Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Abstract

Let 𝐺₀ and 𝐺_∞ be, respectively, bounded and unbounded components of a plane curve Γ satisfying Dini's smoothness condition. In addition to partial sum of Faber series of 𝑓 belonging to weighted Smirnov-Orlicz space 𝐸_𝑀,𝜔 (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 𝐸_𝑀,𝜔(𝐺₀) and 𝐸_𝑀,𝜔(𝐺_∞).

1. Introduction

Let 𝐺₀ and 𝐺_∞ be, respectively, bounded and unbounded components of a closed rectifiable curve Γ of complex plane ℂ. Without loss of generality we may suppose that 0∈𝐺₀. By Riemann conformal mapping theorem [1, page 26], if Γ is connected Jordan curve that consists of more than one point, there exists a conformal mapping 𝜑₀∶𝔻→𝐺₀ of complex unit disc 𝔻∶={𝑤∈ℂ∶|𝑤|=1} onto 𝐺₀. Let 𝛾_𝑟∶=𝜑₀({𝑤∈ℂ∶|𝑤|=𝑟}) for a given 𝑟∈(0,1). We denote by 𝐸^𝑝(𝐺₀), 1≤𝑝≤∞, Smirnov’s classes of analytic functions 𝑓∶𝐺₀→ℂ satisfyingsup_𝑟∈(0,1)_𝛾𝑟ma||x𝑓_𝑧∈𝐺0(𝑧)||^𝑝||𝑑𝑧||<𝑐,if1≤𝑝<∞,||𝑓(𝑧)||<𝐶,if𝑝=∞,(1.1) where positive constant 𝑐 is independent of 𝑟.

It is well known that 𝐸^𝑝(𝐺₀)⊂𝐸¹(𝐺₀) for every 1≤𝑝<∞ and every function 𝑓∈𝐸¹(𝐺₀) has a nontangential boundary values a.e. on Γ, the boundary function belongs to Lebesgue space 𝐿¹(Γ) on Γ. If 1≤𝑝<∞, then 𝐸^𝑝(𝐺₀) is a Banach space with the norm‖𝑓‖_𝑝,Γ∶=12𝜋_Γ||𝑓(𝑧)||^𝑝||𝑑𝑧||^1/𝑝.(1.2) Smirnov classes 𝐸^𝑝(𝐺_∞), 1≤𝑝≤∞, of analytic functions 𝑓∶𝐺_∞→ℂ can be defined similarly and 𝐸^𝑝(𝐺_∞) are fulfilling the same above properties to that of 𝐸^𝑝(𝐺₀).

A smooth Jordan curve Γ will be called Dini-smooth, if the function 𝜃(𝑠), the angle between the tangent line and the positive real axis expressed as a function of arclength 𝑠, has modulus of continuity Ω(𝜃,𝑠) satisfying the Dini condition^𝛿0Ω(𝜃,𝑠)𝑠𝑑𝑠<∞,𝛿>0.(1.3) A Jordan curve Γ will be called Radon curve, if 𝜃(𝑠) has bounded variation and it does not contain cusp point.

Main approximation problems in the spaces 𝐸^𝑝(𝐺₀), 1≤𝑝≤∞, were dealt with by several mathematicians so far. Walsh and Russell gave [2] results in 𝐸^𝑝(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 𝐸^𝑝(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 𝐸^𝑝(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 𝐸^𝑝(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 𝐸_𝑀,𝜔(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 𝐸_𝑀,𝜔(𝐺₀) and 𝐸_𝑀,𝜔(𝐺_∞).

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

2. Approximation Theorems in Weighted Orlicz Space

A function Φ is called Young function if Φ is even, continuous, nonnegative in ℝ, increasing on (0,∞) such thatΦ(0)=0,lim_𝑥→∞Φ(𝑥)=∞.(2.1) A Young function Φ is said to satisfy Δ₂  condition (Φ∈Δ₂) if there is a constant 𝑐>0 such thatΦ(2𝑥)≤𝑐Φ(𝑥)(2.2) for all 𝑥∈ℝ.

Two Young functions Φ and Φ₁ are said to be equivalent if there are 𝑐,𝐶>0 such thatΦ₁(𝑐𝑥)≤Φ(𝑥)≤Φ₁(𝐶𝑥),∀𝑥>0.(2.3)

A function 𝑀∶[0,∞)→[0,∞) is said to be quasiconvex if there exist a convex Young function Φ and a constant 𝑐≥1 such thatΦ(𝑥)≤𝑀(𝑥)≤Φ(𝑐𝑥),∀𝑥≥0,(2.4) holds.

A nonnegative function 𝜔 defined on 𝖳∶=[0,2𝜋] will be called weight if 𝜔 is measurable and a.e. positive. Let 𝑀 be a quasiconvex Young function. We denote by 𝐿_𝑀,𝜔(𝖳) the class of Lebesgue measurable functions 𝑓∶𝖳→ℝ satisfying the condition_𝖳𝑀||𝑓(𝑥)||𝜔(𝑥)𝑑𝑥<∞.(2.5) The linear span of the weighted Orlicz class 𝐿_𝑀,𝜔(𝖳), denoted by 𝐿_𝑀,𝜔(𝖳), becomes a normed space with the Orlicz norm‖𝑓‖_𝑀,𝜔∶=sup_𝖳||𝑓(𝑥)𝑔(𝑥)||𝜔(𝑥)𝑑𝑥∶_𝖳𝑀||𝑔||𝜔(𝑥)𝑑𝑥≤1,(2.6) where 𝑀(𝑦)∶=sup_𝑥≥0(𝑥𝑦−𝑀(𝑥)),𝑦≥0, is the complementary function of 𝑀.

If 𝑀 is quasiconvex and 𝑀 is its complementary function, then Young’s inequality holds𝑥𝑦≤𝑀(𝑥)+𝑀(𝑦),𝑥,𝑦≥0.(2.7) For a quasiconvex function 𝑀 we define the indice 𝑝(𝑀) of 𝑀 as1𝑝(𝑀)∶=inf{𝑝∶𝑝>0,𝑀^𝑝𝑝isquasiconvex},_′(𝑀)∶=𝑝(𝑀)𝑝(𝑀)−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 𝜔∈𝐴_𝑝(𝑀), then it can be easily seen that 𝐿_𝑀,𝜔(𝖳)⊂𝐿¹(𝖳) and 𝐿_𝑀,𝜔(𝖳) becomes a Banach space with the Orlicz norm. The Banach space 𝐿_𝑀,𝜔(𝖳) is called weighted Orlicz space.

We define the Luxemburg functional as‖𝑓‖_(𝑀),𝜔∶=inf𝜏>0∶_𝖳𝑀||𝑓(𝑥)||𝜏𝜔(𝑥)𝑑𝑥≤1.(2.9) There exist [23, page 23] constants 𝑐,𝐶>0 such that𝑐‖𝑓‖_(𝑀),𝜔≤‖𝑓‖_𝑀,𝜔≤𝐶‖𝑓‖_(𝑀),𝜔.(2.10) For a weight 𝜔 we denote by 𝐿^𝑝(𝖳,𝜔) the class of measurable functions on 𝖳 such that 𝜔^1/𝑝𝑓 belongs to Lebesgue space 𝐿^𝑝(𝖳) on 𝖳. We set ‖𝑓‖_𝑝,𝜔∶=‖𝜔^1/𝑝𝑓‖_𝑝 for 𝑓∈𝐿^𝑝(𝖳,𝜔).

A 2𝜋-periodic weight function 𝜔 belongs to the Muckenhoupt class 𝐴_𝑝, 1<𝑝<∞, if||𝐽1||_𝐽𝜔(𝑥)𝑑𝑥1||𝐽||_𝐽𝜔^{−1/(𝑝−1)}(𝑥)𝑑𝑥^𝑝−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 𝑄𝐶^𝜃2(0,1) a class of functions 𝑔 satisfying Δ₂ condition such that 𝑔^𝜃 is quasiconvex for some 𝜃∈(0,1).

In the present section we consider the trigonometric polynomial approximation problems for functions and its fractional derivatives in the spaces 𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀), where 𝑀∈𝑄𝐶^𝜃2(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, 𝑎_𝑘,𝑏_𝑘∈ℝ, 𝑐_𝑘=(𝑎_𝑘−𝑖𝑏_𝑘)/2, 𝑐_−𝑘=(𝑎_𝑘+𝑖𝑏_𝑘)/2, 𝑐₀=𝑎₀/2𝑓(𝑥)∼^∞_𝑘_=−∞𝑐_𝑘𝑒^𝑖𝑘𝑥=^∞_𝑘=0𝑎_𝑘𝑓(𝑥)∼_∞cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥,(2.12)_𝑘=1𝑎_𝑘sin𝑘𝑥−𝑏_𝑘cos𝑘𝑥(2.13)

be the Fourier and the conjugate Fourier series of 𝑓∈𝐿¹(𝖳), respectively. Putting 𝐴_𝑘(𝑥)∶=𝑐_𝑘𝑒^𝑖𝑘𝑥 in (2.12), we define for 𝑛=0,1,2,…𝑆_𝑛(𝑓)∶=𝑆_𝑛(𝑥,𝑓)∶=^𝑛_𝑘=0𝐴_𝑘(𝑥)+𝐴_−𝑘(𝑥)=𝑎⁰2+^𝑛_𝑘=1𝑎_𝑘𝑅cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥_{𝑛⟨𝛼⟩},(𝑓,𝑥)∶=^𝑛_𝑘=01−𝑛+𝑘1^𝛼𝐴^𝑘Θ_{𝑚⟨𝛼⟩}(𝑥)+𝐴_−𝑘(𝑥),𝛼∈ℝ⁺,∶=11−((𝑚+1)/(2𝑚+1))^𝑟𝑅^{⟨𝛼⟩2𝑚}−1((2𝑚+1)/(𝑚+1))^𝑟−1𝑅^{𝑚⟨𝛼⟩}𝑚=1,2,3,….,(2.14)

For a given 𝑓∈𝐿¹(𝖳), assuming_𝖳𝑓(𝑥)𝑑𝑥=0,(2.15) we define 𝛼th fractional (𝛼∈ℝ⁺) integral of 𝑓 as [34, v.2, page 134]𝐼_𝛼(𝑥,𝑓)∶=_{𝑘∈ℤ∗}𝑐_𝑘(𝑖𝑘)^−𝛼𝑒^𝑖𝑘𝑥,(2.16) where(𝑖𝑘)^−𝛼∶=||𝑘||^−𝛼𝑒^{(−1/2)𝜋𝑖𝛼sign𝑘}(2.17) as principal value.

Let 𝛼∈ℝ⁺ be given. We define fractional derivative of a function 𝑓∈𝐿¹(𝖳), satisfying (2.15), as𝑓^(𝛼)(𝑥)∶=𝑑^[𝛼]+1𝑑𝑥^[𝛼]+1𝐼_{1+𝛼−[𝛼]}(𝑥,𝑓)(2.18) provided the right hand side exists.

Setting 𝑥,𝑡∈𝖳, 𝑟∈ℝ⁺, 𝑀∈𝑄𝐶^𝜃2(0,1), 𝜔∈𝐴_𝑝(𝑀), and 𝑓∈𝐿_𝑀,𝜔(𝖳), we define𝜎^𝑟𝑡𝑓(𝑥)∶==𝐼−𝜎_𝑡^𝑟𝑓(𝑥)_∞_𝑘=0(−1)^𝑘⎛⎜⎜⎝𝑘𝑟⎞⎟⎟⎠1(2𝑡)^𝑘^𝑡−𝑡⋯^𝑡−𝑡𝑓𝑥+𝑢₁+⋯𝑢_𝑘𝑑𝑢₁⋯𝑑𝑢_𝑘,(2.19) where (^𝑟𝑘)∶=𝑟(𝑟−1)⋯(𝑟−𝑘+1)/𝑘! for 𝑘≥1 and (^𝑟0)∶=1 are Binomial coefficients, 𝜎_𝑡𝑓(𝑥)∶=(1/2𝑡)∫^𝑡−𝑡𝑓(𝑥+𝑢)𝑑𝑢 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 𝑆_𝑛, 𝜎_ℎ, and 𝑓. If 𝑀∈𝑄𝐶^𝜃2(0,1), 𝜔∈𝐴_𝑝(𝑀), and 𝑓∈𝐿_𝑀,𝜔(𝖳), then there exists a constant 𝑐>0 such that _𝖳𝑀||𝐿𝑓(𝑡)||𝜔(𝑡)𝑑𝑡≤𝑐_𝖳𝑀||𝑓(𝑡)||𝜔(𝑡)𝑑𝑡(2.20) holds.

Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from (2.20) that‖𝐿𝑓‖_𝑀,𝜔≤𝑐‖𝑓‖_𝑀,𝜔(2.21) with a constant 𝑐>0 independent of 𝑓.

By [35, page 14, (1.51)], there exists a constant 𝑐 depending only on 𝑟 such that||||||⎛⎜⎜⎝𝑘𝑟⎞⎟⎟⎠||||||≤𝑐𝑘^𝑟+1,𝑘=1,2,…,(2.22) we have^∞_𝑘=0||||||⎛⎜⎜⎝𝑘𝑟⎞⎟⎟⎠||||||<∞(2.23) and therefore‖‖𝜎^𝑟𝑡𝑓‖‖_𝑀,𝜔≤𝑐‖𝑓‖_𝑀,𝜔<∞(2.24) provided 𝑓∈𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀), where 𝑀∈𝑄𝐶^𝜃2(0,1).

