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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 982360, 41 pages
http://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

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 𝐺0 and 𝐺∞ be, respectively, bounded and unbounded components of a plane curve Ξ“ satisfying Dini's smoothness condition. In addition to partial sum of Faber series of 𝑓 belonging to weighted Smirnov-Orlicz space 𝐸𝑀,πœ” (𝐺0), we prove that interpolating polynomials and Poisson polynomials are near best approximant for 𝑓. 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 𝐸𝑀,πœ”(𝐺0) and 𝐸𝑀,πœ”(𝐺∞).

1. Introduction

Let 𝐺0 and 𝐺∞ be, respectively, bounded and unbounded components of a closed rectifiable curve Ξ“ of complex plane β„‚. Without loss of generality we may suppose that 0∈𝐺0. By Riemann conformal mapping theorem [1, page 26], if Ξ“ is connected Jordan curve that consists of more than one point, there exists a conformal mapping πœ‘0βˆΆπ”»β†’πΊ0 of complex unit disc π”»βˆΆ={π‘€βˆˆβ„‚βˆΆ|𝑀|=1} onto 𝐺0. Let π›Ύπ‘ŸβˆΆ=πœ‘0({π‘€βˆˆβ„‚βˆΆ|𝑀|=π‘Ÿ}) for a given π‘Ÿβˆˆ(0,1). We denote by 𝐸𝑝(𝐺0), 1β‰€π‘β‰€βˆž, Smirnov’s classes of analytic functions π‘“βˆΆπΊ0β†’β„‚ satisfyingsupπ‘Ÿβˆˆ(0,1)ξ€œπ›Ύπ‘Ÿ||||𝑓(𝑧)𝑝||||𝑑𝑧<𝑐,if1≀𝑝<∞,maxπ‘§βˆˆπΊ0||||𝑓(𝑧)<𝐢,if𝑝=∞,(1.1) where positive constant 𝑐 is independent of π‘Ÿ.

It is well known that 𝐸𝑝(𝐺0)βŠ‚πΈ1(𝐺0) for every 1≀𝑝<∞ and every function π‘“βˆˆπΈ1(𝐺0) has a nontangential boundary values a.e. on Ξ“, the boundary function belongs to Lebesgue space 𝐿1(Ξ“) on Ξ“. If 1≀𝑝<∞, then 𝐸𝑝(𝐺0) is a Banach space with the norm‖𝑓‖𝑝,Ξ“ξ‚»1∢=ξ€œ2πœ‹Ξ“||||𝑓(𝑧)𝑝||||𝑑𝑧1/𝑝.(1.2) Smirnov classes 𝐸𝑝(𝐺∞), 1β‰€π‘β‰€βˆž, of analytic functions π‘“βˆΆπΊβˆžβ†’β„‚ can be defined similarly and 𝐸𝑝(𝐺∞) are fulfilling the same above properties to that of 𝐸𝑝(𝐺0).

A smooth Jordan curve Ξ“ will be called Dini-smooth, if the function πœƒ(𝑠), the angle between the tangent line and the positive real axis expressed as a function of arclength 𝑠, has modulus of continuity Ξ©(πœƒ,𝑠) satisfying the Dini conditionξ€œπ›Ώ0Ξ©(πœƒ,𝑠)𝑠𝑑𝑠<∞,𝛿>0.(1.3) A Jordan curve Ξ“ will be called Radon curve, if πœƒ(𝑠) has bounded variation and it does not contain cusp point.

Main approximation problems in the spaces 𝐸𝑝(𝐺0), 1β‰€π‘β‰€βˆž, were dealt with by several mathematicians so far. Walsh and Russell gave [2] results in 𝐸𝑝(𝐺0), 1<𝑝<∞, for algebraic polynomial approximation orders in case of analytic boundary. Al’per proved [3] direct and converse approximation theorems by algebraic polynomials in 𝐸𝑝(𝐺0), 1<𝑝<∞, 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 𝐸𝑝(𝐺0), 1<𝑝<∞. Andersson proved [6] that Kokilashvili’s results also holds in 𝐸1(𝐺0). When the boundary is a regular curve, approximation of functions of 𝐸𝑝(𝐺0), 1<𝑝<∞, 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 𝑓 belonging to weighted Smirnov-Orlicz space 𝐸𝑀,πœ”(𝐺0), interpolating polynomials and Poisson polynomials are near best approximant for 𝑓. 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 𝐸𝑀,πœ”(𝐺0) and 𝐸𝑀,πœ”(𝐺∞).

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

2. Approximation Theorems in Weighted Orlicz Space

A function Ξ¦ is called Young function if Ξ¦ is even, continuous, nonnegative in ℝ, increasing on (0,∞) such thatΞ¦(0)=0,limπ‘₯β†’βˆžΞ¦(π‘₯)=∞.(2.1) A Young function Ξ¦ is said to satisfy Ξ”2  condition (Ξ¦βˆˆΞ”2) if there is a constant 𝑐>0 such thatΞ¦(2π‘₯)≀𝑐Φ(π‘₯)(2.2) for all π‘₯βˆˆβ„.

Two Young functions Ξ¦ and Ξ¦1 are said to be equivalent if there are 𝑐,𝐢>0 such thatΞ¦1(𝑐π‘₯)≀Φ(π‘₯)≀Φ1(𝐢π‘₯),βˆ€π‘₯>0.(2.3)

A function π‘€βˆΆ[0,∞)β†’[0,∞) is said to be quasiconvex if there exist a convex Young function Ξ¦ and a constant 𝑐β‰₯1 such thatΞ¦(π‘₯)≀𝑀(π‘₯)≀Φ(𝑐π‘₯),βˆ€π‘₯β‰₯0,(2.4) holds.

A nonnegative function πœ” defined on π–³βˆΆ=[0,2πœ‹] will be called weight if πœ” is measurable and a.e. positive. Let 𝑀 be a quasiconvex Young function. We denote by 𝐿𝑀,πœ”(𝖳) the class of Lebesgue measurable functions π‘“βˆΆπ–³β†’β„ satisfying the conditionξ€œπ–³π‘€ξ€·||||𝑓(π‘₯)πœ”(π‘₯)𝑑π‘₯<∞.(2.5) The linear span of the weighted Orlicz class 𝐿𝑀,πœ”(𝖳), denoted by 𝐿𝑀,πœ”(𝖳), becomes a normed space with the Orlicz norm‖𝑓‖𝑀,πœ”ξ‚»ξ€œβˆΆ=sup𝖳||||ξ€œπ‘“(π‘₯)𝑔(π‘₯)πœ”(π‘₯)𝑑π‘₯βˆΆπ–³ξ‚‹π‘€ξ€·||𝑔||ξ€Έξ‚Όπœ”(π‘₯)𝑑π‘₯≀1,(2.6) where 𝑀(𝑦)∢=supπ‘₯β‰₯0(π‘₯π‘¦βˆ’π‘€(π‘₯)),𝑦β‰₯0, is the complementary function of 𝑀.

If 𝑀 is quasiconvex and 𝑀 is its complementary function, then Young’s inequality holdsξ‚‹π‘₯𝑦≀𝑀(π‘₯)+𝑀(𝑦),π‘₯,𝑦β‰₯0.(2.7) For a quasiconvex function 𝑀 we define the indice 𝑝(𝑀) of 𝑀 as1𝑝(𝑀)∢=inf{π‘βˆΆπ‘>0,𝑀𝑝𝑝isquasiconvex},β€²(𝑀)∢=𝑝(𝑀).𝑝(𝑀)βˆ’1(2.8) The indice 𝑝(𝑀) was first defined and used by Gogatishvili and Kokilashvili in [22] to obtain weighted inequalities for maximal function. We note that the indice 𝑝(𝑀) is much more convenient than Gustavsson and Peetre’s lower index and Boyd’s upper index. If πœ”βˆˆπ΄π‘(𝑀), then it can be easily seen that 𝐿𝑀,πœ”(𝖳)βŠ‚πΏ1(𝖳) and 𝐿𝑀,πœ”(𝖳) becomes a Banach space with the Orlicz norm. The Banach space 𝐿𝑀,πœ”(𝖳) is called weighted Orlicz space.

We define the Luxemburg functional as‖𝑓‖(𝑀),πœ”ξ‚»ξ€œβˆΆ=inf𝜏>0βˆΆπ–³π‘€ξ‚΅||||𝑓(π‘₯)πœξ‚Άξ‚Όπœ”(π‘₯)𝑑π‘₯≀1.(2.9) There exist [23, page 23] constants 𝑐,𝐢>0 such that𝑐‖𝑓‖(𝑀),πœ”β‰€β€–π‘“β€–π‘€,πœ”β‰€πΆβ€–π‘“β€–(𝑀),πœ”.(2.10) For a weight πœ” we denote by 𝐿𝑝(𝖳,πœ”) the class of measurable functions on 𝖳 such that πœ”1/𝑝𝑓 belongs to Lebesgue space 𝐿𝑝(𝖳) on 𝖳. We set ‖𝑓‖𝑝,πœ”βˆΆ=β€–πœ”1/𝑝𝑓‖𝑝 for π‘“βˆˆπΏπ‘(𝖳,πœ”).

