Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 982360, 41 pagesdoi:10.1155/2012/982360

Research Article

Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Ramazan Akgün

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. Kokilashvili

Copyright © 2010 Ramazan Akgün. 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 , , Smirnov’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. Al’per proved [3] direct and converse approximation theorems by algebraic polynomials in , , for Dini-smooth boundary. Kokilashvili improved [4] to Al’per’s direct and converse results of algebraic polynomial approximation, and then considering Regular curves that Cauchy’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 Kokilashvili’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 Smirnov’s spaces in [9–12]. Approximation properties of Faber series in so-called weighted and unweighted Smirnov-Orlicz spaces are investigated in [13–20]. 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 Section 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   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 Young’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 Peetre’s lower index and Boyd’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 [24–29]. 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 Steklov’s mean operator, and is identity operator.

Theorem A (see [23, page 278, Theorem  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 , , ,  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 Abel’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 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 in the , norm.

A function is called a weight on , if is measurable and has measure zero. We denote by the linear space of Lebesgue measurable functions satisfying the condition for some .

The space becomes a Banach space with the Orlicz norm where is the complementary function of and

The Banach space is called weighted Orlicz space on .

For and let . For fixed , the set of all weights satisfying the relation is denoted by .

We denote by the set of all measurable functions such that belongs to Lebesgue space , , on .

Definition 3.1. Let be a weight on and let , , . The classes of functions and will be called weighted Smirnov-Orlicz classes with respect to domains and , respectively.

In this chapter, we prove that the convergence rate of the interpolating polynomials based on the zeros of the is the same with the best approximating algebraic polynomials in the weighted Smirnov-Orlicz class under the assumption that is a closed Radon curve. This means that interpolating polynomials based on the zeros of the Faber polynomials are near best approximant of belonging to weighted Smirnov-Orlicz class .

In the case that all of the zeros of the Faber polynomial are in , we denote by the interpolating polynomial for based on the zeros of .

Let . Then the functions and defined by are analytic in and , respectively, and .

We denote by the minimal error of approximation by polynomials of , where is the set of algebraic polynomials of degree not greater than .

Let be a rectifiable Jordan curve, , and let be Cauchy’s singular integral of at the point . The linear operator is called the Cauchy singular operator.

If one of the functions or has the nontangential limits a.e. on , then exists a.e. on and also the other one has the nontangential limits a.e. on . Conversely, if exists a.e. on , then both functions and have the nontangential limits a.e. on . In both cases, the formulae hold, and hence holds a.e. on (see, e.g., [1, page 431]).

Lemma 3.2. f is a regular curve, and , then for every one has where the constant depends only on and .

Proof. Assertion (3.20) immediately follows from modular inequality given in (7.5.13) of [23].

Theorem 3.3. If is a closed Radon curve, and , then for every one has where the constant depends only on and .

Proof. First of all we know [16] that all zeros of the Faber polynomials are in . Since interpolating operator is linear and corresponds by a polynomial of degree not more than , we need only to show that, for large values of , is uniformly bounded in weighted Smirnov-Orlicz class . We suppose that is the best approximating algebraic polynomial for in . In this case we have Since we assumed the interpolation nodes as the zeros of the Faber polynomials , using [39, page 59], we have and consequently By Lemma 3.2, we get We set , where is the exterior angle of the point . By the Radon assumption on we get . Then one can find for and therefore From the last inequality we obtain Since we obtain that is uniformly bounded in , namely, Therefore, we conclude that and interpolating polynomial is near best approximant for .

If is Dini-smooth, then [44] there exist constants and such that Similar inequalities hold also for and , in case of and , respectively.

We define Poisson polynomial for function

Theorem 3.4. If is a Dini-smooth curve, and , then for every one has where the constant depends only on and .

Proof. From (3.8) and (3.5), we have where and If is near best approximant for , we get Using we find Taking in the last inequality, the nontangential boundary values from inside of , and using (3.18), we have Since is analytic in , we have and taking nontangential limit in (3.42) we get and hence by transformation we obtain Since one has we can write From equality we have On the other hand, We denote by a subarc of with the center such that it has arc lenght . In this case and, by (1.3), Hence, Inequalities (3.46), (3.48), and (3.52) imply that For every , one has and therefore we get the required inequality of Theorem 3.4.

Theorem 3.4 signifies that Poisson polynomial is near best approximant for .

For , we set If and , then by Theorem A(ii) we have and consequently for any .

Definition 3.5. Let , , and . The function is called th modulus of smoothness of .

It can easily be verified that the function is continuous, nonnegative, subadditive and satisfy for .