Let 𝑀∈𝑄𝐶^𝜃2(0,1). For 𝑟∈ℝ⁺, we define the fractional modulus of smoothness of index 𝑟 for 𝑓∈𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀) asΩ^{𝑟𝑀,𝜔}(𝑓,𝛿)∶=sup_{0<ℎ𝑖,𝑡<𝛿}‖‖‖‖^[𝑟]_𝑖=1𝐼−𝜎_ℎ𝑖𝐼−𝜎_𝑡^{𝑟−[𝑟]}𝑓‖‖‖‖_𝑀,𝜔,(2.25) where [𝑥] denotes the integer part of a real number 𝑥.

Since the operator 𝜎_𝑡 is bounded in 𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀), where 𝑀∈𝑄𝐶^𝜃2(0,1), we have by (2.24) thatΩ^{𝑟𝑀,𝜔}(𝑓,𝛿)≤𝑐‖𝑓‖_𝑀,𝜔,(2.26) where the constant 𝑐>0, dependent only on 𝑟 and 𝑀.

Remark 2.1. The modulus of smoothness Ω^{𝑟𝑀,𝜔}(𝑓,𝛿), where 𝑟∈ℝ⁺, 𝑀∈𝑄𝐶^𝜃2(0,1), 𝜔∈𝐴_𝑝(𝑀), 𝑓∈𝐿_𝑀,𝜔(𝖳) has the following properties:(i)Ω^{𝑟𝑀,𝜔}(𝑓,𝛿) is nonnegative, nondecreasing function of 𝛿≥0 and subadditive,(ii)lim_𝛿→0Ω^{𝑟𝑀,𝜔}(𝑓,𝛿)=0.

For formulations of our results, we need several lemmas.

Lemma A (see [36]). For 𝛼∈ℝ⁺, we suppose that(i)𝑎₁+𝑎₂+⋯+𝑎_𝑛+⋯, (ii)𝑎₁+2^𝛼𝑎₂+⋯+𝑛^𝛼𝑎_𝑛+⋯,
be two series in a Banach space (𝐵,‖⋅‖). Let𝑅^{𝑛⟨𝛼⟩}∶=^𝑛_𝑘=01−𝑛+𝑘1^𝛼𝑎^𝑘𝑅_{𝑛⟨𝛼⟩∗},∶=^𝑛_𝑘=01−𝑛+𝑘1^𝛼𝑘^𝛼𝑎_𝑘(2.27) for 𝑛=1,2,…. Then, ‖‖𝑅^{𝑛⟨𝛼⟩∗}‖‖≤𝑐,𝑛=1,2,…(2.28) for some 𝑐>0 if and only if there exists a 𝑅∈𝐵 such that ‖‖𝑅^{𝑛⟨𝛼⟩}−𝑅‖‖≤𝐶𝑛^𝛼,(2.29) where 𝑐 and 𝐶 are constants depending only on one another.

If 𝑀∈𝑄𝐶^𝜃2(0,1), 𝜔∈𝐴_𝑝(𝑀), and 𝑓∈𝐿_𝑀,𝜔(𝖳), then from Theorem A(ii) and Abel’s transformation we get‖‖𝑅^{𝑛⟨𝛼⟩}(𝑓,⋅)‖‖_𝑀,𝜔≤𝑐‖𝑓‖_𝑀,𝜔,𝑛=1,2,3,…,𝑥∈𝖳(2.30) and therefore from (2.14) and(2.30)‖‖Θ^{𝑛⟨𝛼⟩}(𝑓,⋅)‖‖_𝑀,𝜔≤𝑐‖𝑓‖_𝑀,𝜔,𝑛=1,2,3,…,𝑥∈𝖳.(2.31) From the propertyΘ^{𝑚⟨𝛼⟩}(𝑓)(𝑥)=1∑^{2𝑚𝑘=𝑚+1}(𝑘+1)^𝛼−𝑘^𝛼^2𝑚_𝑘=𝑚+1(𝑘+1)^𝛼−𝑘^𝛼𝑆_𝑘(𝑥,𝑓),𝑥∈𝖳,𝑓∈𝐿¹(𝖳)(2.32) it is known thatΘ^{𝑚⟨𝛼⟩}𝑇_𝑚=𝑇_𝑚(2.33) for 𝑇_𝑚∈𝒯_𝑚, 𝑚=1,2,3,….

Lemma 2.2. Let 𝑇_𝑛∈𝒯_𝑛,𝑛=1,2,3,…,𝑀∈𝑄𝐶^𝜃2(0,1), and 𝜔∈𝐴_𝑝(𝑀). If 𝛼∈ℝ⁺, then there exists a constant 𝑐>0 independent of n such that ‖‖𝑇^𝑛(𝛼)‖‖_𝑀,𝜔≤𝑐𝑛^𝛼‖‖𝑇_𝑛‖‖_𝑀,𝜔(2.34) holds.

Proof. Without loss of generality one can assume that ‖𝑇_𝑛‖_𝑀,𝜔=1. Since 𝑇_𝑛=^𝑛_𝑘=0𝐴_𝑘(𝑥)+𝐴_−𝑘(𝑥),𝑇^𝑛𝑛^𝛼=^𝑛_𝑘=1𝐴^𝑘(𝑥)−𝐴_−𝑘(𝑥)𝑇_𝑛(𝛼)𝑛^𝛼,(𝑖𝑛)^𝛼=^𝑛_𝑘=1𝑘^𝛼𝐴^𝑘(𝑥)−𝐴_−𝑘(𝑥)𝑛^𝛼(2.35) we have by (2.30) and Theorem A(iii) that ‖‖‖‖𝑅_{𝑚⟨𝛼⟩}𝑇𝑛^𝑛𝛼‖‖‖‖_𝑀,𝜔≤𝑐𝑛^𝛼‖‖𝑇_𝑛‖‖_𝑀,𝜔≤𝑐𝑛^𝛼‖‖𝑇_𝑛‖‖_𝑀,𝜔=𝑐𝑛^𝛼(2.36) and from Lemma A ‖‖‖‖𝑅_{𝑚⟨𝛼⟩}(𝑇𝑖𝑛)^{𝑛(𝛼)𝛼}‖‖‖‖_𝑀,𝜔≤𝑐.(2.37) Hence from (2.33) and (2.31), we find ‖‖𝑇^𝑛(𝛼)‖‖_𝑀,𝜔=𝑛^𝛼‖‖‖‖Θ_{𝑚⟨𝛼⟩}(𝑇𝑖𝑛)^{𝑛(𝛼)𝛼}‖‖‖‖_𝑀,𝜔≤𝑐𝑛^𝛼‖‖𝑇_𝑛‖‖_𝑀,𝜔.(2.38) General case follows immediately from this.

Let 𝑀∈𝑄𝐶^𝜃2(0,1). We denote by 𝑊^𝛼𝑀(𝖳,𝜔), 𝛼>0, 𝜔∈𝐴_𝑝(𝑀), the linear space of 2𝜋-periodic real valued functions 𝑓∈𝐿_𝑀,𝜔(𝖳) such that 𝑓^(𝛼)∈𝐿_𝑀,𝜔(𝖳).

Lemma 2.3. Let 𝑀∈𝑄𝐶^𝜃2(0,1). If 𝑓∈𝑊^𝛼𝑀(𝖳,𝜔) with 𝜔∈𝐴_𝑝(𝑀) and 𝛼≥0, then for 𝑛=0,1,2,…, there is a constant 𝑐>0 dependent only on 𝛼 and 𝑀 such that ‖‖𝑓^(𝛼)(⋅)−𝑆^𝑛(𝛼)(⋅,𝑓)‖‖_𝑀,𝜔≤𝑐𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔(2.39) holds.

Proof. If 𝛼=0, then from boundedness (see (2.21)) of the operator 𝑆_𝑛 we get that ‖‖𝑓−𝑆_𝑛𝑓‖‖_𝑀,𝜔≤𝑐𝐸_𝑛(𝑓)_𝑀,𝜔.(2.40) Let 𝑊_𝑛(𝑓)∶=𝑊_𝑛(𝑥,𝑓)∶=(1/(𝑛+1))∑^{2𝑛𝜈=𝑛}𝑆_𝜈(𝑥,𝑓), 𝑛=0,1,2,…. Since 𝑊_𝑛⋅,𝑓^(𝛼)=𝑊^𝑛(𝛼)(⋅,𝑓),(2.41) we have ‖‖𝑓^(𝛼)(⋅)−𝑆^𝑛(𝛼)(⋅,𝑓)‖‖_𝑀,𝜔≤‖‖𝑓^(𝛼)(⋅)−𝑊_𝑛⋅,𝑓^(𝛼)‖‖_𝑀,𝜔+‖‖𝑆^𝑛(𝛼)(⋅,𝑊_𝑛(𝑓))−𝑆^𝑛(𝛼)(⋅,𝑓)‖‖_𝑀,𝜔+‖‖𝑊^𝑛(𝛼)(⋅,𝑓)−𝑆^𝑛(𝛼)(⋅,𝑊_𝑛(𝑓))‖‖_𝑀,𝜔∶=𝐼₁+𝐼₂+𝐼₃.(2.42) From (2.21) we get the boundedness of 𝑊_𝑛 in 𝐿_𝑀,𝜔(𝖳) and we have 𝐼₁≤‖‖𝑓^(𝛼)(⋅)−𝑆_𝑛⋅,𝑓^(𝛼)‖‖_𝑀,𝜔+‖‖𝑆_𝑛(⋅,𝑓^(𝛼))−𝑊_𝑛(⋅,𝑓^(𝛼))‖‖_𝑀,𝜔≤𝑐𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔+‖‖𝑊_𝑛⋅,𝑆_𝑛𝑓^(𝛼)−𝑓^(𝛼)‖‖_𝑀,𝜔≤𝑐𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔.(2.43) From Lemma 2.2 we get 𝐼𝐼₂₃≤𝑐𝑛^𝛼‖‖𝑆_𝑛⋅,𝑊_𝑛(𝑓)−𝑆_𝑛(⋅,𝑓)‖‖_𝑀,𝜔,≤𝑐(2𝑛)^𝛼‖‖𝑊_𝑛(⋅,𝑓)−𝑆_𝑛⋅,𝑊_𝑛(𝑓)‖‖_𝑀,𝜔≤𝑐(2𝑛)^𝛼𝐸_𝑛𝑊_𝑛(𝑓)_𝑀,𝜔.(2.44) Now we have ‖‖𝑆_𝑛⋅,𝑊_𝑛(𝑓)−𝑆_𝑛(⋅,𝑓)‖‖_𝑀,𝜔≤‖‖𝑆_𝑛⋅,𝑊_𝑛(𝑓)−𝑊_𝑛(⋅,𝑓)‖‖_𝑀,𝜔+‖‖𝑊_𝑛(⋅,𝑓)−𝑓(⋅)‖‖_𝑀,𝜔+‖‖𝑓(⋅)−𝑆_𝑛(⋅,𝑓)‖‖_𝑀,𝜔≤𝑐𝐸_𝑛𝑊_𝑛(𝑓)_𝑀,𝜔+𝑐𝐸_𝑛(𝑓)_𝑀,𝜔.(2.45) Since 𝐸_𝑛𝑊_𝑛(𝑓)_𝑀,𝜔≤𝑐𝐸_𝑛(𝑓)_𝑀,𝜔,(2.46) we get ‖‖𝑓^(𝛼)(⋅)−𝑇^𝑛(𝛼)(⋅,𝑓)‖‖_𝑀,𝜔≤𝑐𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔+𝑐𝑛^𝛼𝐸_𝑛𝑊_𝑛(𝑓)_𝑀,𝜔+𝑐𝑛^𝛼𝐸_𝑛(𝑓)_𝑀,𝜔+𝑐(2𝑛)^𝛼𝐸_𝑛𝑊_𝑛(𝑓)_𝑀,𝜔≤𝑐𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔+𝐶𝑛^𝛼𝐸_𝑛(𝑓)_𝑀,𝜔.(2.47) Now we show that 𝐸_𝑛(𝑓)_𝑀,𝜔≤𝑐(𝑛+1)^𝛼𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔.(2.48) For this we set 𝐴_𝑘(𝑥,𝑓)∶=𝑎_𝑘cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥.(2.49) For given 𝑓∈𝐿_𝑀,𝜔(𝖳) and 𝜀>0, by Lemma 3 of [37], there exists a trigonometric polynomial 𝑇 such that _𝖳𝑀||𝑓(𝑥)−𝑇(𝑥)||𝜔(𝑥)𝑑𝑥<𝜀(2.50) which by (2.7) this implies that ‖𝑓−𝑇‖_𝑀,𝜔<𝜀,(2.51) and hence we obtain 𝐸_𝑛(𝑓)_𝑀,𝜔⟶0as𝑛⟶∞.(2.52) In this case from (2.40) we have 𝑓(𝑥)=^∞_𝑘=0𝐴_𝑘(𝑥,𝑓)(2.53) in ‖⋅‖_𝑀,𝜔 norm. If 𝑘=1,2,3,…, then 𝐴_𝑘(𝑥,𝑓)=𝐴_𝑘𝑥+𝛼𝜋𝐴_𝑘2𝑘,𝑓cos𝛼2𝜋+𝐴^𝑘𝑥+𝛼2𝜋𝑘,𝑓sin𝛼2𝜋,𝑥,𝑓^(𝛼)=𝑘^𝛼𝐴_𝑘𝑥+𝛼2𝜋𝑘,𝑓.(2.54) Hence, ^∞_𝑘=0𝐴_𝑘(𝑥,𝑓)=𝐴₀(𝑥,𝑓)+cos𝛼𝜋2^∞_𝑘=1𝐴_𝑘𝑥+𝛼𝜋2𝑘,𝑓+sin𝛼2𝜋^∞_𝑘=1𝐴_𝑘𝑥+𝛼𝜋=𝐴₀2𝑘,𝑓(𝑥,𝑓)+cos𝛼𝜋2^∞_𝑘=1𝑘^−𝛼𝐴_𝑘𝑥,𝑓^(𝛼)+sin𝛼𝜋2^∞_𝑘=1𝑘^−𝛼𝐴_𝑘𝑥,𝑓^(𝛼).(2.55) Therefore, 𝑓(𝑥)−𝑆_𝑛(𝑥,𝑓)=cos𝛼𝜋2^∞_𝑘=𝑛+11𝑘^𝛼𝐴_𝑘𝑥,𝑓^(𝛼)+sin𝛼𝜋2^∞_𝑘=𝑛+11𝑘^𝛼𝐴_𝑘𝑥,𝑓^(𝛼).(2.56) Since ^∞_𝑘=𝑛+1𝑘^−𝛼𝐴_𝑘𝑥,𝑓^(𝛼)=^∞_𝑘_=𝑛+1=𝑘^−𝛼𝑆_𝑘⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)−𝑆_𝑘−1⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)_∞𝑘_=𝑛+1−(𝑛+(𝑘1)^−𝛼_−𝛼−(𝑘+1)^−𝛼)𝑆_𝑘⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)𝑆_𝑛⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)_∞,_𝑘_=𝑛+1𝑘^−𝛼𝐴_𝑘𝑥,𝑓^(𝛼)=^∞_𝑘_=𝑛+1−(𝑛+(𝑘1)^−𝛼_−𝛼−(𝑘+1)^−𝛼)𝑆_𝑘⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)𝑆^𝑛⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅),(2.57) we obtain ‖‖𝑓(⋅)−𝑆_𝑛(⋅,𝑓)‖‖_𝑀,𝜔≤^∞_𝑘_=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)‖‖𝑆_𝑘⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)‖‖_𝑀,𝜔+(𝑛+1)^−𝛼‖‖𝑆_𝑛⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)‖‖_𝑀,𝜔+^∞_𝑘_=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)‖‖𝑆_𝑘⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)‖‖_𝑀,𝜔+(𝑛+1)^−𝛼‖‖𝑆_𝑛⋅,𝑓^(𝛼)−𝑓^(𝛼)(⋅)‖‖_𝑀,𝜔≤𝑐^∞_𝑘_=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)𝐸_𝑘(𝑓)_𝑀,𝜔+(𝑛+1)^−𝛼𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔+𝐶_∞_𝑘_=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)𝐸_𝑘(𝑓)_𝑀,𝜔+(𝑛+1)^−𝛼𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔.(2.58) Consequently, ‖‖𝑓(𝑥)−𝑆_𝑛(𝑥,𝑓)‖‖_𝑀,𝜔≤𝑐𝐸_𝑘𝑓^(𝛼)_𝑀,𝜔^∞_𝑘=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)+(𝑛+1)^−𝛼+𝑐𝐸_𝑛≤𝑐𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔^∞_𝑘=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)+(𝑛+1)^−𝛼𝑓^(𝛼)_𝑀,𝜔^∞_𝑘=𝑛+1(𝑘^−𝛼−(𝑘+1)^−𝛼)+(𝑛+1)^−𝛼≤𝑐(𝑛+1)^𝛼𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔,(2.59) and (2.48) holds. Now (2.47) and (2.48) imply the result.