A 2πœ‹-periodic weight function πœ” belongs to the Muckenhoupt class 𝐴𝑝, 1<𝑝<∞, ifξ‚΅1||𝐽||ξ€œπ½1πœ”(π‘₯)𝑑π‘₯ξ‚Άξ‚΅||𝐽||ξ€œπ½πœ”βˆ’1/(π‘βˆ’1)ξ‚Ά(π‘₯)𝑑π‘₯π‘βˆ’1≀𝑐(2.11) with a finite constant 𝑐 independent of 𝐽, where 𝐽 is any subinterval of 𝖳 and |𝐽| denotes the length of 𝐽.

We will denote by π‘„πΆπœƒ2(0,1) a class of functions 𝑔 satisfying Ξ”2 condition such that π‘”πœƒ is quasiconvex for some πœƒβˆˆ(0,1).

In the present section we consider the trigonometric polynomial approximation problems for functions and its fractional derivatives in the spaces 𝐿𝑀,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀), where π‘€βˆˆπ‘„πΆπœƒ2(0,1). 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 𝑏0=0, π‘Žπ‘˜,π‘π‘˜βˆˆβ„, π‘π‘˜=(π‘Žπ‘˜βˆ’π‘–π‘π‘˜)/2, π‘βˆ’π‘˜=(π‘Žπ‘˜+π‘–π‘π‘˜)/2, 𝑐0=π‘Ž0/2𝑓(π‘₯)βˆΌβˆžξ“π‘˜=βˆ’βˆžπ‘π‘˜π‘’π‘–π‘˜π‘₯=βˆžξ“π‘˜=0ξ€·π‘Žπ‘˜cosπ‘˜π‘₯+π‘π‘˜ξ€Έ,𝑓sinπ‘˜π‘₯(2.12)(π‘₯)βˆΌβˆžξ“π‘˜=1ξ€·π‘Žπ‘˜sinπ‘˜π‘₯βˆ’π‘π‘˜ξ€Έcosπ‘˜π‘₯(2.13)

be the Fourier and the conjugate Fourier series of π‘“βˆˆπΏ1(𝖳), respectively. Putting π΄π‘˜(π‘₯)∢=π‘π‘˜π‘’π‘–π‘˜π‘₯ in (2.12), we define for 𝑛=0,1,2,…𝑆𝑛(𝑓)∢=𝑆𝑛(π‘₯,𝑓)∢=π‘›ξ“π‘˜=0ξ€·π΄π‘˜(π‘₯)+π΄βˆ’π‘˜ξ€Έ=π‘Ž(π‘₯)02+π‘›ξ“π‘˜=1ξ€·π‘Žπ‘˜cosπ‘˜π‘₯+π‘π‘˜ξ€Έ,𝑅sinπ‘˜π‘₯π‘›βŸ¨π›ΌβŸ©(𝑓,π‘₯)∢=π‘›ξ“π‘˜=0ξ‚΅ξ‚€π‘˜1βˆ’ξ‚π‘›+1π›Όξ‚Άξ€·π΄π‘˜(π‘₯)+π΄βˆ’π‘˜ξ€Έ(π‘₯),π›Όβˆˆβ„+,Ξ˜π‘šβŸ¨π›ΌβŸ©1∢=1βˆ’((π‘š+1)/(2π‘š+1))π‘Ÿπ‘…βŸ¨π›ΌβŸ©2π‘šβˆ’1((2π‘š+1)/(π‘š+1))π‘Ÿπ‘…βˆ’1π‘šβŸ¨π›ΌβŸ©,π‘š=1,2,3,….(2.14)

For a given π‘“βˆˆπΏ1(𝖳), assumingξ€œπ–³π‘“(π‘₯)𝑑π‘₯=0,(2.15) we define 𝛼th fractional (π›Όβˆˆβ„+) integral of 𝑓 as [34, v.2, page 134]𝐼𝛼(π‘₯,𝑓)∢=π‘˜βˆˆβ„€βˆ—π‘π‘˜(π‘–π‘˜)βˆ’π›Όπ‘’π‘–π‘˜π‘₯,(2.16) where(π‘–π‘˜)βˆ’π›Ό||π‘˜||∢=βˆ’π›Όπ‘’(βˆ’1/2)πœ‹π‘–π›Όsignπ‘˜(2.17) as principal value.

Let π›Όβˆˆβ„+ be given. We define fractional derivative of a function π‘“βˆˆπΏ1(𝖳), satisfying (2.15), as𝑓(𝛼)𝑑(π‘₯)∢=[𝛼]+1𝑑π‘₯[𝛼]+1𝐼1+π›Όβˆ’[𝛼](π‘₯,𝑓)(2.18) provided the right hand side exists.

Setting π‘₯,π‘‘βˆˆπ–³, π‘Ÿβˆˆβ„+, π‘€βˆˆπ‘„πΆπœƒ2(0,1), πœ”βˆˆπ΄π‘(𝑀), and π‘“βˆˆπΏπ‘€,πœ”(𝖳), we defineπœŽπ‘Ÿπ‘‘ξ€·π‘“(π‘₯)∢=πΌβˆ’πœŽπ‘‘ξ€Έπ‘Ÿ=𝑓(π‘₯)βˆžξ“π‘˜=0(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ‘Ÿπ‘˜βŽžβŽŸβŽŸβŽ 1(2𝑑)π‘˜ξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘π‘“ξ€·π‘₯+𝑒1+β‹―π‘’π‘˜ξ€Έπ‘‘π‘’1β‹―π‘‘π‘’π‘˜,(2.19) where (π‘Ÿπ‘˜)∢=π‘Ÿ(π‘Ÿβˆ’1)β‹―(π‘Ÿβˆ’π‘˜+1)/π‘˜! for π‘˜β‰₯1 and (π‘Ÿ0)∢=1 are Binomial coefficients, πœŽπ‘‘βˆ«π‘“(π‘₯)∢=(1/2𝑑)π‘‘βˆ’π‘‘π‘“(π‘₯+𝑒)𝑑𝑒 is Steklov’s mean operator, and 𝐼 is identity operator.

Theorem A (see [23, page 278, Theorem  6.7.1]). One suppose that 𝐿 is anyone of the operators 𝑆𝑛, πœŽβ„Ž, and 𝑓. If π‘€βˆˆπ‘„πΆπœƒ2(0,1), πœ”βˆˆπ΄π‘(𝑀), and π‘“βˆˆπΏπ‘€,πœ”(𝖳), then there exists a constant 𝑐>0 such that ξ€œπ–³π‘€ξ€·||||ξ€Έξ€œπΏπ‘“(𝑑)πœ”(𝑑)𝑑𝑑≀𝑐𝖳𝑀||||𝑓(𝑑)πœ”(𝑑)𝑑𝑑(2.20) holds.

Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from (2.20) that‖𝐿𝑓‖𝑀,πœ”β‰€π‘β€–π‘“β€–π‘€,πœ”(2.21) with a constant 𝑐>0 independent of 𝑓.

By [35, page 14, (1.51)], there exists a constant 𝑐 depending only on π‘Ÿ such that||||||βŽ›βŽœβŽœβŽπ‘Ÿπ‘˜βŽžβŽŸβŽŸβŽ ||||||β‰€π‘π‘˜π‘Ÿ+1,π‘˜=1,2,…,(2.22) we haveβˆžξ“π‘˜=0||||||βŽ›βŽœβŽœβŽπ‘Ÿπ‘˜βŽžβŽŸβŽŸβŽ ||||||<∞(2.23) and thereforeβ€–β€–πœŽπ‘Ÿπ‘‘π‘“β€–β€–π‘€,πœ”β‰€π‘β€–π‘“β€–π‘€,πœ”<∞(2.24) provided π‘“βˆˆπΏπ‘€,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀), where π‘€βˆˆπ‘„πΆπœƒ2(0,1).

Let π‘€βˆˆπ‘„πΆπœƒ2(0,1). For π‘Ÿβˆˆβ„+, we define the fractional modulus of smoothness of index π‘Ÿ for π‘“βˆˆπΏπ‘€,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀) asΞ©π‘Ÿπ‘€,πœ”(𝑓,𝛿)∢=sup0<β„Žπ‘–,𝑑<𝛿‖‖‖‖[π‘Ÿ]𝑖=1ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έξ€·πΌβˆ’πœŽπ‘‘ξ€Έπ‘Ÿβˆ’[π‘Ÿ]𝑓‖‖‖‖𝑀,πœ”,(2.25) where [π‘₯] denotes the integer part of a real number π‘₯.

Since the operator πœŽπ‘‘ is bounded in 𝐿𝑀,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀), where π‘€βˆˆπ‘„πΆπœƒ2(0,1), we have by (2.24) thatΞ©π‘Ÿπ‘€,πœ”β€–(𝑓,𝛿)≀𝑐𝑓‖𝑀,πœ”,(2.26) where the constant 𝑐>0, dependent only on π‘Ÿ and 𝑀.

Remark 2.1. The modulus of smoothness Ξ©π‘Ÿπ‘€,πœ”(𝑓,𝛿), where π‘Ÿβˆˆβ„+, π‘€βˆˆπ‘„πΆπœƒ2(0,1), πœ”βˆˆπ΄π‘(𝑀), π‘“βˆˆπΏπ‘€,πœ”(𝖳) has the following properties:(i)Ξ©π‘Ÿπ‘€,πœ”(𝑓,𝛿) is nonnegative, nondecreasing function of 𝛿β‰₯0 and subadditive,(ii)lim𝛿→0Ξ©π‘Ÿπ‘€,πœ”(𝑓,𝛿)=0.

For formulations of our results, we need several lemmas.