Let be a Dini-smooth curve and be a weight on . We associate with the following two weights defined on by and let for . Then from (3.33), we have and for . Using the nontangential boundary values of and on , we define for .

We set where , , and is the set of rational functions of the form .

Now we can give several applications of approximation theorems of Section 2.

Theorem 3.6. Let be a Dini-smooth curve, and with . Then there is a constant such that for any natural number where and is the th partial sum of the Faber-Laurent series of .

Corollary 3.7. Let be a Dini-smooth curve, and with . Then there is a constant such that for every natural number where is the th partial sum of the Faber series of .

Corollary 3.8. Let be a Dini-smooth curve, and with . Then there is a constant such that for every natural number where is as in Theorem 3.6.

Theorem 3.9. Let be a Dini-smooth curve, and with . Then for there exists a constant such that hold.

Corollary 3.10. Under the conditions of Corollary 3.7, if then for and

Definition 3.11. Let and . If , then for we set

Corollary 3.12. Let and . If , , and , then .

By Corollaries 3.7 and 3.10 we have the constructive characterization of the class .

Corollary 3.13. Let and , where , be fulfilled. Then the following conditions are equivalent: (a). (b).

The inverse theorem for unbounded domains has the following form.

Theorem 3.14. Let be a Dini-smooth curve, and with . Then there is a constant such that for every natural number holds.

By the similar way to that of , we obtain the following corollaries.

Corollary 3.15. Under the conditions of Corollary 3.8, if then for and

Corollary 3.16. Under the conditions of Theorem 3.14, if then .

By Corollaries 3.8 and 3.15, we have the following.

Corollary 3.17. Let and the conditions of Theorem 3.14 be fulfilled. Then the following conditions are equivalent, (a), (b).

Before the proofs, we need some auxiliary lemmas.

Lemma 3.18. Let be a Dini-smooth curve, and with . Then, and for every .

Proof. Using , we can find a such that , where the inclusion maps being continuous (see, e.g., Lemma 2.13 of [20]). Since by [9], we get and . Using and boundedness of operator in , we obtain from (3.18) that

Lemma 3.19. Let and . Then there exists a constant such that for every natural number where and is th partial sum of the Taylor series of at the origin.

Proof. Using Theorem 2.7 this lemma can be proved by the same method of Theorem 3 of [45].

Let be the set of all polynomials (with no restrictions on the degree), and let be the set of traces of members of on . We define the operators and defined on as

Then it is readily seen that

If , then which, by (3.18), implies that

a.e. on .

Similarly taking from outside of the nontangential limit in the relation we get

a.e. on .

Since is bounded in , we have the following result.

Lemma 3.20. Let be a Dini-smooth curve, and with . Then the linear operators are bounded.

The set of trigonometric polynomials is dense in , which implies density of the algebraic polynomials in . Consequently, from Lemma 3.20, we can extend the operators and from to the spaces and as linear and bounded operators, respectively, and for the extensions and , we have the representations

Lemma 3.21. Let and with . Then, where and is the Poisson kernel.

Proof. There are numbers and such that Since [46, Theorem 10] is a bounded operator in for every , we have by Marcinkiewicz Interpolation Theorem From density of trigonometric polynomials in , we have density of the set of continuous functions on in . Consequently, there is a continuous function on such that, for given and , On the other hand, since the Poisson integral of a continuous function converges to it uniformly on [47, page 239], we have by (2.7) and for . Then, from (3.85), (3.86), and (3.87), we conclude that This completes the proof.

Theorem 3.22. Let be a Dini-smooth curve, and with . Then the linear operators are one-to-one and onto.

Proof. The proof we give, only for the operator . For the operator is the proof goes similarly. Let with the Taylor expansion Since is a Dini-smooth curve, the conditions , and are equivalent.Let . Since is the Poisson integral of its boundary function [48, page 41], we have and using Lemma 3.21, we get , as .Therefore, the boundedness of the operator implies that Since is uniformly convergent on , one has From the last equality and Lemma 3 of [39, page 43] we have and therefore On the other hand, applying (3.33), (2.7), and weighted version of Hölder’s inequality we obtain because .Using here the relation (3.92), we get and then by (3.95), for . If , then , and therefore . This means that the operator is one-to-one.Now we take a function and consider the function . The Cauchy type integral represents analytic functions and in and , respectively. Since , by Lemma 3.18, we have and moreover a.e. on . Since and , we have which proves that the coefficients , also become the Taylor coefficients of the function at the origin, that is, and also Hence the functions and have the same Faber coefficients , and therefore . This proves that the operator is onto.