Lemma 2.4. Let 𝑇_𝑛∈𝒯_𝑛, 𝑛=0,1,2,…, 𝑀∈𝑄𝐶^𝜃2(0,1), and 𝜔∈𝐴_𝑝(𝑀). If 𝛼∈ℝ⁺, then Ω^{𝛼𝑀,𝜔}𝑇^𝑛,𝜋𝑛+1≤(𝑛+𝑐1)^𝛼‖‖𝑇^𝑛(𝛼)‖‖_𝑀,𝜔(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 Ω^{𝛽𝑀,𝜔}𝑇_𝑛,⋅≤𝑐Ω^{𝛼𝑀,𝜔}𝑇_𝑛,⋅.(2.61) It is easily seen that if 𝛼≤𝛽,𝛼,𝛽∈ℤ⁺, then (2.61) holds. Now, we assume 0<𝛼<𝛽≤1. In this case, putting 𝐾(𝑥)∶=𝜎^𝛼𝑡𝑇_𝑛(𝑥), we have 𝜎^{𝑡𝛽−𝛼}𝐾(𝑥)=^∞_𝑗=0(−1)^𝑗⎛⎜⎜⎝𝛽−𝑗𝛼⎞⎟⎟⎠1(2𝑡)^𝑗^𝑡−𝑡⋯^𝑡−𝑡𝐾𝑥+𝑢₁+⋯𝑢_𝑗𝑑𝑢₁⋯𝑑𝑢_𝑗=^∞_𝑗=0(−1)^𝑗⎛⎜⎜⎝𝛽−𝑗𝛼⎞⎟⎟⎠1(2𝑡)^𝑗×^𝑡−𝑡⋯^𝑡−𝑡⎡⎢⎢⎣^∞_𝑘=0(−1)^𝑘⎛⎜⎜⎝𝛼𝑘⎞⎟⎟⎠1(2𝑡)^𝑘×^𝑡−𝑡⋯^𝑡−𝑡𝑇_𝑛𝑥+𝑢₁+⋯𝑢_𝑗+𝑢_𝑗+1+⋯𝑢_𝑗+𝑘𝑑𝑢_𝑗+1⋯𝑑𝑢_𝑗+𝑘⎤⎥⎥⎦𝑑𝑢₁⋯𝑑𝑢_𝑗=^∞_∞𝑗=0_𝑘=0(−1)^𝑗+𝑘⎛⎜⎜⎝𝛽−𝑗𝛼⎞⎟⎟⎠⎛⎜⎜⎝𝛼𝑘⎞⎟⎟⎠1(2𝑡)^𝑗+𝑘^𝑡−𝑡⋯^𝑡−𝑡𝑇_𝑛𝑥+𝑢₁+⋯𝑢_𝑗+𝑘𝑑𝑢₁⋯𝑑𝑢_𝑗+𝑘=_∞_𝜐=0⎧⎪⎨⎪⎩^𝜐_𝜇=0(−1)^𝜐−𝜇⎛⎜⎜⎝𝛽𝜐−−𝜇𝛼⎞⎟⎟⎠⎛⎜⎜⎝𝜇𝛼⎞⎟⎟⎠1(2𝑡)^𝜐^𝑡−𝑡⋯^𝑡−𝑡𝑇_𝑛𝑥+𝑢₁+⋯𝑢_𝜐𝑑𝑢₁⋯𝑑𝑢_𝜐=_∞⎫⎪⎬⎪⎭_𝜐=0(−1)^𝜐1(2𝑡)^𝜐^𝑡−𝑡⋯^𝑡−𝑡𝑇_𝑛𝑥+𝑢₁+…𝑢_𝜐𝑑𝑢₁⋯𝑑𝑢_𝜐𝜐_𝜇=0=⎛⎜⎜⎝𝛽𝜐−−𝜇_∞𝛼⎞⎟⎟⎠⎛⎜⎜⎝𝜇𝛼⎞⎟⎟⎠_𝜐=0(−1)^𝜐⎛⎜⎜⎝𝛽𝜐⎞⎟⎟⎠1(2𝑡)^𝜐^𝑡−𝑡⋯^𝑡−𝑡𝑇_𝑛𝑥+𝑢₁+…𝑢_𝜐𝑑𝑢₁⋯𝑑𝑢_𝜐=𝜎^𝛽𝑡𝑇_𝑛(𝑥).(2.62) Then, ‖‖𝜎^𝛽𝑡𝑇_𝑛‖‖_𝑀,𝜔=‖‖𝜎^{𝑡𝛽−𝛼}𝐾‖‖_𝑀,𝜔≤𝑐‖‖𝜎^𝛼𝑡𝑇_𝑛‖‖_𝑀,𝜔,(2.63) and hence (2.61) holds. We note that if 𝑟₁,𝑟₂∈ℤ⁺,𝛼₁,𝛽₁∈(0,1) taking 𝛼∶=𝑟₁+𝛼₁,𝛽∶=𝑟₂+𝛽₁ for the remaining cases 𝑟₁=𝑟₂,𝛼₁<𝛽₁ or 𝑟₁<𝑟₂,𝛼₁<𝛽₁ or 𝑟₁<𝑟₂,𝛼₁>𝛽₁, it can easily be obtained from the last inequality that the required inequality (2.61) holds. Now we will show that if 𝜚∈𝑊^𝑀(2𝑟)(𝖳,𝜔), 𝑟=1,2,3,…, then Ω^{𝑟𝑀,𝜔}(𝜚,𝛿)≤𝑐𝛿^2𝑟‖‖𝜚^(2𝑟)‖‖_𝑀,𝜔.(2.64) Putting 𝑔(𝑥)∶=^𝑟_𝑖=2𝐼−𝜎_ℎ𝑖𝜚(𝑥),(2.65) we have 𝐼−𝜎_ℎ1𝑔(𝑥)=^𝑟_𝑖=1𝐼−𝜎_ℎ𝑖𝜚(𝑥),_𝑟_𝑖=1𝐼−𝜎_ℎ𝑖𝜚(𝑥)=21ℎ₁^ℎ1_−ℎ1(𝑔(𝑥)−𝑔(𝑥+𝑡))𝑑𝑡=−18ℎ₁^ℎ1₀^𝑡0^𝑢−𝑢𝑔^′(𝑥+𝑠)𝑑𝑠𝑑𝑢𝑑𝑡.(2.66) Therefore, ‖‖‖‖^𝑟_𝑖=1𝐼−𝜎_ℎ𝑖𝜚(𝑥)‖‖‖‖_𝑀,𝜔=18ℎ₁sup_𝖳||||^ℎ₁₀^𝑡0^𝑢−𝑢≤1𝑔^(𝑥+𝑠)𝑑𝑠𝑑𝑢𝑑𝑡||||||𝑣(𝑥)||𝜔(𝑥)𝑑𝑥∶_𝖳𝑀||𝑣(𝑥)||𝜔(𝑥)𝑑𝑥≤18ℎ₁^ℎ1₀^𝑡02𝑢‖‖‖12𝑢^𝑢−𝑢≤𝑐𝑔^(𝑥+𝑠)𝑑𝑠‖‖‖_𝑀,𝜔𝑑𝑢𝑑𝑡8ℎ₁^ℎ1₀^𝑡02𝑢‖‖𝑔^‖‖_𝑀,𝜔𝑑𝑢𝑑𝑡=𝑐ℎ²¹‖‖𝑔^‖‖_𝑀,𝜔.(2.67) Since 𝑔^(𝑥)=^𝑟_𝑖=2𝐼−𝜎_ℎ𝑖𝜚^(𝑥),(2.68) we obtain that Ω^{𝑟𝑀,𝜔}(𝜚,𝛿)≤sup_{0<ℎ𝑖≤𝛿𝑖=1,2,…,𝑟}𝑐ℎ²¹‖‖𝑔^‖‖_𝑀,𝜔=𝑐𝛿²‖‖‖‖^𝑟_𝑖=2𝐼−𝜎_ℎ𝑖𝜚^(𝑥)‖‖‖‖_𝑀,𝜔=𝑐𝛿²sup_{0<ℎ𝑖≤𝛿𝑖=2,…,𝑟}‖‖‖‖^𝑟_𝑖=2𝐼−𝜎_ℎ𝑖𝜚^≤𝑐𝛿₄(𝑥)‖‖‖‖_𝑀,𝜔=𝑐𝛿²Ω^{𝑟−1𝑀,𝜔}𝜚^,𝛿Ω^{𝑟−2𝑀,𝜔}𝜚⁽⁴⁾,𝛿≤⋯≤𝐶𝛿^2𝑟‖‖𝜚^(2𝑟)‖‖_𝑀,𝜔.(2.69) Using (2.61), (2.64), and Lemma 2.2, we get Ω^{𝛼𝑀,𝜔}𝑇^𝑛,𝜋𝑛+1≤𝑐Ω^{[𝛼]𝑀,𝜔}𝑇^𝑛,𝜋𝑛+1≤𝑐𝑛+𝜋1^2[𝛼]‖‖𝑇^{𝑛(2[𝛼])}‖‖_𝑀,𝜔≤𝑐(𝑛+1)^2[𝛼](𝑛+1)^{[𝛼]−(𝛼−[𝛼])}‖‖𝑇^𝑛(𝛼)‖‖_𝑀,𝜔=𝑐(𝑛+1)^𝛼‖‖𝑇^𝑛(𝛼)‖‖_𝑀,𝜔(2.70) which is the required result (2.60) for 𝛼≥1. On the other hand in case of 0<𝛼<1 the inequality (2.60) can be obtained by Marcinkiewicz Multiplier Theorem for 𝐿_𝑀,𝜔(𝖳) where 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝜔∈𝐴_𝑝(𝑀).