Lemma A (see [36]). For π›Όβˆˆβ„+, we suppose that(i)π‘Ž1+π‘Ž2+β‹―+π‘Žπ‘›+β‹―, (ii)π‘Ž1+2π›Όπ‘Ž2+β‹―+π‘›π›Όπ‘Žπ‘›+β‹―,
be two series in a Banach space (𝐡,β€–β‹…β€–). Letπ‘…π‘›βŸ¨π›ΌβŸ©βˆΆ=π‘›ξ“π‘˜=0ξ‚΅ξ‚€π‘˜1βˆ’ξ‚π‘›+1π›Όξ‚Άπ‘Žπ‘˜,π‘…π‘›βŸ¨π›ΌβŸ©βˆ—βˆΆ=π‘›ξ“π‘˜=0ξ‚΅ξ‚€π‘˜1βˆ’ξ‚π‘›+1π›Όξ‚Άπ‘˜π›Όπ‘Žπ‘˜(2.27) for 𝑛=1,2,…. Then, β€–β€–π‘…π‘›βŸ¨π›ΌβŸ©βˆ—β€–β€–β‰€π‘,𝑛=1,2,…(2.28) for some 𝑐>0 if and only if there exists a π‘…βˆˆπ΅ such that β€–β€–π‘…π‘›βŸ¨π›ΌβŸ©β€–β€–β‰€πΆβˆ’π‘…π‘›π›Ό,(2.29) where 𝑐 and 𝐢 are constants depending only on one another.

If π‘€βˆˆπ‘„πΆπœƒ2(0,1), πœ”βˆˆπ΄π‘(𝑀), and π‘“βˆˆπΏπ‘€,πœ”(𝖳), then from Theorem A(ii) and Abel’s transformation we getβ€–β€–π‘…π‘›βŸ¨π›ΌβŸ©β€–β€–(𝑓,β‹…)𝑀,πœ”β‰€π‘β€–π‘“β€–π‘€,πœ”,𝑛=1,2,3,…,π‘₯βˆˆπ–³(2.30) and therefore from (2.14) and(2.30)β€–β€–Ξ˜π‘›βŸ¨π›ΌβŸ©β€–β€–(𝑓,β‹…)𝑀,πœ”β‰€π‘β€–π‘“β€–π‘€,πœ”,𝑛=1,2,3,…,π‘₯βˆˆπ–³.(2.31) From the propertyΞ˜π‘šβŸ¨π›ΌβŸ©1(𝑓)(π‘₯)=βˆ‘2π‘šπ‘˜=π‘š+1ξ€Ί(π‘˜+1)π›Όβˆ’π‘˜π›Όξ€»2π‘šξ“π‘˜=π‘š+1ξ€Ί(π‘˜+1)π›Όβˆ’π‘˜π›Όξ€»π‘†π‘˜(π‘₯,𝑓),π‘₯βˆˆπ–³,π‘“βˆˆπΏ1(𝖳)(2.32) it is known thatΞ˜π‘šβŸ¨π›ΌβŸ©ξ€·π‘‡π‘šξ€Έ=π‘‡π‘š(2.33) for π‘‡π‘šβˆˆπ’―π‘š, π‘š=1,2,3,….

Lemma 2.2. Let π‘‡π‘›βˆˆπ’―π‘›,𝑛=1,2,3,…,π‘€βˆˆπ‘„πΆπœƒ2(0,1), and πœ”βˆˆπ΄π‘(𝑀). If π›Όβˆˆβ„+, then there exists a constant 𝑐>0 independent of n such that ‖‖𝑇𝑛(𝛼)‖‖𝑀,πœ”β‰€π‘π‘›π›Όβ€–β€–π‘‡π‘›β€–β€–π‘€,πœ”(2.34) holds.

Proof. Without loss of generality one can assume that ‖𝑇𝑛‖𝑀,πœ”=1. Since 𝑇𝑛=π‘›ξ“π‘˜=0ξ€·π΄π‘˜(π‘₯)+π΄βˆ’π‘˜ξ€Έ,𝑇(π‘₯)𝑛𝑛𝛼=π‘›ξ“π‘˜=1ξƒ¬ξ€·π΄π‘˜(π‘₯)βˆ’π΄βˆ’π‘˜ξ€Έ(π‘₯)𝑛𝛼,𝑇𝑛(𝛼)(𝑖𝑛)𝛼=π‘›ξ“π‘˜=1π‘˜π›Όξƒ¬ξ€·π΄π‘˜(π‘₯)βˆ’π΄βˆ’π‘˜ξ€Έ(π‘₯)𝑛𝛼(2.35) we have by (2.30) and Theorem A(iii) that β€–β€–β€–β€–π‘…π‘šβŸ¨π›ΌβŸ©ξƒ©ξ‚π‘‡π‘›π‘›π›Όξƒͺ‖‖‖‖𝑀,πœ”β‰€π‘π‘›π›Όβ€–β€–ξ‚π‘‡π‘›β€–β€–π‘€,πœ”β‰€π‘π‘›π›Όβ€–β€–π‘‡π‘›β€–β€–π‘€,πœ”=𝑐𝑛𝛼(2.36) and from Lemma A β€–β€–β€–β€–π‘…π‘šβŸ¨π›ΌβŸ©ξƒ©π‘‡π‘›(𝛼)(𝑖𝑛)𝛼ξƒͺ‖‖‖‖𝑀,πœ”β‰€π‘.(2.37) Hence from (2.33) and (2.31), we find ‖‖𝑇𝑛(𝛼)‖‖𝑀,πœ”=π‘›π›Όβ€–β€–β€–β€–Ξ˜π‘šβŸ¨π›ΌβŸ©ξƒ©π‘‡π‘›(𝛼)(𝑖𝑛)𝛼ξƒͺ‖‖‖‖𝑀,πœ”β‰€π‘π‘›π›Όβ€–β€–π‘‡π‘›β€–β€–π‘€,πœ”.(2.38) General case follows immediately from this.

Let π‘€βˆˆπ‘„πΆπœƒ2(0,1). We denote by π‘Šπ›Όπ‘€(𝖳,πœ”), 𝛼>0, πœ”βˆˆπ΄π‘(𝑀), the linear space of 2πœ‹-periodic real valued functions π‘“βˆˆπΏπ‘€,πœ”(𝖳) such that 𝑓(𝛼)βˆˆπΏπ‘€,πœ”(𝖳).

Lemma 2.3. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1). If π‘“βˆˆπ‘Šπ›Όπ‘€(𝖳,πœ”) with πœ”βˆˆπ΄π‘(𝑀) and 𝛼β‰₯0, then for 𝑛=0,1,2,…, there is a constant 𝑐>0 dependent only on 𝛼 and 𝑀 such that ‖‖𝑓(𝛼)(β‹…)βˆ’π‘†π‘›(𝛼)β€–β€–(β‹…,𝑓)𝑀,πœ”β‰€π‘πΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”(2.39) holds.

