 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

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 , , Smirnovβs 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. Alβper proved [3] direct and converse approximation theorems by algebraic polynomials in , , for Dini-smooth boundary. Kokilashvili improved [4] to Alβperβs direct and converse results of algebraic polynomial approximation, and then considering Regular curves that Cauchyβs Singular Integral Operator is bounded (corners are permitted), he obtained [5] improved direct and converse approximation theorems in Smirnov spaces , . Andersson proved [6] that Kokilashviliβs 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 Smirnovβs spaces in [9β12]. Approximation properties of Faber series in so-called weighted and unweighted Smirnov-Orlicz spaces are investigated in [13β20]. 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 Sectionβ3 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 ββ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 Youngβs 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 Peetreβs lower index and Boydβs 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 [24β29]. 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 Steklovβs mean operator, and is identity operator.

Theorem A (see [23, page 278, Theoremββ6.7.1]). 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 , , , βhas 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 Abelβs 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