Definition 2.5. For 𝑓∈𝐿_𝑀,𝜔(𝖳),𝛿>0, and 𝑟=1,2,3,…, the Peetre𝐾-functional is defined as 𝐾𝛿,𝑓;𝐿_𝑀,𝜔(𝖳),𝑊^𝛼𝑀(𝖳,𝜔)∶=inf_{𝑔∈𝑊𝛼𝑀(𝖳,𝜔)}‖𝑓−𝑔‖^𝑀,𝜔+𝛿‖‖𝑔^(𝑟)‖‖_𝑀,𝜔.(2.71)

Proposition 2.6. Let 𝑀∈𝑄𝐶^𝜃2(0,1), 𝜔∈𝐴_𝑝(𝑀), and 𝑓∈𝐿_𝑀,𝜔(𝖳). Then the 𝐾-functional 𝐾(𝛿^2𝑟,𝑓;𝐿_𝑀,𝜔(𝖳),𝑊^𝑀2𝑟(𝖳,𝜔)) in (2.71) and the modulus Ω^{𝑟𝑀,𝜔}(𝑓,𝛿),𝑟=1,2,3,…, are equivalent.

Proof. If ℎ∈𝑊^𝑀2𝑟(𝖳,𝜔), then we have Ω^{𝑟𝑀,𝜔}(𝑓,𝛿)≤𝑐‖𝑓−ℎ‖_𝑀,𝜔+𝑐𝛿^2𝑟‖‖ℎ^(2𝑟)‖‖_𝑀,𝜔≤𝑐𝐾𝛿^2𝑟,𝑓;𝐿_𝑀,𝜔(𝖳),𝑊^𝑀2𝑟(𝖳,𝜔).(2.72)
Putting𝐿_𝛿𝑓(𝑥)∶=3𝛿⁻³^𝛿0^𝑢0^𝑡−𝑡𝑓(𝑥+𝑠)𝑑𝑠𝑑𝑡𝑑𝑢,𝑥∈𝖳,(2.73) we have 𝑑²𝑑𝑥²𝐿_𝛿𝑓=𝑐𝛿²𝐼−𝜎_𝛿𝑓,(2.74) and hence 𝑑^2𝑟𝑑𝑥^2𝑟𝐿^𝑟𝛿𝑓=𝑐𝛿^2𝑟𝐼−𝜎_𝛿^𝑟,𝑟=1,2,3,….(2.75) On the other hand, we find ‖‖𝐿_𝛿𝑓‖‖_𝑀,𝜔≤3𝛿⁻³^𝛿0^𝑢02𝑡‖‖𝜎_𝑡𝑓‖‖_𝑀,𝜔𝑑𝑡𝑑𝑢≤𝑐‖𝑓‖_𝑀,𝜔.(2.76) Now, let 𝐴^𝑟𝛿∶=𝐼−(𝐼−𝐿^𝑟𝛿)^𝑟. Then 𝐴^𝑟𝛿𝑓∈𝑊^𝑀2𝑟(𝖳,𝜔) and ‖‖‖𝑑^2𝑟𝑑𝑥^2𝑟𝐴^𝑟𝛿𝑓‖‖‖_𝑀,𝜔≤𝑐‖‖‖𝑑^2𝑟𝑑𝑥^2𝑟𝐿^𝑟𝛿𝑓‖‖‖_𝑀,𝜔=𝑐𝛿^2𝑟‖‖𝐼−𝜎_𝛿^𝑟‖‖_𝑀,𝜔≤𝑐𝛿^2𝑟Ω^{𝑟𝑀,𝜔}(𝑓,𝛿).(2.77) Since 𝐼−𝐿^𝑟𝛿=𝐼−𝐿_𝛿^𝑟−1_𝑗=0𝐿^𝑗𝛿,(2.78) we get ‖‖𝐼−𝐿^𝑟𝛿𝑔‖‖_𝑀,𝜔≤𝑐‖‖𝐼−𝐿_𝛿𝑔‖‖_𝑀,𝜔≤3𝑐𝛿⁻³^𝛿0^𝑢0≤𝑐sup_{0<𝑡≤𝛿}2𝑡‖‖𝐼−𝜎_𝑡𝑔‖‖_𝑀,𝜔𝑑𝑡𝑑𝑢‖‖𝐼−𝜎_𝑡𝑔‖‖_𝑀,𝜔.(2.79) Taking into account ‖‖𝑓−𝐴^𝑟𝛿𝑓‖‖_𝑀,𝜔=‖‖𝐼−𝐿^𝑟𝛿^𝑟𝑓‖‖_𝑀,𝜔(2.80) by a recursive procedure, we obtain ‖‖𝑓−𝐴^𝑟𝛿𝑓‖‖_𝑀,𝜔≤𝑐sup_{0<𝑡1≤𝛿}‖‖𝐼−𝜎_𝑡1𝐼−𝐿^𝑟𝛿^𝑟−1𝑓‖‖_𝑀,𝜔≤𝑐sup_{0<𝑡1≤𝛿}sup_{0<𝑡2≤𝛿}‖‖𝐼−𝜎_𝑡1𝐼−𝜎_𝑡2𝐼−𝐿^𝑟𝛿^𝑟−2𝑓‖‖_𝑀,𝜔≤⋯≤𝑐sup_{0<𝑡𝑖≤𝛿𝑖=1,2,…,𝑟}‖‖‖‖^𝑟_𝑖=1𝐼−𝜎_𝑡𝑖𝑓(𝑥)‖‖‖‖_𝑀,𝜔=𝑐Ω^{𝑟𝑀,𝜔}(𝑓,𝛿).(2.81)
Now we can formulate the results.

Theorem 2.7. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑟∈ℝ⁺. If 𝑓∈𝐿_𝑀,𝜔(𝖳) with 𝜔∈𝐴_𝑝(𝑀), then there is a constant 𝑐>0 dependent only on 𝑟 and 𝑀 such that for 𝑛=0,1,2,3,…𝐸_𝑛(𝑓)_𝑀,𝜔≤𝑐Ω^{𝑟𝑀,𝜔}𝑓,1𝑛+1(2.82) holds.

Proof. We put 𝑘−1<𝑟≤𝑘, 𝑘∈ℤ⁺. From Remark 2.1(i), (2.64), (2.71), Proposition 2.6, and (2.61), we get for every 𝑔∈𝑊^{2𝑘𝑀,𝜔}(𝕋) and 𝑛=0,1,2,3,…𝐸_𝑛(𝑓)_𝑀,𝜔≤≤𝐸𝑐𝐾^𝑛(𝑓(−𝑛+𝑔)1)^𝑀,𝜔_−2𝑘+𝐸_𝑛(𝑔)_𝑀,𝜔≤𝑐‖𝑓−𝑔‖_𝑀,𝜔+(𝑛+1)^−2𝑘‖‖𝑔^(2𝑘)‖‖_𝑀,𝜔,𝑓;𝐿_𝑀,𝜔(𝖳),𝑊^𝑀2𝑘(𝖳,𝜔)≤𝑐Ω^{𝑘𝑀,𝜔}≤𝑐Ω_{𝑟𝑀,𝜔}𝑓,(𝑛+11)𝑓,(𝑛+11).(2.83)

Theorem 2.8. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑟∈ℝ⁺. If 𝑓∈𝐿_𝑀,𝜔(𝖳) with 𝜔∈𝐴_𝑝(𝑀), then there is a constant 𝑐>0 dependent only on 𝑟 and 𝑀 such that for 𝑛=0,1,2,3,…Ω^{𝑟𝑀,𝜔}𝑓,𝜋𝑛+1≤(𝑛+𝑐1)^𝑟𝑛_𝜈=0(𝜈+1)^𝑟−1𝐸_𝜈(𝑓)_𝑀,𝜔(2.84) holds.

Proof. Let 𝑇_𝑛∈𝒯_𝑛 be the best approximating polynomial of 𝑓∈𝐿_𝑀,𝜔(𝖳) and let 𝑚∈ℤ⁺. Then, Ω^{𝑟𝑀,𝜔}𝑓,𝜋𝑛+1≤Ω^{𝑟𝑀,𝜔}𝑓−𝑇^2𝑚,𝜋(𝑛+1)+Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1)≤𝑐𝐸^2𝑚(𝑓)_𝑀,𝜔+Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1).(2.85) By Lemma 2.4 we have Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1)≤𝑐𝑛+11^𝑟‖‖𝑇^2(𝑟)_𝑚‖‖_𝑀,𝜔.(2.86) Since 𝑇^2(𝑟)_𝑚(𝑥)=𝑇^1(𝑟)(𝑥)+^𝑚−1_𝜈=0𝑇^{2(𝑟)𝜈+1}(𝑥)−𝑇^2(𝑟)_𝜈(𝑥),(2.87) we get Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1)≤(𝑛+𝑐1)^𝑟‖‖𝑇^1(𝑟)‖‖_𝑀,𝜔+^𝑚−1_𝜈=0‖‖𝑇^2(𝑟)_𝜈+1−𝑇^2(𝑟)_𝜈‖‖_𝑀,𝜔.(2.88) Fractional Bernstein inequality of Lemma 2.2 gives ‖‖𝑇^2(𝑟)_𝜈+1−‖‖𝑇𝑇^2(𝑟)_𝜈‖‖_{𝑀,𝜔1(𝑟)}≤𝑐2^𝜈𝑟‖‖𝑇_2𝜈+1−𝑇₂^𝜈‖‖_𝑀,𝜔≤𝑐2^𝜈𝑟+1𝐸₂^𝜈(𝑓)_𝑀,𝜔,‖‖_𝑀,𝜔=‖‖𝑇^1(𝑟)−𝑇^0(𝑟)‖‖_𝑀,𝜔≤𝑐𝐸₀(𝑓)_𝑀,𝜔.(2.89) Hence, Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1)≤(𝑛+𝑐1)^𝑟𝐸⁰(𝑓)_𝑀,𝜔+^𝑚−1_𝜈=02^(𝜈+1)𝑟𝐸₂^𝜈(𝑓)_𝑀,𝜔.(2.90) It is easily seen that 2^(𝜈+1)𝑟𝐸₂^𝜈(𝑓)_𝑀,𝜔≤𝑐^∗2𝜈_𝜇=_2𝜈−1+1𝜇^𝑟−1𝐸_𝜇(𝑓)_𝑀,𝜔,𝜈=1,2,3,…,(2.91) where 𝑐^∗=2^𝑟+12,0<𝑟<1,_2𝑟,𝑟≥1.(2.92) Therefore, Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1)≤(𝑛+𝑐1)^𝑟⎧⎪⎨⎪⎩𝐸₀(𝑓)_𝑀,𝜔+2^𝑟𝐸₁(𝑓)_𝑀,𝜔+𝐶^𝑚_2𝜈=1^𝜈_{𝜇=2𝜈−1+1}𝜇^𝑟−1𝐸_𝜇(𝑓)_𝑀,𝜔≤𝑐⎫⎪⎬⎪⎭(𝑛+1)^𝑟⎧⎪⎨⎪⎩𝐸₀(𝑓)_𝑀,𝜔+^2𝑚_𝜇=1𝜇^𝑟−1𝐸_𝜇(𝑓)_𝑀,𝜔≤𝑐⎫⎪⎬⎪⎭(𝑛+1)^{𝑟2𝑚−1}_𝜈=0(𝜈+1)^𝑟−1𝐸_𝜈(𝑓)_𝑀,𝜔.(2.93) If we choose 2^𝑚≤𝑛+1≤2^𝑚+1, then Ω^{𝑟𝑀,𝜔}𝑇^2𝑚,𝜋(𝑛+1)≤(𝑛+𝑐1)^𝑟𝑛_𝜈=0𝐸_2𝑚(𝜈+1)^𝑟−1𝐸_𝜈(𝑓)_𝑀,𝜔,(𝑓)_𝑀,𝜔≤𝐸₂^𝑚−1(𝑓)_𝑀,𝜔≤𝑐(𝑛+1)^𝑟𝑛_𝜈=0(𝜈+1)^𝑟−1𝐸_𝜈(𝑓)_𝑀,𝜔.(2.94) Last two inequalities complete the proof.

From Theorems 2.7 and 2.8 we have the following corollaries.

Corollary 2.9. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑟∈ℝ⁺. If 𝑓∈𝐿_𝑀,𝜔(𝖳) with 𝜔∈𝐴_𝑝(𝑀) and 𝐸_𝑛(𝑓)_𝑀,𝜔=𝒪(𝑛^−𝜎),𝜎>0,𝑛=1,2,…,(2.95) then Ω^{𝑟𝑀,𝜔}(𝑓,𝛿)=⎧⎪⎨⎪⎩𝒪(𝛿^𝜎𝒪𝛿);𝑟>𝜎,_𝜎|||log1𝒪(𝛿_𝛼𝛿|||;𝑟=𝜎,);𝑟<𝜎,(2.96) hold.

Definition 2.10. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑟∈ℝ⁺. If 𝑓∈𝐿_𝑀,𝜔(𝖳) and 𝜔∈𝐴_𝑝(𝑀) then for 0<𝜎<𝑟 we set Lip𝜎(𝑟,𝑀,𝜔)∶={𝑓∈𝐿_𝑀,𝜔(𝖳)∶Ω^{𝑟𝑀,𝜔}(𝑓,𝛿)=𝒪(𝛿^𝜎),𝛿>0}.

Corollary 2.11. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑟∈ℝ⁺. If 𝑓∈𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀), 0<𝜎<𝑟 and 𝐸_𝑛(𝑓)_𝑀,𝜔=𝒪(𝑛^−𝜎),𝑛=1,2,…, then 𝑓∈Lip𝜎(𝑟,𝑀,𝜔).