Proof. If 𝛼=0, then from boundedness (see (2.21)) of the operator 𝑆𝑛 we get that β€–β€–π‘“βˆ’π‘†π‘›π‘“β€–β€–π‘€,πœ”β‰€π‘πΈπ‘›(𝑓)𝑀,πœ”.(2.40) Let π‘Šπ‘›(𝑓)∢=π‘Šπ‘›βˆ‘(π‘₯,𝑓)∢=(1/(𝑛+1))2π‘›πœˆ=π‘›π‘†πœˆ(π‘₯,𝑓), 𝑛=0,1,2,…. Since π‘Šπ‘›ξ€·β‹…,𝑓(𝛼)ξ€Έ=π‘Šπ‘›(𝛼)(β‹…,𝑓),(2.41) we have ‖‖𝑓(𝛼)(β‹…)βˆ’π‘†π‘›(𝛼)β€–β€–(β‹…,𝑓)𝑀,πœ”β‰€β€–β€–π‘“(𝛼)(β‹…)βˆ’π‘Šπ‘›ξ€·β‹…,𝑓(𝛼)‖‖𝑀,πœ”+‖‖𝑆𝑛(𝛼)(β‹…,π‘Šπ‘›(𝑓))βˆ’π‘†π‘›(𝛼)β€–β€–(β‹…,𝑓)𝑀,πœ”+β€–β€–π‘Šπ‘›(𝛼)(β‹…,𝑓)βˆ’π‘†π‘›(𝛼)(β‹…,π‘Šπ‘›β€–β€–(𝑓))𝑀,πœ”βˆΆ=𝐼1+𝐼2+𝐼3.(2.42) From (2.21) we get the boundedness of π‘Šπ‘› in 𝐿𝑀,πœ”(𝖳) and we have 𝐼1≀‖‖𝑓(𝛼)(β‹…)βˆ’π‘†π‘›ξ€·β‹…,𝑓(𝛼)‖‖𝑀,πœ”+‖‖𝑆𝑛(β‹…,𝑓(𝛼))βˆ’π‘Šπ‘›(β‹…,𝑓(𝛼))‖‖𝑀,πœ”β‰€π‘πΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”+β€–β€–π‘Šπ‘›ξ€·β‹…,𝑆𝑛𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)‖‖𝑀,πœ”β‰€π‘πΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”.(2.43) From Lemma 2.2 we get 𝐼2≀𝑐𝑛𝛼‖‖𝑆𝑛⋅,π‘Šπ‘›ξ€Έ(𝑓)βˆ’π‘†π‘›β€–β€–(β‹…,𝑓)𝑀,πœ”,𝐼3≀𝑐(2𝑛)π›Όβ€–β€–π‘Šπ‘›(β‹…,𝑓)βˆ’π‘†π‘›ξ€·β‹…,π‘Šπ‘›ξ€Έβ€–β€–(𝑓)𝑀,πœ”β‰€π‘(2𝑛)π›ΌπΈπ‘›ξ€·π‘Šπ‘›ξ€Έ(𝑓)𝑀,πœ”.(2.44) Now we have ‖‖𝑆𝑛⋅,π‘Šπ‘›ξ€Έ(𝑓)βˆ’π‘†π‘›β€–β€–(β‹…,𝑓)𝑀,πœ”β‰€β€–β€–π‘†π‘›ξ€·β‹…,π‘Šπ‘›ξ€Έ(𝑓)βˆ’π‘Šπ‘›β€–β€–(β‹…,𝑓)𝑀,πœ”+β€–β€–π‘Šπ‘›β€–β€–(β‹…,𝑓)βˆ’π‘“(β‹…)𝑀,πœ”+‖‖𝑓(β‹…)βˆ’π‘†π‘›β€–β€–(β‹…,𝑓)𝑀,πœ”β‰€π‘πΈπ‘›ξ€·π‘Šπ‘›ξ€Έ(𝑓)𝑀,πœ”+𝑐𝐸𝑛(𝑓)𝑀,πœ”.(2.45) Since πΈπ‘›ξ€·π‘Šπ‘›ξ€Έ(𝑓)𝑀,πœ”β‰€π‘πΈπ‘›(𝑓)𝑀,πœ”,(2.46) we get ‖‖𝑓(𝛼)(β‹…)βˆ’π‘‡π‘›(𝛼)β€–β€–(β‹…,𝑓)𝑀,πœ”β‰€π‘πΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”+π‘π‘›π›ΌπΈπ‘›ξ€·π‘Šπ‘›ξ€Έ(𝑓)𝑀,πœ”+𝑐𝑛𝛼𝐸𝑛(𝑓)𝑀,πœ”+𝑐(2𝑛)π›ΌπΈπ‘›ξ€·π‘Šπ‘›ξ€Έ(𝑓)𝑀,πœ”β‰€π‘πΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”+𝐢𝑛𝛼𝐸𝑛(𝑓)𝑀,πœ”.(2.47) Now we show that 𝐸𝑛(𝑓)𝑀,πœ”β‰€π‘(𝑛+1)𝛼𝐸𝑛𝑓(𝛼)𝑀,πœ”.(2.48) For this we set π΄π‘˜(π‘₯,𝑓)∢=π‘Žπ‘˜cosπ‘˜π‘₯+π‘π‘˜sinπ‘˜π‘₯.(2.49) For given π‘“βˆˆπΏπ‘€,πœ”(𝖳) and πœ€>0, by Lemma 3 of [37], there exists a trigonometric polynomial 𝑇 such that ξ€œπ–³π‘€ξ€·||||𝑓(π‘₯)βˆ’π‘‡(π‘₯)πœ”(π‘₯)𝑑π‘₯<πœ€(2.50) which by (2.7) this implies that β€–π‘“βˆ’π‘‡β€–π‘€,πœ”<πœ€,(2.51) and hence we obtain 𝐸𝑛(𝑓)𝑀,πœ”βŸΆ0asπ‘›βŸΆβˆž.(2.52) In this case from (2.40) we have 𝑓(π‘₯)=βˆžξ“π‘˜=0π΄π‘˜(π‘₯,𝑓)(2.53) in ‖⋅‖𝑀,πœ” norm. If π‘˜=1,2,3,…, then π΄π‘˜(π‘₯,𝑓)=π΄π‘˜ξ‚€π‘₯+π›Όπœ‹ξ‚2π‘˜,𝑓cosπ›Όπœ‹2+π΄π‘˜ξ‚€π‘₯+π›Όπœ‹,𝑓2π‘˜sinπ›Όπœ‹2,π΄π‘˜ξ€·π‘₯,𝑓(𝛼)ξ€Έ=π‘˜π›Όπ΄π‘˜ξ‚€π‘₯+π›Όπœ‹ξ‚.2π‘˜,𝑓(2.54) Hence, βˆžξ“π‘˜=0π΄π‘˜(π‘₯,𝑓)=𝐴0(π‘₯,𝑓)+cosπ›Όπœ‹2βˆžξ“π‘˜=1π΄π‘˜ξ‚€π‘₯+π›Όπœ‹ξ‚2π‘˜,𝑓+sinπ›Όπœ‹2βˆžξ“π‘˜=1π΄π‘˜ξ‚€π‘₯+π›Όπœ‹,𝑓2π‘˜=𝐴0(π‘₯,𝑓)+cosπ›Όπœ‹2βˆžξ“π‘˜=1π‘˜βˆ’π›Όπ΄π‘˜ξ€·π‘₯,𝑓(𝛼)ξ€Έ+sinπ›Όπœ‹2βˆžξ“π‘˜=1π‘˜βˆ’π›Όπ΄π‘˜ξ‚€ξ‚π‘“π‘₯,(𝛼).(2.55) Therefore, 𝑓(π‘₯)βˆ’π‘†π‘›(π‘₯,𝑓)=cosπ›Όπœ‹2βˆžξ“π‘˜=𝑛+11π‘˜π›Όπ΄π‘˜ξ€·π‘₯,𝑓(𝛼)ξ€Έ+sinπ›Όπœ‹2βˆžξ“π‘˜=𝑛+11π‘˜π›Όπ΄π‘˜ξ‚€ξ‚π‘“π‘₯,(𝛼).(2.56) Since βˆžξ“π‘˜=𝑛+1π‘˜βˆ’π›Όπ΄π‘˜ξ€·π‘₯,𝑓(𝛼)ξ€Έ=βˆžξ“π‘˜=𝑛+1π‘˜βˆ’π›Όπ‘†ξ€Ίξ€·π‘˜ξ€·β‹…,𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)(ξ€Έβˆ’ξ€·π‘†β‹…)π‘˜βˆ’1ξ€·β‹…,𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)(=β‹…)ξ€Έξ€»βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)ξ€·π‘†π‘˜ξ€·β‹…,𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)ξ€Έ(β‹…)βˆ’(𝑛+1)βˆ’π›Όξ€·π‘†π‘›ξ€·β‹…,𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)(ξ€Έ,β‹…)βˆžξ“π‘˜=𝑛+1π‘˜βˆ’π›Όπ΄π‘˜ξ‚€ξ‚π‘“π‘₯,(𝛼)=βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)ξ‚€π‘†π‘˜ξ‚€ξ‚π‘“β‹…,(𝛼)ξ‚βˆ’ξ‚π‘“(𝛼)(⋅)βˆ’(𝑛+1)βˆ’π›Όξ‚€π‘†π‘›ξ‚€ξ‚π‘“β‹…,(𝛼)ξ‚βˆ’ξ‚π‘“(𝛼),(β‹…)(2.57) we obtain ‖‖𝑓(β‹…)βˆ’π‘†π‘›(β€–β€–β‹…,𝑓)𝑀,πœ”β‰€βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)β€–β€–π‘†π‘˜ξ€·β‹…,𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)(β€–β€–β‹…)𝑀,πœ”+(𝑛+1)βˆ’π›Όβ€–β€–π‘†π‘›ξ€·β‹…,𝑓(𝛼)ξ€Έβˆ’π‘“(𝛼)β€–β€–(β‹…)𝑀,πœ”+βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)β€–β€–π‘†π‘˜ξ‚€ξ‚π‘“β‹…,(𝛼)ξ‚βˆ’ξ‚π‘“(𝛼)(β€–β€–β‹…)𝑀,πœ”+(𝑛+1)βˆ’π›Όβ€–β€–π‘†π‘›ξ‚€ξ‚π‘“β‹…,(𝛼)ξ‚βˆ’ξ‚π‘“(𝛼)β€–β€–(β‹…)𝑀,πœ”ξƒ¬β‰€π‘βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)πΈπ‘˜(𝑓)𝑀,πœ”+(𝑛+1)βˆ’π›ΌπΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”ξƒ­ξƒ¬+πΆβˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)πΈπ‘˜(𝑓)𝑀,πœ”+(𝑛+1)βˆ’π›ΌπΈπ‘›ξ‚€ξ‚π‘“(𝛼)𝑀,πœ”ξƒ­.(2.58) Consequently, ‖‖𝑓(π‘₯)βˆ’π‘†π‘›β€–β€–(π‘₯,𝑓)𝑀,πœ”β‰€π‘πΈπ‘˜ξ€·π‘“(𝛼)𝑀,πœ”ξƒ¬βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)+(𝑛+1)βˆ’π›Όξƒ­+𝑐𝐸𝑛𝑓(𝛼)𝑀,πœ”ξƒ¬βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)+(𝑛+1)βˆ’π›Όξƒ­β‰€π‘πΈπ‘›ξ€·π‘“(𝛼)𝑀,πœ”ξƒ¬βˆžξ“π‘˜=𝑛+1(π‘˜βˆ’π›Όβˆ’(π‘˜+1)βˆ’π›Ό)+(𝑛+1)βˆ’π›Όξƒ­β‰€π‘(𝑛+1)𝛼𝐸𝑛𝑓(𝛼)𝑀,πœ”,(2.59) and (2.48) holds. Now (2.47) and (2.48) imply the result.