Proof of Theorem 3.6. We prove that the rational function satisfies the required inequality of Theorem 3.6. This inequality is true if we can show that because a.e. on .First we prove (3.106). Let . Then and . According to (3.101), a.e. on . On the other hand, which implies the inequality a.e. on .Let . Using (3.7) and (3.110), we have Hence, taking the nontangential limit , inside of , we obtain a.e. on .Using (3.19), (3.110), Minkowski’s inequality, and the boundedness of , we get On the other hand, from the proof of Theorem 3.22 we know that the Faber-Laurent coefficients of the function and the Taylor coefficients of the function at the origin are the same. Then taking Lemma 3.19 into account, we conclude that and (3.106) is proved.  The proof of relation (3.107) goes similarly; we use the relations (3.6) and (3.108) instead of (3.7) and (3.110), respectively. Hence (3.19), (3.106), and (3.107) complete the proof.

Proof of Theorem 3.9. Let . Then we have . Since by Theorem 3.22 the operator is linear, bounded, one-to-one and onto, the operator is also linear and bounded. We take a as the best approximating algebraic polynomial to in , that is, Then, , and therefore because the operator is bounded.Theorem 2.8 and (3.116) imply that

References

G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, vol. 26 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1969.

J. L. Walsh and H. G. Russell, “Integrated continuity conditions and degree of approximation by polynomials or by bounded analytic functions,” Transactions of the American Mathematical Society, vol. 92, pp. 355–370, 1959.

S. Ya. Al'per, “Approximation in the mean of analytic functions of class ,” in Investigations on the Modern Problems of the Function Theory of a Complex Variable, pp. 273–286, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, Russia, 1960.

V. M. Kokilashvili, “Approximation in the mean of analytic functions of class ,” Soviet Mathematics. Doklady, vol. 8, pp. 1393–1397, 1967.

W. Kokilashvili, “A direct theorem on mean approximation of analytic functions by polynomials,” Soviet Mathematics. Doklady, vol. 10, pp. 411–414, 1969.

J.-E. Andersson, “On the degree of polynomial approximation in ,” Journal of Approximation Theory, vol. 19, no. 1, pp. 61–68, 1977.

D. M. Israfilov, “Approximate properties of the generalized Faber series in an integral metric,” Izvestiya Akademii Nauk Azerbaidzhanskoi SSR, vol. 2, pp. 10–14, 1987 (Russian).

A. Çavuş and D. M. Israfilov, “Approximation by Faber-Laurent rational functions in the mean of functions of class with ,” Approximation Theory and its Applications, vol. 11, no. 1, pp. 105–118, 1995.

D. M. Israfilov, “Approximation by -Faber polynomials in the weighted Smirnov class and the Bieberbach polynomials,” Constructive Approximation, vol. 17, no. 3, pp. 335–351, 2001.

D. M. Israfilov, “Approximation by -Faber-Laurent rational functions in the weighted Lebesgue spaces,” Czechoslovak Mathematical Journal, vol. 54, no. 3, pp. 751–765, 2004.

D. M. Israfilov and A. Guven, “Approximation in weighted Smirnov classes,” East Journal on Approximations, vol. 11, no. 1, pp. 91–102, 2005.

A. Guven and D. M. Israfilov, “Improved inverse theorems in weighted Lebesgue and Smirnov spaces,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 14, no. 4, pp. 681–692, 2007.

W. Kokilashvili, “On analytic functions of Smirnov-Orlicz classes,” Studia Mathematica, vol. 31, pp. 43–59, 1968.

A. Guven and D. M. Israfilov, “Polynomial approximation in Smirnov-Orlicz classes,” Computational Methods and Function Theory, vol. 2, no. 2, pp. 509–517, 2002.

D. M. Israfilov, B. Oktay, and R. Akgun, “Approximation in Smirnov-Orlicz classes,” Glasnik Matematički III, vol. 40, no. 1, pp. 87–102, 2005.

R. Akgün and D. M. Israfilov, “Approximation by interpolating polynomials in Smirnov-Orlicz class,” Journal of the Korean Mathematical Society, vol. 43, no. 2, pp. 413–424, 2006.

D. M. Israfilov and R. Akgün, “Approximation in weighted Smirnov-Orlicz classes,” Journal of Mathematics of Kyoto University, vol. 46, no. 4, pp. 755–770, 2006.

R. Akgün and D. M. Israfilov, “Polynomial approximation in weighted Smirnov-Orlicz space,” Proceedings of A. Razmadze Mathematical Institute, vol. 139, pp. 89–92, 2005.

D. M. Israfilov and R. Akgün, “Approximation by polynomials and rational functions in weighted rearrangement invariant spaces,” Journal of Mathematical Analysis and Applications, vol. 346, no. 2, pp. 489–500, 2008.