Corollary 2.12. Let 0<𝜎<𝑟 and let 𝑓∈𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀), where 𝑀∈𝑄𝐶^𝜃2(0,1). Then the following conditions are equivalent: ((ba))𝑓𝐸_𝑛∈Lip𝜎(𝑟,𝑀,𝜔),(𝑓)_𝑀,𝜔=𝒪(𝑛^−𝜎),𝑛=1,2,….(2.97)

Theorem 2.13. Let 𝑓∈𝐿_𝑀,𝜔(𝖳), 𝜔∈𝐴_𝑝(𝑀), where 𝑀∈𝑄𝐶^𝜃2(0,1). If 𝛼∈(0,∞) and ^∞_𝜈=1𝜈^𝛼−1𝐸_𝜈(𝑓)_𝑀,𝜔<∞,(2.98) then 𝐸_𝑛𝑓^(𝛼)_𝑀,𝜔≤𝑐(𝑛+1)^𝛼𝐸_𝑛(𝑓)_𝑀,𝜔+^∞_𝜈=_𝑛+1𝜈^𝛼−1𝐸_𝜈(𝑓)_𝑀,𝜔(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)∈𝐿_𝑀,𝜔(𝖳). Since ‖‖𝑓^(𝛼)−𝑆_𝑛𝑓^(𝛼)‖‖_𝑀,𝜔≤‖‖𝑆₂^𝑚+2𝑓^(𝛼)−𝑆_𝑛𝑓^(𝛼)‖‖_𝑀,𝜔+^∞_𝑘=_𝑚+2‖‖𝑆_2𝑘+1𝑓^(𝛼)−𝑆₂^𝑘𝑓^(𝛼)‖‖_𝑀,𝜔,(2.100) we have for 2^𝑚<𝑛<2^𝑚+1‖‖𝑆_2𝑚+2𝑓^(𝛼)−𝑆_𝑛𝑓^(𝛼)‖‖_𝑀,𝜔≤𝑐2^(𝑚+2)𝛼𝐸_𝑛(𝑓)_𝑀,𝜔≤𝐶(𝑛+1)^𝛼𝐸_𝑛(𝑓)_𝑀,𝜔.(2.101) On the other hand, we find ^∞_𝑘_=𝑚+2‖‖𝑆_2𝑘+1𝑓^(𝛼)−𝑆₂^𝑘𝑓^(𝛼)‖‖_𝑀,𝜔≤𝑐^∞_𝑘=_𝑚+22^(𝑘+1)𝛼𝐸₂^𝑘(𝑓)_𝑀,𝜔≤𝐶^∞_2𝑘=𝑚+2^𝑘_{𝜇=2𝑘−1+1}𝜇^𝛼−1𝐸_𝜇(𝑓)_𝑀,𝜔=𝑐^∞_𝜈=2_𝑚+1+1𝜈^𝛼−1𝐸_𝜈(𝑓)_𝑀,𝜔≤𝑐^∞_𝜈=_𝑛+1𝜈^𝛼−1𝐸_𝜈(𝑓)_𝑀,𝜔,(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 𝑓∈𝑊^𝛼𝑀(𝖳,𝜔), 𝜔∈𝐴_𝑝(𝑀), 𝑟∈(0,∞), and ^∞_𝜈=1𝜈^𝛼−1𝐸_𝜈(𝑓)_𝑀,𝜔<∞(2.103) for some 𝛼>0. In this case for 𝑛=0,1,2,…, there exists a constant 𝑐>0 dependent only on 𝛼, 𝑟, and 𝑀 such that Ω^{𝑟𝑀,𝜔}𝑓^(𝛼),𝜋𝑛+1≤(𝑛+𝑐1)^𝑟𝑛_𝜈=0(𝜈+1)^{𝛼+𝑟−1}𝐸_𝜈(𝑓)_𝑀,𝜔+𝑐^∞_𝜈=_𝑛+1𝜈^𝛼−1𝐸_𝜈(𝑓)_𝑀,𝜔(2.104) hold.

3. Near Best Approximants in Weighted Smirnov-Orlicz Space

Let 𝑤=𝜑(𝑧) and 𝑤=𝜑₁(𝑧) be the conformal mappings of 𝐺_∞ and 𝐺₀ onto the complement 𝔻_∞ of 𝔻, normalized by the conditions𝜑𝜑(∞)=∞,lim_𝑧→∞𝜑(𝑧)/𝑧>0,₁(0)=∞,lim_𝑧→0𝑧𝜑₁(𝑧)>0,(3.1) respectively. We denote by 𝜓 and 𝜓₁ the inverse mappings of 𝜑 and 𝜑₁, respectively, and 𝕋∶=𝜕𝔻. These mappings 𝜓 and 𝜓₁ have in some deleted neighborhood of ∞ the representations𝜓(𝑤)=𝛼𝑤+𝛼₀+^∞_𝑘=1𝛼_𝑘𝑤^𝑘,𝛼>0,𝜓₁(𝑤)=^∞_𝑙=1𝛽_𝑙𝑤^𝑙,𝛽₁>0.(3.2) Therefore, the functions𝜓^′(𝑤)𝜓(𝑤)−𝑧,𝑧∈𝐺⁰,𝜓^1(𝑤)𝜓₁(𝑤)−𝑧,𝑧∈𝐺^∞(3.3) are analytic in 𝔻_∞ and have, respectively, simple zero and zero of order 2 at ∞. Hence they have expansions𝜓^(𝑤)𝜓(𝑤)−𝑧=^∞_𝑘=0𝐹_𝑘(𝑧)𝑤^𝑘+1𝜓_1,𝑧∈𝐺₀,𝑤∈𝔻_∞,(𝑤)𝜓₁(𝑤)−𝑧=^∞_𝑘=1𝐹_𝑘(1/𝑧)𝑤^𝑘+1,𝑧∈𝐺_∞,𝑤∈𝔻_∞,(3.4) where 𝐹_𝑘(𝑧) and 𝐹_𝑘(1/𝑧) are, respectively, Faber Polynomials of degree 𝑘 for continuums 𝐺₀ and ℂ⧵𝐺₀, with the integral representations [38, pp. 35, 255]𝐹_𝑘(𝑧)=12𝜋𝑖_𝕋𝑤^𝑘𝜓^(𝑤)𝜓(𝑤)−𝑧𝑑𝑤,𝑧∈𝐺⁰𝐹_𝑘,1𝑧=21𝜋𝑖_𝕋𝑤^𝑘𝜓^1(𝑤)𝜓₁(𝑤)−𝑧𝑑𝑤,𝑧∈𝐺^∞𝐹,(3.5)_𝑘(𝑧)=𝜑^𝑘(𝑧)+12𝜋𝑖_Γ𝜑^𝑘(𝜍)𝜍−𝑧𝑑𝜍,𝑧∈𝐺^∞𝐹,𝑘=0,1,2,…,(3.6)_𝑘1𝑧=𝜑^𝑘1(𝑧)−21𝜋𝑖_Γ𝜑^𝑘1(𝜍)𝜍−𝑧𝑑𝜍,𝑧∈𝐺⁰⧵{0}.(3.7) We put𝑎_𝑘∶=𝑎_𝑘(𝑓)∶=12𝜋𝑖_𝕋𝑓₀(𝑤)𝑤^𝑘+1𝑎_𝑘̃𝑑𝑤,𝑘=0,1,2,…,∶=𝑎_𝑘̃(𝑓)∶=12𝜋𝑖_𝕋𝑓₁(𝑤)𝑤^𝑘+1𝑑𝑤,𝑘=1,2,…(3.8) and correspond the series^∞_𝑘=0𝑎_𝑘𝐹_𝑘(𝑧)+^∞_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧(3.9) with the function 𝑓∈𝐿¹(Γ), that is,𝑓(𝑧)∼^∞_𝑘=0𝑎_𝑘𝐹_𝑘(𝑧)+^∞_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧.(3.10) This series is called the Faber-Laurent series of the function 𝑓 and the coefficients 𝑎_𝑘 and 𝑎_𝑘̃ 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 Γ∈𝐶(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 𝐸^𝑝(𝐺₀), 1<𝑝<∞, have the same order of convergence. In [42] considering Γ∈𝐶(1,𝛼) and choosing the interpolation nodes as the zeros of the Faber polynomials, Zhu obtain similar result.

In the above-cited works, Γ does not admit corners, whereas many domains in the complex plain may have corners. When Γ 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 𝑓 in the 𝐸^𝑝(𝐺₀), 1<𝑝<∞ norm.

A function 𝜔∶Γ→[0,∞] is called a weight on Γ, if 𝜔 is measurable and 𝜔⁻¹({0,∞}) has measure zero. We denote by 𝐿_𝑀,𝜔(Γ) the linear space of Lebesgue measurable functions 𝑓∶Γ→ℂ satisfying the condition_Γ𝑀𝛼||𝑓(𝑧)||𝜔(𝑧)||𝑑𝑧||<∞(3.11) for some 𝛼>0.

The space 𝐿_𝑀,𝜔(Γ) becomes a Banach space with the Orlicz norm ‖𝑓‖_𝑀,Γ,𝜔∶=sup_Γ||𝑓(𝑧)𝑔(𝑧)||𝜔(𝑧)||𝑑𝑧||∶𝑔∈𝐿_𝑁,𝜔(Γ);𝜌(𝑔;𝑁)≤1,(3.12) where 𝑁 is the complementary function of 𝑀 and𝜌(𝑔;𝑁)∶=_Γ𝑁||𝑔(𝑧)||𝜔(𝑧)||𝑑𝑧||.(3.13)

The Banach space 𝐿_𝑀,𝜔(Γ) is called weighted Orlicz space on Γ.

For 𝑧∈Γ and 𝜖>0 let Γ(𝑧,𝜖)∶={𝑡∈Γ∶|𝑡−𝑧|<𝜖}. For fixed 𝑝∈[1,∞), the set of all weights 𝜔∶Γ→[0,∞] satisfying the relationsup_𝑧∈Γsup_𝜖>01𝜖_{Γ(𝑧,𝜖)}𝜔(𝜏)||𝑑𝜏||1𝜖_{Γ(𝑧,𝜖)}sup_𝜖>0𝜔(𝜏)^{−1/(𝑝−1)}||𝑑𝜏||^𝑝−1<∞,if𝑝>1,1𝜖_{Γ(𝑧,𝜖)}𝜔(𝜏)||𝑑𝜏||≤𝑐𝜔(𝑧),∀𝑧∈Γ,if𝑝=1(3.14) is denoted by 𝐴_𝑝(Γ).

We denote by 𝐿^𝑝(Γ,𝜔) the set of all measurable functions 𝑓∶Γ→ℂ such that |𝑓|𝜔^1/𝑝 belongs to Lebesgue space 𝐿^𝑝(Γ), 1≤𝑝<∞, on Γ.

Definition 3.1. Let 𝜔 be a weight on Γ and let 𝐸_𝑀,𝜔(𝐺₀)∶={𝑓∈𝐸¹(𝐺₀)∶𝑓∈𝐿_𝑀,𝜔(Γ)}, 𝐸_𝑀,𝜔(𝐺_∞)∶={𝑓∈𝐸¹(𝐺_∞)∶𝑓∈𝐿_𝑀,𝜔(Γ)}, 𝐸_𝑀,𝜔(𝐺_∞)∶={𝑓∈𝐸_𝑀,𝜔(𝐺_∞)∶𝑓(∞)=0}. The classes of functions 𝐸_𝑀,𝜔(𝐺₀) and 𝐸_𝑀,𝜔(𝐺_∞) will be called weighted Smirnov-Orlicz classes with respect to domains 𝐺₀ and 𝐺_∞, respectively.

In this chapter, we prove that the convergence rate of the interpolating polynomials based on the zeros of the 𝐹_𝑛 is the same with the best approximating algebraic polynomials in the weighted Smirnov-Orlicz class 𝐸_𝑀,𝜔(𝐺₀) under the assumption that Γ is a closed Radon curve. This means that interpolating polynomials based on the zeros of the Faber polynomials are near best approximant of 𝑓 belonging to weighted Smirnov-Orlicz class 𝐸_𝑀,𝜔(𝐺₀).

In the case that all of the zeros of the 𝑛th Faber polynomial 𝐹_𝑛 are in 𝐺₀, we denote by 𝐿_𝑛(𝑓,⋅) the (𝑛−1)th interpolating polynomial for 𝑓∈𝐸_𝑀,𝜔(𝐺₀) based on the zeros of 𝐹_𝑛.

Let 𝑓∈𝐿¹(Γ). Then the functions 𝑓⁺ and 𝑓⁻ defined by𝑓⁺(𝑧)=12𝜋𝑖_Γ𝑓(𝜍)𝜍−𝑧𝑑𝜍,𝑧∈𝐺⁰,𝑓⁻(𝑧)=12𝜋𝑖_Γ𝑓(𝜍)𝜍−𝑧𝑑𝜍,𝑧∈𝐺^∞(3.15) are analytic in 𝐺₀ and 𝐺_∞, respectively, and 𝑓⁻(∞)=0.

We denote by𝐸_𝑛(𝑓)_𝑀,Γ,𝜔∶=inf‖𝑓−𝑝‖_𝑀,Γ,𝜔∶𝑝∈𝒫_𝑛(3.16) the minimal error of approximation by polynomials of 𝑓, where 𝒫_𝑛 is the set of algebraic polynomials of degree not greater than 𝑛.

Let Γ be a rectifiable Jordan curve, 𝑓∈𝐿¹(Γ), and let𝑆_Γ𝑓(𝑡)∶=lim_𝜀→012𝜋𝑖_{Γ⧵Γ(𝑡,𝜀)}𝑓(𝜍)𝜍−𝑡𝑑𝜍,𝑡∈Γ(3.17) be Cauchy’s singular integral of 𝑓 at the point 𝑡. The linear operator 𝑆_Γ∶𝑓→𝑆_Γ𝑓 is called the Cauchy singular operator.