Lemma 2.4. Let π‘‡π‘›βˆˆπ’―π‘›, 𝑛=0,1,2,…, π‘€βˆˆπ‘„πΆπœƒ2(0,1), and πœ”βˆˆπ΄π‘(𝑀). If π›Όβˆˆβ„+, then Ω𝛼𝑀,πœ”ξ‚€π‘‡π‘›,πœ‹ξ‚β‰€π‘π‘›+1(𝑛+1)𝛼‖‖𝑇𝑛(𝛼)‖‖𝑀,πœ”(2.60) hold, where the constant 𝑐>0 is dependent only on 𝛼 and 𝑀.

Proof. First we prove that if 0<𝛼<𝛽, then Ω𝛽𝑀,πœ”ξ€·π‘‡π‘›ξ€Έ,⋅≀𝑐Ω𝛼𝑀,πœ”ξ€·π‘‡π‘›ξ€Έ,β‹….(2.61) It is easily seen that if 𝛼≀𝛽,𝛼,π›½βˆˆβ„€+, then (2.61) holds. Now, we assume 0<𝛼<𝛽≀1. In this case, putting 𝐾(π‘₯)∢=πœŽπ›Όπ‘‘π‘‡π‘›(π‘₯), we have πœŽπ‘‘π›½βˆ’π›ΌπΎ(π‘₯)=βˆžξ“π‘—=0(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ 1π›½βˆ’π›Ό(2𝑑)π‘—ξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘πΎξ€·π‘₯+𝑒1+⋯𝑒𝑗𝑑𝑒1⋯𝑑𝑒𝑗=βˆžξ“π‘—=0(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ 1π›½βˆ’π›Ό(2𝑑)π‘—Γ—ξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘βŽ‘βŽ’βŽ’βŽ£βˆžξ“π‘˜=0(βˆ’1)π‘˜βŽ›βŽœβŽœβŽπ›Όπ‘˜βŽžβŽŸβŽŸβŽ 1(2𝑑)π‘˜Γ—ξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘π‘‡π‘›ξ€·π‘₯+𝑒1+⋯𝑒𝑗+𝑒𝑗+1+⋯𝑒𝑗+π‘˜ξ€Έπ‘‘π‘’π‘—+1⋯𝑑𝑒𝑗+π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘‘π‘’1⋯𝑑𝑒𝑗=βˆžξ“βˆžπ‘—=0ξ“π‘˜=0(βˆ’1)𝑗+π‘˜βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Όπ‘˜βŽžβŽŸβŽŸβŽ ξ‚Έ1π›½βˆ’π›Ό(2𝑑)𝑗+π‘˜ξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘π‘‡π‘›ξ€·π‘₯+𝑒1+⋯𝑒𝑗+π‘˜ξ€Έπ‘‘π‘’1⋯𝑑𝑒𝑗+π‘˜ξ‚Ή=βˆžξ“πœ=0⎧βŽͺ⎨βŽͺβŽ©πœξ“πœ‡=0(βˆ’1)πœβˆ’πœ‡βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Όπœ‡βŽžβŽŸβŽŸβŽ 1π›½βˆ’π›Όπœβˆ’πœ‡(2𝑑)πœξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘π‘‡π‘›ξ€½π‘₯+𝑒1+β‹―π‘’πœξ€Ύπ‘‘π‘’1β‹―π‘‘π‘’πœβŽ«βŽͺ⎬βŽͺ⎭=βˆžξ“πœ=0(βˆ’1)𝜐1(2𝑑)πœξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘π‘‡π‘›ξ€·π‘₯+𝑒1+β€¦π‘’πœξ€Έπ‘‘π‘’1β‹―π‘‘π‘’πœπœξ“πœ‡=0βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ›Όπœ‡βŽžβŽŸβŽŸβŽ =π›½βˆ’π›Όπœβˆ’πœ‡βˆžξ“πœ=0(βˆ’1)πœβŽ›βŽœβŽœβŽπ›½πœβŽžβŽŸβŽŸβŽ 1(2𝑑)πœξ€œπ‘‘βˆ’π‘‘β‹―ξ€œπ‘‘βˆ’π‘‘π‘‡π‘›ξ€·π‘₯+𝑒1+β€¦π‘’πœξ€Έπ‘‘π‘’1β‹―π‘‘π‘’πœ=πœŽπ›½π‘‘π‘‡π‘›(π‘₯).(2.62) Then, β€–β€–πœŽπ›½π‘‘π‘‡π‘›β€–β€–π‘€,πœ”=β€–β€–πœŽπ‘‘π›½βˆ’π›ΌπΎβ€–β€–π‘€,πœ”β€–β€–πœŽβ‰€π‘π›Όπ‘‘π‘‡π‘›β€–β€–π‘€,πœ”,(2.63) and hence (2.61) holds. We note that if π‘Ÿ1,π‘Ÿ2βˆˆβ„€+,𝛼1,𝛽1∈(0,1) taking π›ΌβˆΆ=π‘Ÿ1+𝛼1,π›½βˆΆ=π‘Ÿ2+𝛽1 for the remaining cases π‘Ÿ1=π‘Ÿ2,𝛼1<𝛽1 or π‘Ÿ1<π‘Ÿ2,𝛼1<𝛽1 or π‘Ÿ1<π‘Ÿ2,𝛼1>𝛽1, it can easily be obtained from the last inequality that the required inequality (2.61) holds. Now we will show that if πœšβˆˆπ‘Šπ‘€(2π‘Ÿ)(𝖳,πœ”), π‘Ÿ=1,2,3,…, then Ξ©π‘Ÿπ‘€,πœ”(𝜚,𝛿)≀𝑐𝛿2π‘Ÿβ€–β€–πœš(2π‘Ÿ)‖‖𝑀,πœ”.(2.64) Putting 𝑔(π‘₯)∢=π‘Ÿξ‘π‘–=2ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έπœš(π‘₯),(2.65) we have ξ€·πΌβˆ’πœŽβ„Ž1𝑔(π‘₯)=π‘Ÿξ‘π‘–=1ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έπœš(π‘₯),π‘Ÿξ‘π‘–=1ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έ1𝜚(π‘₯)=2β„Ž1ξ€œβ„Ž1βˆ’β„Ž1(1𝑔(π‘₯)βˆ’π‘”(π‘₯+𝑑))𝑑𝑑=βˆ’8β„Ž1ξ€œβ„Ž10ξ€œπ‘‘0ξ€œπ‘’βˆ’π‘’π‘”β€²ξ…ž(π‘₯+𝑠)𝑑𝑠𝑑𝑒𝑑𝑑.(2.66) Therefore, β€–β€–β€–β€–π‘Ÿξ‘π‘–=1ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έβ€–β€–β€–β€–πœš(π‘₯)𝑀,πœ”=18β„Ž1ξƒ―ξ€œsup𝖳||||ξ€œβ„Ž10ξ€œπ‘‘0ξ€œπ‘’βˆ’π‘’π‘”ξ…žξ…ž||||||||ξ€œ(π‘₯+𝑠)𝑑𝑠𝑑𝑒𝑑𝑑𝑣(π‘₯)πœ”(π‘₯)𝑑π‘₯βˆΆπ–³ξ‚‹π‘€ξ€·||||≀1𝑣(π‘₯)πœ”(π‘₯)𝑑π‘₯≀18β„Ž1ξ€œβ„Ž10ξ€œπ‘‘0β€–β€–β€–12π‘’ξ€œ2π‘’π‘’βˆ’π‘’π‘”ξ…žξ…žβ€–β€–β€–(π‘₯+𝑠)𝑑𝑠𝑀,πœ”β‰€π‘π‘‘π‘’π‘‘π‘‘8β„Ž1ξ€œβ„Ž10ξ€œπ‘‘0‖‖𝑔2π‘’ξ…žξ…žβ€–β€–π‘€,πœ”π‘‘π‘’π‘‘π‘‘=π‘β„Ž21β€–β€–π‘”ξ…žξ…žβ€–β€–π‘€,πœ”.(2.67) Since π‘”ξ…žξ…ž(π‘₯)=π‘Ÿξ‘π‘–=2ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έπœšξ…žξ…ž(π‘₯),(2.68) we obtain that Ξ©π‘Ÿπ‘€,πœ”(𝜚,𝛿)≀sup0<β„Žπ‘–β‰€π›Ώπ‘–=1,2,…,π‘Ÿπ‘β„Ž21β€–β€–π‘”ξ…žξ…žβ€–β€–π‘€,πœ”=𝑐𝛿2β€–β€–β€–β€–π‘Ÿξ‘π‘–=2ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έπœšξ…žξ…žβ€–β€–β€–β€–(π‘₯)𝑀,πœ”=𝑐𝛿2sup0<β„Žπ‘–β‰€π›Ώπ‘–=2,…,π‘Ÿβ€–β€–β€–β€–π‘Ÿξ‘π‘–=2ξ€·πΌβˆ’πœŽβ„Žπ‘–ξ€Έπœšξ…žξ…žβ€–β€–β€–β€–(π‘₯)𝑀,πœ”=𝑐𝛿2Ξ©π‘Ÿβˆ’1𝑀,πœ”ξ€·πœšξ…žξ…žξ€Έ,𝛿≀𝑐𝛿4Ξ©π‘Ÿβˆ’2𝑀,πœ”ξ€·πœš(4)ξ€Έ,𝛿≀⋯≀𝐢𝛿2π‘Ÿβ€–β€–πœš(2π‘Ÿ)‖‖𝑀,πœ”.(2.69) Using (2.61), (2.64), and Lemma 2.2, we get Ω𝛼𝑀,πœ”ξ‚€π‘‡π‘›,πœ‹ξ‚π‘›+1≀𝑐Ω[𝛼]𝑀,πœ”ξ‚€π‘‡π‘›,πœ‹ξ‚ξ‚€πœ‹π‘›+1≀𝑐𝑛+12[𝛼]‖‖𝑇𝑛(2[𝛼])‖‖𝑀,πœ”β‰€π‘(𝑛+1)2[𝛼](𝑛+1)[𝛼]βˆ’(π›Όβˆ’[𝛼])‖‖𝑇𝑛(𝛼)‖‖𝑀,πœ”=𝑐(𝑛+1)𝛼‖‖𝑇𝑛(𝛼)‖‖𝑀,πœ”(2.70) which is the required result (2.60) for 𝛼β‰₯1. On the other hand in case of 0<𝛼<1 the inequality (2.60) can be obtained by Marcinkiewicz Multiplier Theorem for 𝐿𝑀,πœ”(𝖳) where π‘€βˆˆπ‘„πΆπœƒ2(0,1) and πœ”βˆˆπ΄π‘(𝑀).