R. Akgün and D. M. Israfilov, “Approximation and moduli of smoothness of fractional order in Smirov-Orlicz spaces,” Glasnik Matematicki III, vol. 42, no. 1, pp. 121–136, 2008.

A. I. Shvai, “Approximation on analytic functions by Poisson polynomials,” Ukrainian Mathematical Journal, vol. 25, no. 6, pp. 710–713, 1973.

A. Gogatishvili and V. Kokilashvili, “Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces,” Georgian Mathematical Journal, vol. 1, no. 6, pp. 641–673, 1994.

I. Genebashvili, A. Gogatishvili, V. Kokilashvili, and M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, vol. 92 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, UK, 1998.

R. Akgün and D. M. Israfilov, “Simultaneous and converse approximation theorems in weighted Orlicz spaces,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 17, no. 1, pp. 13–28, 2010.

E. A. Gadjieva, Investigation of the Properties of Functions with Quasimonotone Fourier Coeffcients in Generalized Nikolskii-Besov Spaces, Author's Summary of Dissertation, Tbilisi, Georgia, 1986.

N. X. Ky, “On approximation by trigonometric polynomials in -spaces,” Studia Scientiarum Mathematicarum Hungarica, vol. 28, no. 1-2, pp. 183–188, 1993.

N. X. Ky, “Moduli of mean smoothness and approximation with -weights,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica, vol. 40, pp. 37–48, 1997.

L. Ephremidze, V. Kokilashvili, and Y. E. Yildirir, “On the inverse inequalities for trigonometric polynomial approximations in weighted Lorentz spaces,” Proceedings of A. Razmadze Mathematical Institute, vol. 144, pp. 132–136, 2007.

V. Kokilashvili and Y. E. Yildirir, “On the approximation in weighted Lebesgue spaces,” Proceedings of A. Razmadze Mathematical Institute, vol. 143, pp. 103–113, 2007.

G. Mastroianni and V. Totik, “Best approximation and moduli of smoothness for doubling weights,” Journal of Approximation Theory, vol. 110, no. 2, pp. 180–199, 2001.

Z. Ditzian and D. S. Lubinsky, “Jackson and smoothness theorems for Freud weights in ,” Constructive Approximation, vol. 13, no. 1, pp. 99–152, 1997.

Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9 of Springer Series in Computational Mathematics, Springer, New York, NY, USA, 1987.

H. N. Mhaskar, Introduction to the Theory of Weighted Polynomial Approximation, vol. 7 of Series in Approximations and Decompositions, World Scientific, River Edge, NJ, USA, 1996.

A. Zygmund, Trigonometric Series, vol. 1-2, Cambridge University Press, New York, NY, USA, 2nd edition, 1959.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.

I. Joó, “Saturation theorems for Hermite-Fourier series,” Acta Mathematica Hungarica, vol. 57, no. 1-2, pp. 169–179, 1991.

M. Khabazi, “The mean convergence of trigonometric Fourier series in weighted Orlicz classes,” Proceedings of A. Razmadze Mathematical Institute, vol. 129, pp. 65–75, 2002.

P. K. Suetin, Series of Faber Polynomials, vol. 1 of Analytical Methods and Special Functions, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1998.

D. Gaier, Lectures on Complex Approximation, Birkhäuser, Boston, Mass, USA, 1987.

V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable: Constructive Theory, Translated from the Russian by Scripta Technica, The M.I.T. Press, Cambridge, Mass, USA, 1968.

X. C. Shen and L. Zhong, “On Lagrange Interpolation in for ,” Advanced Mathematics, vol. 18, pp. 342–345, 1989 (Chinese).

L. Y. Zhu, “A new class of interpolation nodes,” Advances in Mathematics, vol. 24, no. 4, pp. 327–334, 1995 (Chinese).

L. Zhong and L. Zhu, “Convergence of interpolants based on the roots of Faber polynomials,” Acta Mathematica Hungarica, vol. 65, no. 3, pp. 273–283, 1994.

S. Warschawski, “Über das randverhalten der ableitung der abbildungsfunktion bei konformer abbildung,” Mathematische Zeitschrift, vol. 35, no. 1, pp. 321–456, 1932.

D. M. Israfilov and A. Guven, “Approximation by trigonometric polynomials in weighted Orlicz spaces,” Studia Mathematica, vol. 174, no. 2, pp. 147–168, 2006.

B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Transactions of the American Mathematical Society, vol. 165, pp. 207–226, 1972.

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, NY, USA, 3rd edition, 1987.

P. L. Duren, Theory of Hp Spaces, vol. 38 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1970.