If one of the functions 𝑓⁺ or 𝑓⁻ has the nontangential limits a.e. on Γ, then 𝑆_Γ𝑓(𝑧) exists a.e. on Γ and also the other one has the nontangential limits a.e. on Γ. Conversely, if 𝑆_Γ𝑓(𝑧) exists a.e. on Γ, then both functions 𝑓⁺ and 𝑓⁻ have the nontangential limits a.e. on Γ. In both cases, the formulae𝑓⁺(𝑧)=𝑆_Γ𝑓(𝑧)+𝑓(𝑧)2,𝑓⁻(𝑧)=𝑆_Γ𝑓(𝑧)−𝑓(𝑧)2(3.18) hold, and hence𝑓=𝑓⁺−𝑓⁻(3.19) holds a.e. on Γ (see, e.g., [1, page 431]).

Lemma 3.2. f Γ is a regular curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝜔∈𝐴_𝑝(𝑀)(Γ), then for every 𝑓∈𝐸_𝑀,𝜔(𝐺₀) one has ‖‖𝑆_Γ(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝑐‖𝑓‖_𝑀,Γ,𝜔,(3.20) where the constant c depends only on Γ and 𝑀.

Proof. Assertion (3.20) immediately follows from modular inequality _Γ𝑀||𝑆_Γ(𝑓,𝑡)||𝜔(𝑡)𝑑𝑡≤𝑐_Γ𝑀||𝑓(𝑡)||𝜔(𝑡)𝑑𝑡(3.21) given in (7.5.13) of [23].

Theorem 3.3. If Γ is a closed Radon curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝜔∈𝐴_𝑝(𝑀)(Γ), then for every 𝑓∈𝐸_𝑀,𝜔(𝐺₀) one has ‖‖𝑓−𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝑐𝐸_𝑛(𝑓)_𝑀,Γ,𝜔,(3.22) where the constant 𝑐 depends only on Γ and 𝑀.

Proof. First of all we know [16] that all zeros of the Faber polynomials are in 𝐺₀. Since interpolating operator 𝐿_𝑛(𝑓,⋅) is linear and corresponds 𝑓 by a polynomial of degree not more than 𝑛−1, we need only to show that, for large values of 𝑛, 𝐿_𝑛(𝑓,⋅) is uniformly bounded in weighted Smirnov-Orlicz class 𝐸_𝑀,𝜔(𝐺₀). We suppose that 𝑃_𝑛−1 is the (𝑛−1)th best approximating algebraic polynomial for 𝑓 in 𝐸_𝑀,𝜔(𝐺₀). In this case we have ‖‖𝑓−𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔=‖‖𝑓−𝑃_𝑛−1−𝐿_𝑛𝑓−𝑃_𝑛−1,⋅‖‖_𝑀,Γ,𝜔≤1+‖‖𝐿_𝑛‖‖‖‖𝑓−𝑃_𝑛−1‖‖_𝑀,Γ,𝜔.(3.23) Since we assumed the interpolation nodes as the zeros of the Faber polynomials 𝐹_𝑛, using [39, page 59], we have 𝑓(𝑧)−𝐿_𝑛(𝑓,𝑧)=𝐹^𝑛(𝑧)2𝜋𝑖_Γ𝑓(𝜍)𝐹_𝑛(𝜍)(𝜍−𝑧)𝑑𝜍=𝐹^𝑛(𝑧)𝑆_Γ𝐹𝑓_𝑛(𝑧),𝑧∈𝐺⁰(3.24) and consequently ‖‖𝑓(𝑧)−𝐿_𝑛(𝑓,𝑧)‖‖_𝑀,Γ,𝜔=‖‖‖𝐹_𝑛(⋅)𝑆_Γ𝐹𝑓_𝑛(⋅)‖‖‖_𝑀,Γ,𝜔≤max_𝑧∈Γ||𝐹_𝑛(𝑧)||‖‖‖𝑆_Γ𝐹𝑓_𝑛‖‖‖_𝑀,Γ,𝜔.(3.25) By Lemma 3.2, we get ‖‖𝑓−𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝑐max_𝑧∈Γ||𝐹_𝑛(𝑧)||‖‖‖𝑓𝐹_𝑛‖‖‖_𝑀,Γ,𝜔≤𝑐max_{𝑧,𝜍∈Γ}||||𝐹_𝑛(𝑧)𝐹_𝑛(𝜍)||||‖𝑓‖^𝑀,Γ,𝜔.(3.26) We set 𝜅∶=max_𝑧∈Γ|𝜗_𝑧−1|, where 𝜗_𝑧𝜋 is the exterior angle of the point 𝑧∈Γ. By the Radon assumption on Γ we get 0≤𝜅<1. Then one can find for 𝑧∈Γ0,5−0,5⋅𝜅<||𝐹_𝑛(𝑧)||<1,5+0,5⋅𝜅,(3.27) and therefore max_{𝑧,𝜍∈Γ}||||𝐹_𝑛(𝑧)𝐹_𝑛(𝜍)||||≤31+−𝜅𝜅.(3.28) From the last inequality we obtain ‖‖𝑓−𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝑐3+𝜅1−𝜅‖𝑓‖^𝑀,Γ,𝜔.(3.29) Since ‖‖𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤‖𝑓‖_𝑀,Γ,𝜔+‖‖𝑓−𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤1+𝑐3+𝜅1−𝜅‖𝑓‖^𝑀,Γ,𝜔,(3.30) we obtain that 𝐿_𝑛(𝑓,⋅) is uniformly bounded in 𝐸_𝑀,𝜔(𝐺₀), namely, ‖‖𝐿_𝑛‖‖≤𝑐.(3.31) Therefore, we conclude that ‖‖𝑓−𝐿_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝑐‖‖𝑓−𝑃_𝑛−1‖‖_𝑀,Γ,𝜔=𝑐𝐸_𝑛(𝑓)_𝑀,Γ,𝜔(3.32) and interpolating polynomial 𝐿_𝑛(𝑓,⋅) is near best approximant for 𝑓.

If Γ is Dini-smooth, then [44] there exist constants 𝑐 and 𝐶 such that0<𝑐<|||𝜓^′(𝑤)|||<𝐶<∞,|𝑤|≥1.(3.33) Similar inequalities hold also for 𝜓^1 and 𝜑^1, in case of |𝑤|=1 and 𝑧∈Γ, respectively.

We define Poisson polynomial for function 𝑓∈𝐸_𝑀,𝜔(𝐺₀)𝑉_𝑛(𝑓,𝑧)∶=^𝑛_𝑘=0𝑐_𝑘𝐹_𝑘(𝑧)+^2𝑛−1_𝑘_=𝑛+12−𝑘𝑛𝑐^𝑘𝐹_𝑘(𝑧),𝑧∈𝐺₀.(3.34)

Theorem 3.4. If Γ is a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝜔∈𝐴_𝑝(𝑀)(Γ), then for every 𝑓∈𝐸_𝑀,𝜔(𝐺₀) one has ‖‖𝑓−𝑉_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝑐𝐸_𝑛(𝑓)_𝑀,Γ,𝜔,(3.35) where the constant 𝑐 depends only on Γ and 𝑀.

Proof. From (3.8) and (3.5), we have 𝑉_𝑛(𝑓,𝑧)=12𝜋^02𝜋𝑓𝜓𝑒^𝑖𝑡𝑑𝑡2𝜋𝑖_Γ∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝜑^𝑘(𝜍)𝑒^{−𝑖𝑘𝑡}𝜍−𝑧𝑑𝜍,(3.36) where 𝑧∈𝐺₀ and 𝜆_|𝑘|∶=12,−𝑘0≤𝑘≤𝑛,𝑛,𝑛+1≤𝑘≤2𝑛−1.(3.37) If 𝑃_𝑛∈𝒫_𝑛 is near best approximant for 𝑓∈𝐸_𝑀,𝜔(𝐺₀), we get ‖‖𝑓−𝑉_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝐸_𝑛(𝑓)_𝑀,Γ,𝜔+‖‖𝑃_𝑛−𝑉_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔.(3.38) Using 𝑃_𝑛(𝑧)=12𝜋^02𝜋𝑃_𝑛𝜓𝑒^𝑖𝑡𝑑𝑡2𝜋𝑖_Γ∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆^𝑘||𝑘||𝜑(𝜍)𝑒^{−𝑖𝑘𝑡}𝜍−𝑧𝑑𝜍,𝑧∈𝐺⁰,(3.39) we find 𝑃_𝑛(𝑧=)−1𝑉_𝑛(𝑓,𝑧)2𝜋^02𝜋𝑃_𝑛𝜓𝑒^𝑖𝑡−𝑓𝜓𝑒^𝑖𝑡2𝑑𝜋𝑡𝑖_Γ∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆^𝑘||𝑘||𝜑(𝜍)𝑒^{−𝑖𝑘𝑡}𝜍−𝑧𝑑𝜍,𝑧∈𝐺⁰.(3.40) Taking in the last inequality, the nontangential boundary values from inside of Γ, 𝑧→𝑧₀∈Γ and using (3.18), we have 𝑃_𝑛𝑧₀−𝑉_𝑛𝑓,𝑧₀=1×2𝜋⎡⎢⎢⎣1^02𝜋𝑃_𝑛𝜓𝑒^𝑖𝑡−𝑓𝜓𝑒^𝑖𝑡𝑑𝑡2^2𝑛−1_{𝑘=−(2𝑛−1)}𝜆^𝑘||𝑘||𝜑𝑧₀𝑒^{−𝑖𝑘𝑡}+12𝜋𝑖_Γ∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆^𝑘||𝑘||𝜑(𝜍)𝑒^{−𝑖𝑘𝑡}𝜍−𝑧₀𝑑𝜍⎤⎥⎥⎦.(3.41) Since 𝜑^−2𝑛(𝜍)∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝜑^𝑘(𝜍)𝑒^{−𝑖𝑘𝑡} is analytic in 𝐺_∞, we have 12𝜋𝑖_Γ∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝜑^𝑘(𝜍)𝑒^{−𝑖𝑘𝑡}(𝜍−𝑧)𝜑^2𝑛(𝜍)𝑑𝜍=0,𝑧∈𝐺⁰,(3.42) and taking nontangential limit in (3.42) we get 12^2𝑛−1_{𝑘=−(2𝑛−1)}𝜆_|𝑘|𝜑^𝑘𝑧₀𝑒^{−𝑖𝑘𝑡}=−12𝜋𝑖_Γ𝜑^2𝑛𝑧₀𝜑^2𝑛(𝜍)∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝜑^𝑘(𝜍)𝑒^{−𝑖𝑘𝑡}𝜍−𝑧₀𝑑𝜍,(3.43) and hence by transformation 𝑧₀=𝜓(𝑤₀) we obtain 𝑃_𝑛𝑧=₀−1𝑉_𝑛𝑓,𝑧₀2𝜋^02𝜋𝑃_𝑛𝜓𝑒^𝑖𝑡−𝑓𝜓𝑒^𝑖𝑡2𝑑𝜋𝑡𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}𝜓(𝑤)−𝜓𝑤₀𝜓^(𝑤)𝑑𝑤.(3.44) Since one has 12𝜋𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛∑^{2𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}𝜓(𝑤)−𝜓𝑤₀𝜓^=1(𝑤)𝑑𝑤2𝜋𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛^2𝑛−1_{𝑘=−(2𝑛−1)}𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}𝜓^(𝑤)+1𝜓(𝑤)−𝜓𝑤₀−𝑤−1𝑤₀𝑑𝑤2𝜋𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛𝑤−1𝑤_{02𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡},(3.45) we can write 𝑃_𝑛𝑧=₀−1𝑉_𝑛𝑓,𝑧₀×2𝜋1^02𝜋𝑃_𝑛𝜓𝑒^𝑖𝑡−𝑓𝜓𝑒^𝑖𝑡𝑑𝑡2𝜋𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛^2𝑛−1_{𝑘=−(2𝑛−1)}𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}𝜓^(𝑤)+1𝜓(𝑤)−𝜓𝑤₀−𝑤−1𝑤₀𝑑𝑤2𝜋^02𝜋𝑃_𝑛𝜓𝑒^𝑖𝑡−𝑓𝜓𝑒^𝑖𝑡2𝑑𝜋𝑡𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛𝑤−1𝑤₀×^2𝑛−1_𝑘=−_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}∶=𝐼₁+𝐼₂.(3.46) From equality 12𝜋𝑖_𝕋1−𝑤𝑤^02𝑛2𝑛𝑤−1𝑤_{02𝑛−1𝑘=−(2𝑛−1)}𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}𝑑𝑤=^2𝑛−1_𝑘=−_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘0𝑒^{−𝑖𝑘𝑡},(3.47) we have ‖‖𝐼₂‖‖_𝑀,Γ,𝜔≤𝐸_𝑛(𝑓)_𝑀,Γ,𝜔12𝜋^02𝜋|||||_{2𝑛−1𝑘=−}_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘0𝑒^{−𝑖𝑘𝑡}|||||𝑑𝑡.(3.48) On the other hand, ‖‖𝐼₁‖‖_𝑀,Γ,𝜔≤𝐸_𝑛(𝑓)_𝑀,Γ,𝜔14𝜋²^02𝜋|||||_{2𝑛−1𝑘=−}_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}|||||_𝕋||||1−𝑤^02𝑛𝑤^2𝑛||||||||𝜓^(𝑤)𝜓(𝑤)−𝜓𝑤₀−𝑤−1𝑤₀||||||𝑑𝑤||.(3.49) We denote by 𝐴 a subarc of 𝕋 with the center 𝑤₀ such that it has arc lenght 𝒪(1/𝑛). In this case _𝐴||||1−𝑤^02𝑛𝑤^2𝑛||||||||𝜓^(𝑤)𝜓(𝑤)−𝜓𝑤₀−𝑤−1𝑤₀≤||||||𝑑𝑤_𝐴||||𝑤^2𝑛−𝑤^02𝑛||||||||𝜓^(𝑤)||||𝜓(𝑤)−𝜓𝑤₀||−||𝑤−1𝑤₀||||||||𝑑𝑤||≤𝑐(3.50) and, by (1.3), _𝕋⧵𝐴||||1−𝑤^02𝑛𝑤^2𝑛||||||||𝜓^(𝑤)𝜓(𝑤)−𝜓𝑤₀−𝑤−1𝑤₀≤2_𝕋⧵𝐴||||||𝑑𝑤||||𝜓^(𝑤)𝑤−𝑤₀−𝜓(𝑤)−𝜓𝑤₀||≤𝑐_11/𝑛||𝜓(𝑤)−𝜓𝑤₀||||𝑤−𝑤₀||||𝑑𝑤||𝜔𝜓^,𝑡𝑡𝑑𝑡≤𝑐.(3.51) Hence, ‖‖𝐼₁‖‖_𝑀,Γ,𝜔≤𝐸_𝑛(𝑓)_𝑀,Γ,𝜔^02𝜋|||||_{2𝑛−1𝑘=−}_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}|||||𝑑𝑡.(3.52) Inequalities (3.46), (3.48), and (3.52) imply that ‖‖𝑃_𝑛−𝑉_𝑛(𝑓,⋅)‖‖_𝑀,Γ,𝜔≤𝐸_𝑛(𝑓)_𝑀,Γ,𝜔⎧⎪⎨⎪⎩^02𝜋|||||_{2𝑛−1𝑘=−}_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘0𝑒^{−𝑖𝑘𝑡}|||||+_02𝜋|||||_{2𝑛−1𝑘=−}_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}|||||⎫⎪⎬⎪⎭.(3.53) For every 𝑤∈𝕋, one has ^02𝜋|||||_{2𝑛−1𝑘=−}_(2𝑛−1)𝜆_|𝑘|𝑤^𝑘𝑒^{−𝑖𝑘𝑡}|||||≤𝑐,(3.54) and therefore we get the required inequality of Theorem 3.4.