Definition 2.5. For π‘“βˆˆπΏπ‘€,πœ”(𝖳),𝛿>0, and π‘Ÿ=1,2,3,…, the Peetre𝐾-functional is defined as 𝐾𝛿,𝑓;𝐿𝑀,πœ”(𝖳),π‘Šπ›Όπ‘€ξ€Έ(𝖳,πœ”)∢=infπ‘”βˆˆπ‘Šπ›Όπ‘€(𝖳,πœ”)ξ‚†β€–π‘“βˆ’π‘”β€–π‘€,πœ”β€–β€–π‘”+𝛿(π‘Ÿ)‖‖𝑀,πœ”ξ‚‡.(2.71)

Proposition 2.6. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1), πœ”βˆˆπ΄π‘(𝑀), and π‘“βˆˆπΏπ‘€,πœ”(𝖳). Then the 𝐾-functional 𝐾(𝛿2π‘Ÿ,𝑓;𝐿𝑀,πœ”(𝖳),π‘Šπ‘€2π‘Ÿ(𝖳,πœ”)) in (2.71) and the modulus Ξ©π‘Ÿπ‘€,πœ”(𝑓,𝛿),π‘Ÿ=1,2,3,…, are equivalent.

Proof. If β„Žβˆˆπ‘Šπ‘€2π‘Ÿ(𝖳,πœ”), then we have Ξ©π‘Ÿπ‘€,πœ”(𝑓,𝛿)β‰€π‘β€–π‘“βˆ’β„Žβ€–π‘€,πœ”+𝑐𝛿2π‘Ÿβ€–β€–β„Ž(2π‘Ÿ)‖‖𝑀,πœ”ξ€·π›Ώβ‰€π‘πΎ2π‘Ÿ,𝑓;𝐿𝑀,πœ”(𝖳),π‘Šπ‘€2π‘Ÿξ€Έ(𝖳,πœ”).(2.72)
Putting𝐿𝛿𝑓(π‘₯)∢=3π›Ώβˆ’3ξ€œπ›Ώ0ξ€œπ‘’0ξ€œπ‘‘βˆ’π‘‘π‘“(π‘₯+𝑠)𝑑𝑠𝑑𝑑𝑑𝑒,π‘₯βˆˆπ–³,(2.73) we have 𝑑2𝑑π‘₯2𝐿𝛿𝑐𝑓=𝛿2ξ€·πΌβˆ’πœŽπ›Ώξ€Έπ‘“,(2.74) and hence 𝑑2π‘Ÿπ‘‘π‘₯2π‘ŸπΏπ‘Ÿπ›Ώπ‘π‘“=𝛿2π‘Ÿξ€·πΌβˆ’πœŽπ›Ώξ€Έπ‘Ÿ,π‘Ÿ=1,2,3,….(2.75) On the other hand, we find ‖‖𝐿𝛿𝑓‖‖𝑀,πœ”β‰€3π›Ώβˆ’3ξ€œπ›Ώ0ξ€œπ‘’0β€–β€–πœŽ2𝑑𝑑𝑓‖‖𝑀,πœ”π‘‘π‘‘π‘‘π‘’β‰€π‘β€–π‘“β€–π‘€,πœ”.(2.76) Now, let π΄π‘Ÿπ›ΏβˆΆ=πΌβˆ’(πΌβˆ’πΏπ‘Ÿπ›Ώ)π‘Ÿ. Then π΄π‘Ÿπ›Ώπ‘“βˆˆπ‘Šπ‘€2π‘Ÿ(𝖳,πœ”) and ‖‖‖𝑑2π‘Ÿπ‘‘π‘₯2π‘Ÿπ΄π‘Ÿπ›Ώπ‘“β€–β€–β€–π‘€,πœ”β€–β€–β€–π‘‘β‰€π‘2π‘Ÿπ‘‘π‘₯2π‘ŸπΏπ‘Ÿπ›Ώπ‘“β€–β€–β€–π‘€,πœ”=𝑐𝛿2π‘Ÿβ€–β€–ξ€·πΌβˆ’πœŽπ›Ώξ€Έπ‘Ÿβ€–β€–π‘€,πœ”β‰€π‘π›Ώ2π‘ŸΞ©π‘Ÿπ‘€,πœ”(𝑓,𝛿).(2.77) Since πΌβˆ’πΏπ‘Ÿπ›Ώ=ξ€·πΌβˆ’πΏπ›Ώξ€Έπ‘Ÿβˆ’1𝑗=0𝐿𝑗𝛿,(2.78) we get β€–β€–ξ€·πΌβˆ’πΏπ‘Ÿπ›Ώξ€Έπ‘”β€–β€–π‘€,πœ”β€–β€–ξ€·β‰€π‘πΌβˆ’πΏπ›Ώξ€Έπ‘”β€–β€–π‘€,πœ”β‰€3π‘π›Ώβˆ’3ξ€œπ›Ώ0ξ€œπ‘’0β€–β€–ξ€·2π‘‘πΌβˆ’πœŽπ‘‘ξ€Έπ‘”β€–β€–π‘€,πœ”π‘‘π‘‘π‘‘π‘’β‰€π‘sup0<π‘‘β‰€π›Ώβ€–β€–ξ€·πΌβˆ’πœŽπ‘‘ξ€Έπ‘”β€–β€–π‘€,πœ”.(2.79) Taking into account β€–β€–π‘“βˆ’π΄π‘Ÿπ›Ώπ‘“β€–β€–π‘€,πœ”=β€–β€–ξ€·πΌβˆ’πΏπ‘Ÿπ›Ώξ€Έπ‘Ÿπ‘“β€–β€–π‘€,πœ”(2.80) by a recursive procedure, we obtain β€–β€–π‘“βˆ’π΄π‘Ÿπ›Ώπ‘“β€–β€–π‘€,πœ”β‰€π‘sup0<𝑑1β‰€π›Ώβ€–β€–ξ€·πΌβˆ’πœŽπ‘‘1ξ€Έξ€·πΌβˆ’πΏπ‘Ÿπ›Ώξ€Έπ‘Ÿβˆ’1𝑓‖‖𝑀,πœ”β‰€π‘sup0<𝑑1≀𝛿sup0<𝑑2β‰€π›Ώβ€–β€–ξ€·πΌβˆ’πœŽπ‘‘1ξ€Έξ€·πΌβˆ’πœŽπ‘‘2ξ€Έξ€·πΌβˆ’πΏπ‘Ÿπ›Ώξ€Έπ‘Ÿβˆ’2𝑓‖‖𝑀,πœ”β‰€β‹―β‰€π‘sup0<𝑑𝑖≀𝛿𝑖=1,2,…,π‘Ÿβ€–β€–β€–β€–π‘Ÿξ‘π‘–=1ξ€·πΌβˆ’πœŽπ‘‘π‘–ξ€Έβ€–β€–β€–β€–π‘“(π‘₯)𝑀,πœ”=π‘Ξ©π‘Ÿπ‘€,πœ”(𝑓,𝛿).(2.81)
Now we can formulate the results.

Theorem 2.7. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1) and π‘Ÿβˆˆβ„+. If π‘“βˆˆπΏπ‘€,πœ”(𝖳) with πœ”βˆˆπ΄π‘(𝑀), then there is a constant 𝑐>0 dependent only on π‘Ÿ and 𝑀 such that for 𝑛=0,1,2,3,…𝐸𝑛(𝑓)𝑀,πœ”β‰€π‘Ξ©π‘Ÿπ‘€,πœ”ξ‚€1𝑓,𝑛+1(2.82) holds.

Proof. We put π‘˜βˆ’1<π‘Ÿβ‰€π‘˜, π‘˜βˆˆβ„€+. From Remark 2.1(i), (2.64), (2.71), Proposition 2.6, and (2.61), we get for every π‘”βˆˆπ‘Š2π‘˜π‘€,πœ”(𝕋) and 𝑛=0,1,2,3,…𝐸𝑛(𝑓)𝑀,πœ”β‰€πΈπ‘›(π‘“βˆ’π‘”)𝑀,πœ”+𝐸𝑛(𝑔)𝑀,πœ”ξ‚ƒβ€–β‰€π‘π‘“βˆ’π‘”β€–π‘€,πœ”+(𝑛+1)βˆ’2π‘˜β€–β€–π‘”(2π‘˜)‖‖𝑀,πœ”ξ‚„ξ€·β‰€π‘πΎ(𝑛+1)βˆ’2π‘˜,𝑓;𝐿𝑀,πœ”(𝖳),π‘Šπ‘€2π‘˜ξ€Έ(𝖳,πœ”)β‰€π‘Ξ©π‘˜π‘€,πœ”ξ‚΅1𝑓,ξ‚Ά(𝑛+1)β‰€π‘Ξ©π‘Ÿπ‘€,πœ”ξ‚΅1𝑓,ξ‚Ά.(𝑛+1)(2.83)

Theorem 2.8. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1) and π‘Ÿβˆˆβ„+. If π‘“βˆˆπΏπ‘€,πœ”(𝖳) with πœ”βˆˆπ΄π‘(𝑀), then there is a constant 𝑐>0 dependent only on π‘Ÿ and 𝑀 such that for 𝑛=0,1,2,3,β€¦Ξ©π‘Ÿπ‘€,πœ”ξ‚€πœ‹π‘“,≀𝑐𝑛+1(𝑛+1)π‘Ÿπ‘›ξ“πœˆ=0(𝜈+1)π‘Ÿβˆ’1𝐸𝜈(𝑓)𝑀,πœ”(2.84) holds.

Proof. Let π‘‡π‘›βˆˆπ’―π‘› be the best approximating polynomial of π‘“βˆˆπΏπ‘€,πœ”(𝖳) and let π‘šβˆˆβ„€+. Then, Ξ©π‘Ÿπ‘€,πœ”ξ‚€πœ‹π‘“,𝑛+1β‰€Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘“βˆ’π‘‡2π‘š,πœ‹ξ‚Ά(𝑛+1)+Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹ξ‚Ά(𝑛+1)≀𝑐𝐸2π‘š(𝑓)𝑀,πœ”+Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹(ξ‚Ά.𝑛+1)(2.85) By Lemma 2.4 we have Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹ξ‚Άξ‚€1(𝑛+1)≀𝑐𝑛+1π‘Ÿβ€–β€–π‘‡2(π‘Ÿ)π‘šβ€–β€–π‘€,πœ”.(2.86) Since 𝑇2(π‘Ÿ)π‘š(π‘₯)=𝑇1(π‘Ÿ)(π‘₯)+π‘šβˆ’1ξ“πœˆ=0𝑇2(π‘Ÿ)𝜈+1(π‘₯)βˆ’π‘‡2(π‘Ÿ)πœˆξ‚‡(π‘₯),(2.87) we get Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹ξ‚Άβ‰€π‘(𝑛+1)(𝑛+1)π‘Ÿξƒ―β€–β€–π‘‡1(π‘Ÿ)‖‖𝑀,πœ”+π‘šβˆ’1ξ“πœˆ=0‖‖𝑇2(π‘Ÿ)𝜈+1βˆ’π‘‡2(π‘Ÿ)πœˆβ€–β€–π‘€,πœ”ξƒ°.(2.88) Fractional Bernstein inequality of Lemma 2.2 gives ‖‖𝑇2(π‘Ÿ)𝜈+1βˆ’π‘‡2(π‘Ÿ)πœˆβ€–β€–π‘€,πœ”β‰€π‘2πœˆπ‘Ÿβ€–β€–π‘‡2𝜈+1βˆ’π‘‡2πœˆβ€–β€–π‘€,πœ”β‰€π‘2πœˆπ‘Ÿ+1𝐸2𝜈(𝑓)𝑀,πœ”,‖‖𝑇1(π‘Ÿ)‖‖𝑀,πœ”=‖‖𝑇1(π‘Ÿ)βˆ’π‘‡0(π‘Ÿ)‖‖𝑀,πœ”β‰€π‘πΈ0(𝑓)𝑀,πœ”.(2.89) Hence, Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹ξ‚Άβ‰€π‘(𝑛+1)(𝑛+1)π‘Ÿξƒ―πΈ0(𝑓)𝑀,πœ”+π‘šβˆ’1ξ“πœˆ=02(𝜈+1)π‘ŸπΈ2𝜈(𝑓)𝑀,πœ”ξƒ°.(2.90) It is easily seen that 2(𝜈+1)π‘ŸπΈ2𝜈(𝑓)𝑀,πœ”β‰€π‘βˆ—2πœˆξ“πœ‡=2πœˆβˆ’1+1πœ‡π‘Ÿβˆ’1πΈπœ‡(𝑓)𝑀,πœ”,𝜈=1,2,3,…,(2.91) where π‘βˆ—=ξ‚»2π‘Ÿ+12,0<π‘Ÿ<1,2π‘Ÿ,π‘Ÿβ‰₯1.(2.92) Therefore, Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹ξ‚Άβ‰€π‘(𝑛+1)(𝑛+1)π‘ŸβŽ§βŽͺ⎨βŽͺ⎩𝐸0(𝑓)𝑀,πœ”+2π‘ŸπΈ1(𝑓)𝑀,πœ”+πΆπ‘šξ“2𝜈=1πœˆξ“πœ‡=2πœˆβˆ’1+1πœ‡π‘Ÿβˆ’1πΈπœ‡(𝑓)𝑀,πœ”βŽ«βŽͺ⎬βŽͺβŽ­β‰€π‘(𝑛+1)π‘ŸβŽ§βŽͺ⎨βŽͺ⎩𝐸0(𝑓)𝑀,πœ”+2π‘šξ“πœ‡=1πœ‡π‘Ÿβˆ’1πΈπœ‡(𝑓)𝑀,πœ”βŽ«βŽͺ⎬βŽͺβŽ­β‰€π‘(𝑛+1)π‘Ÿ2π‘šβˆ’1ξ“πœˆ=0(𝜈+1)π‘Ÿβˆ’1𝐸𝜈(𝑓)𝑀,πœ”.(2.93) If we choose 2π‘šβ‰€π‘›+1≀2π‘š+1, then Ξ©π‘Ÿπ‘€,πœ”ξ‚΅π‘‡2π‘š,πœ‹ξ‚Άβ‰€π‘(𝑛+1)(𝑛+1)π‘Ÿπ‘›ξ“πœˆ=0(𝜈+1)π‘Ÿβˆ’1𝐸𝜈(𝑓)𝑀,πœ”,𝐸2π‘š(𝑓)𝑀,πœ”β‰€πΈ2π‘šβˆ’1(𝑓)𝑀,πœ”β‰€π‘(𝑛+1)π‘Ÿπ‘›ξ“πœˆ=0(𝜈+1)π‘Ÿβˆ’1𝐸𝜈(𝑓)𝑀,πœ”.(2.94) Last two inequalities complete the proof.

From Theorems 2.7 and 2.8 we have the following corollaries.

Corollary 2.9. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1) and π‘Ÿβˆˆβ„+. If π‘“βˆˆπΏπ‘€,πœ”(𝖳) with πœ”βˆˆπ΄π‘(𝑀) and 𝐸𝑛(𝑓)𝑀,πœ”=π’ͺ(π‘›βˆ’πœŽ),𝜎>0,𝑛=1,2,…,(2.95) then Ξ©π‘Ÿπ‘€,πœ”βŽ§βŽͺ⎨βŽͺ⎩(𝑓,𝛿)=π’ͺ(π›ΏπœŽπ’ͺ𝛿);π‘Ÿ>𝜎,𝜎|||ξ‚€1log𝛿|||;π‘Ÿ=𝜎,π’ͺ(𝛿𝛼);π‘Ÿ<𝜎,(2.96) hold.

Definition 2.10. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1) and π‘Ÿβˆˆβ„+. If π‘“βˆˆπΏπ‘€,πœ”(𝖳) and πœ”βˆˆπ΄π‘(𝑀) then for 0<𝜎<π‘Ÿ we set Lip𝜎(π‘Ÿ,𝑀,πœ”)∢={π‘“βˆˆπΏπ‘€,πœ”(𝖳)βˆΆΞ©π‘Ÿπ‘€,πœ”(𝑓,𝛿)=π’ͺ(π›ΏπœŽ),𝛿>0}.

Corollary 2.11. Let π‘€βˆˆπ‘„πΆπœƒ2(0,1) and π‘Ÿβˆˆβ„+. If π‘“βˆˆπΏπ‘€,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀), 0<𝜎<π‘Ÿ and 𝐸𝑛(𝑓)𝑀,πœ”=π’ͺ(π‘›βˆ’πœŽ),𝑛=1,2,…, then π‘“βˆˆLip𝜎(π‘Ÿ,𝑀,πœ”).

Corollary 2.12. Let 0<𝜎<π‘Ÿ and let π‘“βˆˆπΏπ‘€,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀), where π‘€βˆˆπ‘„πΆπœƒ2(0,1). Then the following conditions are equivalent: (a)π‘“βˆˆLip𝜎(π‘Ÿ,𝑀,πœ”),(b)𝐸𝑛(𝑓)𝑀,πœ”=π’ͺ(π‘›βˆ’πœŽ),𝑛=1,2,….(2.97)

Theorem 2.13. Let π‘“βˆˆπΏπ‘€,πœ”(𝖳), πœ”βˆˆπ΄π‘(𝑀), where π‘€βˆˆπ‘„πΆπœƒ2(0,1). If π›Όβˆˆ(0,∞) and βˆžξ“πœˆ=1πœˆπ›Όβˆ’1𝐸𝜈(𝑓)𝑀,πœ”<∞,(2.98) then 𝐸𝑛𝑓(𝛼)𝑀,πœ”ξƒ©β‰€π‘(𝑛+1)𝛼𝐸𝑛(𝑓)𝑀,πœ”+βˆžξ“πœˆ=𝑛+1πœˆπ›Όβˆ’1𝐸𝜈(𝑓)𝑀,πœ”ξƒͺ(2.99) hold where the constant 𝑐>0 is dependent only on 𝛼 and 𝑀.