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

For 𝑔∈𝐿_𝑀,𝜔(𝕋), we set𝜎_ℎ(𝑔)(𝑤)∶=12ℎ^ℎ−ℎ𝑔𝑤𝑒^𝑖𝑡𝑑𝑡,0<ℎ<𝜋,𝑤∈𝕋.(3.55) If 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝜔∈𝐴_𝑝(𝑀)(𝕋), then by Theorem A(ii) we have‖‖𝜎_ℎ(𝑔)‖‖_{𝑀,𝕋,𝜔}≤𝑐‖𝑔‖_{𝑀,𝕋,𝜔},(3.56) and consequently 𝜎_ℎ(𝑔)∈𝐿_𝑀,𝜔(𝕋) for any 𝑔∈𝐿_𝑀,𝜔(𝕋).

Definition 3.5. Let 𝑀∈𝑄𝐶^𝜃2(0,1), 𝜔∈𝐴_𝑝(𝑀)(𝕋), and 𝑟>0. The function Ω^{𝑟𝑀,𝕋,𝜔}(𝑔,𝛿)∶=sup_{0<ℎ𝑖,𝑡≤𝛿}‖‖‖‖^[𝑟]_𝑖=1𝐼−𝜎_ℎ𝑖𝐼−𝜎_𝑡^{𝑟−[𝑟]}𝑔‖‖‖‖_{𝑀,𝕋,𝜔},𝛿>0(3.57) is called 𝑟th modulus of smoothness of 𝑔∈𝐿_𝑀,𝜔(𝕋).

It can easily be verified that the function Ω^{𝑟𝑀,𝕋,𝜔}(𝑔,⋅) is continuous, nonnegative, subadditive and satisfy lim_𝛿→0Ω^{𝑟𝑀,𝕋,𝜔}(𝑔,𝛿)=0 for 𝑔∈𝐿_𝑀,𝜔(𝕋).

Let Γ be a Dini-smooth curve and 𝜔 be a weight on Γ. We associate with 𝜔 the following two weights defined on 𝕋 by𝜔₀∶=𝜔∘𝜓,𝜔₁∶=𝜔∘𝜓₁(3.58) and let 𝑓₀∶=𝑓∘𝜓,𝑓₁∶=𝑓∘𝜓₁ for 𝑓∈𝐿_𝑀,𝜔(Γ). Then from (3.33), we have 𝑓₀∈𝐿_𝑀,𝜔0(𝕋) and 𝑓₁∈𝐿_𝑀,𝜔1(𝕋) for 𝑓∈𝐿_𝑀,𝜔(Γ). Using the nontangential boundary values of 𝑓⁺⁰ and 𝑓⁺¹ on 𝕋, we defineΩΩ^{𝑟𝑀,Γ,𝜔}(𝑓,𝛿)∶=Ω^{𝑟𝑀,𝕋,𝜔}₀𝑓⁺⁰,𝛿_{𝑟𝑀,Γ,𝜔},(𝑓,𝛿)∶=Ω^{𝑟𝑀,𝕋,𝜔}₁𝑓⁺¹,𝛿(3.59) for 𝑟,𝛿>0.

We set𝐸_𝑛(𝑓)_{𝑀,𝕋,𝜔}∶=inf_{𝑃∈𝒫𝑛(𝔻)}‖𝑓−𝑃‖_{𝑀,𝕋,𝜔},𝐸_𝑛(𝑔)_𝑀,Γ,𝜔∶=inf_{𝑅∈ℛ𝑛}‖𝑔−𝑅‖_𝑀,Γ,𝜔,(3.60) where 𝑓∈𝐸_𝑀,𝜔(𝔻), 𝑔∈𝐸_𝑀,𝜔(𝐺_∞), and ℛ_𝑛 is the set of rational functions of the form ∑^𝑛𝑘=0𝑎_𝑘𝑧^−𝑘.

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

Theorem 3.6. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐿_𝑀,𝜔(Γ) with 𝜔∈𝐴₁(Γ). Then there is a constant 𝑐>0 such that for any natural number 𝑛‖‖𝑓−𝑅_𝑛(⋅,𝑓)‖‖_𝑀,Γ,𝜔≤𝑐Ω^{𝑟𝑀,Γ,𝜔}𝑓,(𝑛+11)+Ω^{𝑟𝑀,Γ,𝜔}𝑓,(𝑛+11),(3.61) where 𝑟>0 and 𝑅_𝑛(⋅,𝑓) is the 𝑛th partial sum of the Faber-Laurent series of 𝑓.

Corollary 3.7. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐸_𝑀,𝜔(𝐺₀) with 𝜔∈𝐴₁(Γ). Then there is a constant 𝑐>0 such that for every natural number 𝑛‖‖𝑓−𝑃_𝑛(⋅,𝑓)‖‖_𝑀,Γ,𝜔≤𝑐Ω^{𝑟𝑀,Γ,𝜔}𝑓,(𝑛+11),𝑟>0,(3.62) where 𝑃_𝑛(⋅,𝑓) is the 𝑛th partial sum of the Faber series of 𝑓.

Corollary 3.8. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐸_𝑀,𝜔(𝐺_∞) with 𝜔∈𝐴₁(Γ). Then there is a constant 𝑐>0 such that for every natural number 𝑛‖‖𝑓−𝑅_𝑛(⋅,𝑓)‖‖_𝑀,Γ,𝜔≤𝑐₈Ω^{𝑟𝑀,Γ,𝜔}𝑓,(𝑛+11),𝑟>0,(3.63) where 𝑅_𝑛(⋅,𝑓) is as in Theorem 3.6.

Theorem 3.9. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐸_𝑀,𝜔(𝐺₀) with 𝜔∈𝐴₁(Γ). Then for 𝑟>0 there exists a constant 𝑐>0 such that Ω^{𝑟𝑀,Γ,𝜔}𝑓,1𝑛≤𝑛𝑐^𝑟𝐸⁰(𝑓)_𝑀,Γ,𝜔+^𝑛_𝑘=1𝑘^𝑟−1𝐸_𝑘(𝑓)_𝑀,Γ,𝜔(3.64) hold.

Corollary 3.10. Under the conditions of Corollary 3.7, if 𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=𝒪(𝑛^−𝛼),𝛼>0,𝑛=1,2,3,…,(3.65) then for 𝑓∈𝐸_𝑀,𝜔(𝐺₀) and 𝑟>0Ω^{𝑟𝑀,Γ,𝜔}(𝑓,𝛿)=⎧⎪⎨⎪⎩𝒪(𝛿^𝛼𝒪𝛿),𝑟>𝛼,_𝛼|||log1𝒪(𝛿_𝑟𝛿|||,𝑟=𝛼,),𝑟<𝛼.(3.66)

Definition 3.11. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝛼∈ℝ⁺. If 𝑓∈𝐸_𝑀,𝜔(𝐺₀), then for 0<𝜎<𝛼 we set LLiipp𝜎𝜎((𝛼𝛼,,𝑀𝑀,,ΓΓ,,𝜔𝜔))∶∶==𝑓𝑓∈∈𝐸𝐸_{𝑀,𝜔𝑀,𝜔}𝐺_∞∶Ω^{𝛼𝑀,Γ,𝜔}(𝑓,𝛿)=𝒪(𝛿^𝛼),𝐺₀∶Ω^{𝛼𝑀,Γ,𝜔}(𝑓,𝛿)=𝒪(𝛿^𝜎),𝛿>0.(3.67)

Corollary 3.12. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝛼∈ℝ⁺. If 𝑓∈𝐸_𝑀,𝜔(𝐺₀), 𝜔∈𝐴₁(Γ), 0<𝜎<𝛼 and 𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=𝒪(𝑛^−𝜎),𝑛=1,2,…, then 𝑓∈Lip𝜎(𝛼,𝑀,Γ,𝜔).

By Corollaries 3.7 and 3.10 we have the constructive characterization of the class Lip𝜎(𝛼,𝑀,Γ,𝜔).

Corollary 3.13. Let 0<𝜎<𝛼 and 𝑓∈𝐸_𝑀,𝜔(𝐺₀),𝜔∈𝐴₁(Γ), where 𝑀∈𝑄𝐶^𝜃2(0,1), be fulfilled. Then the following conditions are equivalent:(a)𝑓∈Lip𝜎(𝛼,𝑀,Γ,𝜔). (b)𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=𝒪(𝑛^−𝜎),𝑛=1,2,….

The inverse theorem for unbounded domains has the following form.

Theorem 3.14. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐸_𝑀,𝜔(𝐺_∞) with 𝜔∈𝐴₁(Γ). Then there is a constant 𝑐>0 such that for every natural number 𝑛Ω^{𝑟𝑀,Γ,𝜔}𝑓,1𝑛≤𝑛𝑐^𝑟𝐸⁰(𝑓)_𝑀,Γ,𝜔+^𝑛_𝑘=1𝑘^𝑟−1𝐸_𝑘(𝑓)_𝑀,Γ,𝜔,𝑟>0(3.68) holds.

By the similar way to that of 𝐸_𝑀,𝜔(𝐺₀), we obtain the following corollaries.

Corollary 3.15. Under the conditions of Corollary 3.8, if 𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=𝒪(𝑛^−𝛼),𝛼>0,𝑛=1,2,3,…,(3.69) then for𝑓∈𝐸_𝑀,𝜔(𝐺_∞) and 𝑟>0Ω^{𝑟𝑀,Γ,𝜔}(𝑓,𝛿)=⎧⎪⎨⎪⎩𝒪(𝛿^𝛼𝒪𝛿),𝑟>𝛼,_𝛼|||log1𝒪(𝛿_𝑟𝛿|||,𝑟=𝛼,),𝑟<𝛼.(3.70)

Corollary 3.16. Under the conditions of Theorem 3.14, if 𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=𝒪(𝑛^−𝛼),𝛼>0,𝑛=1,2,3,…,(3.71) then 𝑓∈Lip𝜎(𝛼,𝑀,Γ,𝜔).

By Corollaries 3.8 and 3.15, we have the following.

Corollary 3.17. Let 𝛼>0 and the conditions of Theorem 3.14 be fulfilled. Then the following conditions are equivalent,(a)𝑓∈Lip𝜎(𝛼,𝑀,Γ,𝜔), (b)𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=𝒪(𝑛^−𝛼),𝑛=1,2,3,….

Before the proofs, we need some auxiliary lemmas.

Lemma 3.18. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐿_𝑀,𝜔(Γ) with 𝜔∈𝐴₁(Γ). Then, 𝑓⁺∈𝐸_𝑀,𝜔(𝐺₀) and 𝑓⁻∈𝐸_𝑀,𝜔(𝐺_∞) for every 𝑓∈𝐿_𝑀,𝜔(Γ).