Proof of Theorem 2.13. The condition (2.98) and Lemma 2.3 implies that 𝑓(𝛼) exist and 𝑓(𝛼)βˆˆπΏπ‘€,πœ”(𝖳). Since ‖‖𝑓(𝛼)βˆ’π‘†π‘›ξ€·π‘“(𝛼)‖‖𝑀,πœ”β‰€β€–β€–π‘†2π‘š+2𝑓(𝛼)ξ€Έβˆ’π‘†π‘›ξ€·π‘“(𝛼)‖‖𝑀,πœ”+βˆžξ“π‘˜=π‘š+2‖‖𝑆2π‘˜+1𝑓(𝛼)ξ€Έβˆ’π‘†2π‘˜ξ€·π‘“(𝛼)‖‖𝑀,πœ”,(2.100) we have for 2π‘š<𝑛<2π‘š+1‖‖𝑆2π‘š+2𝑓(𝛼)ξ€Έβˆ’π‘†π‘›ξ€·π‘“(𝛼)‖‖𝑀,πœ”β‰€π‘2(π‘š+2)𝛼𝐸𝑛(𝑓)𝑀,πœ”β‰€πΆ(𝑛+1)𝛼𝐸𝑛(𝑓)𝑀,πœ”.(2.101) On the other hand, we find βˆžξ“π‘˜=π‘š+2‖‖𝑆2π‘˜+1𝑓(𝛼)ξ€Έβˆ’π‘†2π‘˜ξ€·π‘“(𝛼)‖‖𝑀,πœ”β‰€π‘βˆžξ“π‘˜=π‘š+22(π‘˜+1)𝛼𝐸2π‘˜(𝑓)𝑀,πœ”β‰€πΆβˆžξ“2π‘˜=π‘š+2π‘˜ξ“πœ‡=2π‘˜βˆ’1+1πœ‡π›Όβˆ’1πΈπœ‡(𝑓)𝑀,πœ”=π‘βˆžξ“πœˆ=2π‘š+1+1πœˆπ›Όβˆ’1𝐸𝜈(𝑓)𝑀,πœ”β‰€π‘βˆžξ“πœˆ=𝑛+1πœˆπ›Όβˆ’1𝐸𝜈(𝑓)𝑀,πœ”,(2.102) 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 π‘“βˆˆπ‘Šπ›Όπ‘€(𝖳,πœ”), πœ”βˆˆπ΄π‘(𝑀), π‘Ÿβˆˆ(0,∞), and βˆžξ“πœˆ=1πœˆπ›Όβˆ’1𝐸𝜈(𝑓)𝑀,πœ”<∞(2.103) for some 𝛼>0. In this case for 𝑛=0,1,2,…, there exists a constant 𝑐>0 dependent only on 𝛼, π‘Ÿ, and 𝑀 such that Ξ©π‘Ÿπ‘€,πœ”ξ‚€π‘“(𝛼),πœ‹ξ‚β‰€π‘π‘›+1(𝑛+1)π‘Ÿπ‘›ξ“πœˆ=0(𝜈+1)𝛼+π‘Ÿβˆ’1𝐸𝜈(𝑓)𝑀,πœ”+π‘βˆžξ“πœˆ=𝑛+1πœˆπ›Όβˆ’1𝐸𝜈(𝑓)𝑀,πœ”(2.104) hold.

3. Near Best Approximants in Weighted Smirnov-Orlicz Space

Let 𝑀=πœ‘(𝑧) and 𝑀=πœ‘1(𝑧) be the conformal mappings of 𝐺∞ and 𝐺0 onto the complement π”»βˆž of 𝔻, normalized by the conditionsπœ‘(∞)=∞,limπ‘§β†’βˆžπœ‘πœ‘(𝑧)/𝑧>0,1(0)=∞,lim𝑧→0π‘§πœ‘1(𝑧)>0,(3.1) respectively. We denote by πœ“ and πœ“1 the inverse mappings of πœ‘ and πœ‘1, respectively, and π•‹βˆΆ=πœ•π”». These mappings πœ“ and πœ“1 have in some deleted neighborhood of ∞ the representationsπœ“(𝑀)=𝛼𝑀+𝛼0+βˆžξ“π‘˜=1π›Όπ‘˜π‘€π‘˜,𝛼>0,πœ“1(𝑀)=βˆžξ“π‘™=1𝛽𝑙𝑀𝑙,𝛽1>0.(3.2) Therefore, the functionsπœ“β€²(𝑀)πœ“(𝑀)βˆ’π‘§,π‘§βˆˆπΊ0,πœ“ξ…ž1(𝑀)πœ“1(𝑀)βˆ’π‘§,π‘§βˆˆπΊβˆž(3.3) are analytic in π”»βˆž and have, respectively, simple zero and zero of order 2 at ∞. Hence they have expansionsπœ“ξ…ž(𝑀)=πœ“(𝑀)βˆ’π‘§βˆžξ“π‘˜=0πΉπ‘˜(𝑧)π‘€π‘˜+1,π‘§βˆˆπΊ0,π‘€βˆˆπ”»βˆž,πœ“ξ…ž1(𝑀)πœ“1=(𝑀)βˆ’π‘§βˆžξ“π‘˜=1ξ‚πΉπ‘˜(1/𝑧)π‘€π‘˜+1,π‘§βˆˆπΊβˆž,π‘€βˆˆπ”»βˆž,(3.4) where πΉπ‘˜(𝑧) and ξ‚πΉπ‘˜(1/𝑧) are, respectively, Faber Polynomials of degree π‘˜ for continuums 𝐺0 and ℂ⧡𝐺0, with the integral representations [38, pp. 35, 255]πΉπ‘˜1(𝑧)=ξ€œ2πœ‹π‘–π•‹π‘€π‘˜πœ“ξ…ž(𝑀)πœ“(𝑀)βˆ’π‘§π‘‘π‘€,π‘§βˆˆπΊ0,ξ‚πΉπ‘˜ξ‚€1𝑧=1ξ€œ2πœ‹π‘–π•‹π‘€π‘˜πœ“ξ…ž1(𝑀)πœ“1(𝑀)βˆ’π‘§π‘‘π‘€,π‘§βˆˆπΊβˆž,𝐹(3.5)π‘˜(𝑧)=πœ‘π‘˜1(𝑧)+ξ€œ2πœ‹π‘–Ξ“πœ‘π‘˜(𝜍)πœβˆ’π‘§π‘‘πœ,π‘§βˆˆπΊβˆžξ‚πΉ,π‘˜=0,1,2,…,(3.6)π‘˜ξ‚€1𝑧=πœ‘π‘˜11(𝑧)βˆ’ξ€œ2πœ‹π‘–Ξ“πœ‘π‘˜1(𝜍)πœβˆ’π‘§π‘‘πœ,π‘§βˆˆπΊ0⧡{0}.(3.7) We putπ‘Žπ‘˜βˆΆ=π‘Žπ‘˜(1𝑓)∢=ξ€œ2πœ‹π‘–π•‹π‘“0(𝑀)π‘€π‘˜+1𝑑𝑀,π‘˜=0,1,2,…,Μƒπ‘Žπ‘˜βˆΆ=Μƒπ‘Žπ‘˜1(𝑓)∢=ξ€œ2πœ‹π‘–π•‹π‘“1(𝑀)π‘€π‘˜+1𝑑𝑀,π‘˜=1,2,…(3.8) and correspond the seriesβˆžξ“π‘˜=0π‘Žπ‘˜πΉπ‘˜(𝑧)+βˆžξ“π‘˜=1Μƒπ‘Žπ‘˜ξ‚πΉπ‘˜ξ‚€1𝑧(3.9) with the function π‘“βˆˆπΏ1(Ξ“), that is,𝑓(𝑧)βˆΌβˆžξ“π‘˜=0π‘Žπ‘˜πΉπ‘˜(𝑧)+βˆžξ“π‘˜=1Μƒπ‘Žπ‘˜ξ‚πΉπ‘˜ξ‚€1𝑧.(3.10) This series is called the Faber-Laurent series of the function 𝑓 and the coefficients π‘Žπ‘˜ and Μƒπ‘Žπ‘˜ are said to be the Faber-Laurent coefficients of 𝑓. 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 Ξ“βˆˆπΆ(2,𝛼), 0<𝛼<1, Shen and Zhong obtain a series of interpolation nodes in 𝐺0 and show that interpolating polynomials and best approximating polynomial in 𝐸𝑝(𝐺0), 1<𝑝<∞, have the same order of convergence. In [42] considering Ξ“βˆˆπΆ(1,𝛼) and choosing the interpolation nodes as the zeros of the Faber polynomials, Zhu obtain similar result.

In the above-cited works, Ξ“ does not admit corners, whereas many domains in the complex plain may have corners. When Ξ“ 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