Proof. Using 𝑀∈Δ₂, we can find a 𝑝∈(1,∞) such that 𝐿_𝑀,𝜔(Γ)⊂𝐿^𝑝(Γ,𝜔), where the inclusion maps being continuous (see, e.g., Lemma 2.13 of [20]). Since 𝜔∈𝐴_𝑝(Γ) by [9], we get 𝑓⁺∈𝐸¹(𝐺₀) and 𝑓⁻∈𝐸¹(𝐺_∞). Using 𝜔∈𝐴_𝑝(𝑀)(Γ) and boundedness of operator 𝑆_Γ in 𝐿_𝑀,𝜔(Γ), we obtain from (3.18) that 𝑓⁺∈𝐿_𝑀,𝜔(Γ),𝑓⁻∈𝐿_𝑀,𝜔(Γ).(3.72)

Lemma 3.19. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝜔∈𝐴_𝑝(𝑀)(𝕋). Then there exists a constant 𝑐>0 such that for every natural number 𝑛‖‖𝑔−𝑇_𝑛𝑔‖‖_{𝑀,𝕋,𝜔}≤𝑐Ω^{𝑟𝑀,𝕋,𝜔}𝑔,(𝑛+11),𝑔∈𝐸^𝑀,𝜔(𝔻),(3.73) where 𝑟>0 and 𝑇_𝑛𝑔 is 𝑛th partial sum of the Taylor series of g at the origin.

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

Let 𝒫 be the set of all polynomials (with no restrictions on the degree), and let 𝒫(𝔻) be the set of traces of members of 𝒫 on 𝔻. We define the operators 𝑇∶𝒫(𝔻)→𝐸_𝑀,𝜔(𝐺₀) and 𝑇∶𝒫(𝔻)→𝐸_𝑀,𝜔(𝐺_∞) defined on 𝒫(𝔻) as𝑇(𝑃)(𝑧)∶=12𝜋𝑖_𝕋𝑃(𝑤)𝜓^′(𝑤)𝜓(𝑤)−𝑧𝑑𝑤,𝑧∈𝐺⁰𝑇(𝑃)(𝑧)∶=1,2𝜋𝑖_𝕋𝑃(𝑤)𝜓^1(𝑤)𝜓₁(𝑤)−𝑧𝑑𝑤,𝑧∈𝐺^∞.(3.74)

Then it is readily seen that𝑇^𝑛_𝑘=0𝑏_𝑘𝑤^𝑘=^𝑛_𝑘=0𝑏_𝑘𝐹_𝑘(𝑧),𝑇^𝑛_𝑘=0𝑑_𝑘𝑤^𝑘=^𝑛_𝑘=0𝑑_𝑘𝐹_𝑘1𝑧.(3.75)

If 𝑧^∈𝐺₀, then𝑇(𝑃)𝑧^=12𝜋𝑖_𝕋𝑃(𝑤)𝜓^(𝑤)𝜓(𝑤)−𝑧^𝑑𝑤=12𝜋𝑖_Γ(𝑃∘𝜑)(𝜍)𝜍−𝑧^𝑑𝜍=(𝑃∘𝜑)⁺𝑧^,(3.76) which, by (3.18), implies that𝑇(𝑃)(𝑧)=𝑆_Γ(𝑃∘𝜑)(𝑧)+12(𝑃∘𝜑)(𝑧)(3.77)

a.e. on Γ.

Similarly taking from outside of Γ the nontangential limit 𝑧^→𝑧∈Γ in the relation𝑇(𝑃)𝑧^=12𝜋𝑖_Γ𝑃𝜑₁(𝜍)𝜍−𝑧^𝑑𝜍=𝑃∘𝜑₁⁻𝑧^,𝑧^∈𝐺_∞,(3.78) we get𝑇(𝑃)(𝑧)=−12𝑃∘𝜑¹(𝑧)+𝑆_Γ𝑃∘𝜑₁(𝑧)(3.79)

a.e. on Γ.

Since 𝑆_Γ is bounded in 𝐿_𝑀,𝜔(Γ), we have the following result.

Lemma 3.20. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐿_𝑀,𝜔(Γ) with 𝜔∈𝐴_𝑝(𝑀)(Γ). Then the linear operators 𝑇∶𝒫(𝔻)⟶𝐸_𝑀,𝜔𝐺₀,𝑇∶𝒫(𝔻)⟶𝐸_𝑀,𝜔𝐺_∞(3.80) are bounded.

The set of trigonometric polynomials is dense in 𝐿_𝑀,𝜔(𝖳), which implies density of the algebraic polynomials in 𝐸_𝑀,𝜔(𝔻). Consequently, from Lemma 3.20, we can extend the operators 𝑇 and 𝑇 from 𝒫(𝔻) to the spaces 𝐸_𝑀,𝜔0(𝔻) and 𝐸_𝑀,𝜔1(𝔻) as linear and bounded operators, respectively, and for the extensions 𝑇∶𝐸_𝑀,𝜔0(𝔻)→𝐸_𝑀,𝜔(𝐺₀) and 𝑇∶𝐸_𝑀,𝜔1(𝔻)→𝐸_𝑀,𝜔(𝐺_∞), we have the representations𝑇(𝑔)(𝑧)=12𝜋𝑖_𝕋𝑔(𝑤)𝜓^(𝑤)𝜓(𝑤)−𝑧𝑑𝑤,𝑧∈𝐺⁰𝑇(𝑔)(𝑧)=1,𝑔∈𝐸_𝑀,𝜔0(𝔻),2𝜋𝑖_𝕋𝑔(𝑤)𝜓^1(𝑤)𝜓₁(𝑤)−𝑧𝑑𝑤,𝑧∈𝐺^∞,𝑔∈𝐸_𝑀,𝜔1(𝔻).(3.81)

Lemma 3.21. Let 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐿_𝑀,𝜔(𝕋) with 𝜔∈𝐴_𝑝(𝑀)(𝕋). Then, ‖‖𝑃_𝑟(𝑓)−𝑓‖‖_{𝑀,𝕋,𝜔}⟶0,as𝑟⟶1⁻,(3.82) where 𝑃_𝑟(𝑓)𝑒^𝑖𝜃∶=12𝜋^02𝜋𝑃(𝑟,𝜃−𝑡)𝑓𝑒^𝑖𝑡𝑑𝑡,0<𝑟<1(3.83) and 𝑃(𝑟,𝜃−𝑡) is the Poisson kernel.

Proof. There are numbers 𝑝 and 𝑞 such that 1<𝑝<𝑝(𝑀)<𝑞<∞,𝜔∈𝐴_𝑝(𝕋).(3.84) Since [46, Theorem 10] 𝑃_𝑟 is a bounded operator in 𝐿^𝑝(𝕋,𝜔) for every 1<𝑝<∞, we have by Marcinkiewicz Interpolation Theorem ‖‖𝑃_𝑟(𝑓)‖‖_{𝑀,𝕋,𝜔}≤𝑐‖𝑓‖_{𝑀,𝕋,𝜔}.(3.85) From density of trigonometric polynomials in 𝐿_𝑀,𝜔(𝖳), we have density of the set of continuous functions on 𝕋 in 𝐿_𝑀,𝜔(𝕋). Consequently, there is a continuous function 𝑓^∗ on 𝕋 such that, for given 𝜖>0 and 𝑓∈𝐿_𝑀,𝜔(𝕋), ‖𝑓−𝑓^∗‖_{𝑀,𝕋,𝜔}<𝜖.(3.86)
On the other hand, since the Poisson integral of a continuous function converges to it uniformly on 𝕋 [47, page 239], we have by (2.7) and 𝜔∈𝐴_𝑝(𝕋)‖‖𝑃_𝑟𝑓^∗−𝑓^∗‖‖_{𝑀,𝕋,𝜔}=sup_𝜌(𝑔,𝑀_)≤1_𝕋<𝜖𝑀(1)_𝕋||𝑃_𝑟𝑓^∗(𝑤)−𝑓^∗(𝑤)||||𝑔(𝑤)||𝜔(𝑤)||𝑑𝑤||𝜔(𝑤)||𝑑𝑤||+1≤𝐶𝜖(3.87) for 0<1−𝑟<𝛿(𝜖). Then, from (3.85), (3.86), and (3.87), we conclude that ‖‖𝑃_𝑟(𝑓)−𝑓‖‖_{𝑀,𝕋,𝜔}≤‖‖𝑃_𝑟(𝑓)−𝑃_𝑟𝑓^∗‖‖_{𝑀,𝕋,𝜔}+‖‖𝑃_𝑟𝑓^∗−𝑓^∗‖‖_{𝑀,𝕋,𝜔}+‖𝑓^∗−𝑓‖_{𝑀,𝕋,𝜔}=‖‖𝑃_𝑟𝑓−𝑓^∗‖‖_{𝑀,𝕋,𝜔}+‖‖𝑃_𝑟𝑓^∗−𝑓^∗‖‖_{𝑀,𝕋,𝜔}+‖𝑓^∗−𝑓‖_{𝑀,𝕋,𝜔}≤𝑐‖𝑓^∗−𝑓‖_{𝑀,𝕋,𝜔}+‖‖𝑃_𝑟𝑓^∗−𝑓^∗‖‖_{𝑀,𝕋,𝜔}<{𝑐+𝐶}𝜖.(3.88) This completes the proof.

Theorem 3.22. Let Γ be a Dini-smooth curve, 𝑀∈𝑄𝐶^𝜃2(0,1) and 𝑓∈𝐿_𝑀,𝜔(Γ) with 𝜔∈𝐴₁(Γ). Then the linear operators 𝑇∶𝐸_𝑀,𝜔0(𝔻)⟶𝐸_𝑀,𝜔𝐺₀,𝑇∶𝐸_𝑀,𝜔1(𝔻)⟶𝐸_𝑀,𝜔𝐺_∞(3.89) are one-to-one and onto.

Proof. The proof we give, only for the operator 𝑇. For the operator 𝑇 is the proof goes similarly. Let 𝑔∈𝐸_𝑀,𝜔0(𝔻) with the Taylor expansion 𝑔(𝑤)∶=^∞_𝑘=0𝛼_𝑘𝑤^𝑘,𝑤∈𝔻.(3.90) Since Γ is a Dini-smooth curve, the conditions 𝜔∈𝐴₁(Γ),𝜔₀∈𝐴₁(𝕋), and 𝜔₁∈𝐴₁(𝕋) are equivalent.
Let 𝑔_𝑟(𝑤)∶=𝑔(𝑟𝑤),0<𝑟<1. Since 𝑔∈𝐸¹(𝔻) is the Poisson integral of its boundary function [48, page 41], we have‖‖𝑔_𝑟−𝑔‖‖_{𝑀,𝕋,𝜔0}=‖‖𝑃_𝑟(𝑔)−𝑔‖‖_{𝑀,𝕋,𝜔0},(3.91) and using Lemma 3.21, we get ‖𝑔_𝑟−𝑔‖_{𝑀,𝕋,𝜔0}→0, as 𝑟→1⁻.
Therefore, the boundedness of the operator 𝑇 implies that‖‖𝑇𝑔_𝑟−𝑇(𝑔)‖‖_𝑀,Γ,𝜔⟶0,as𝑟⟶1⁻.(3.92) Since ∑^∞𝑘=0𝛼_𝑘𝑟^𝑘𝑤^𝑘 is uniformly convergent on 𝕋, one has 𝑇𝑔_𝑟𝑧^′=12𝜋𝑖_𝕋𝑔_𝑟(𝑤)𝜓^(𝑤)𝜓(𝑤)−𝑧^𝑑𝑤=^∞_𝑚₌₀𝛼_𝑚𝑟^𝑚12𝜋𝑖_𝕋𝑤^𝑚𝜓^(𝑤)𝜓(𝑤)−𝑧^=𝑑𝑤_∞𝑚₌₀𝛼_𝑚𝑟^𝑚𝐹_𝑚𝑧^,𝑧^∈𝐺₀.(3.93) From the last equality and Lemma 3 of [39, page 43] we have 𝑎_𝑘𝑇𝑔_𝑟=12𝜋𝑖_𝕋𝑇𝑔_𝑟(𝜓(𝑤))𝑤^𝑘+1𝑑𝑤=12𝜋𝑖_𝕋∑^∞𝑚=0𝛼_𝑚𝑟^𝑚𝐹_𝑚(𝜓(𝑤))𝑤^𝑘+1=𝑑𝑤_∞𝑚₌₀𝛼_𝑚𝑟^𝑚12𝜋𝑖_𝕋𝐹_𝑚(𝜓(𝑤))𝑤^𝑘+1𝑑𝑤=𝛼_𝑘𝑟^𝑘,(3.94) and therefore 𝑎_𝑘𝑇𝑔_𝑟⟶𝛼_𝑘,as𝑟⟶1⁻.(3.95) On the other hand, applying (3.33), (2.7), and weighted version of Hölder’s inequality _Γ||𝑢(𝑧)𝑣(𝑧)||𝜔(𝑧)||𝑑𝑧||≤𝑐‖𝑢‖_𝑀,Γ,𝜔‖𝑣‖_{𝑀,Γ,𝜔},𝑢∈𝐿_𝑀,𝜔(Γ),𝑣∈𝐿_𝑀,𝜔(Γ),(3.96) we obtain ||𝑎_𝑘=𝑇||||𝑔1_𝑟−𝑎_𝑘(𝑇(𝑔))||2𝜋𝑖_𝕋𝑇𝑔_𝑟−𝑇(𝑔)(𝜓(𝑤))𝑤^𝑘+1≤1𝑑𝑤||||2𝜋_𝕋||𝑇𝑔_𝑟−𝑇(𝑔)(𝜓(𝑤))||||𝑑𝑤||=12𝜋_Γ≤𝑐||𝑇𝑔_𝑟−𝑇(𝑔)(𝑧)|||||𝜑^′(𝑧)|||||𝑑𝑧||2𝜋_Γ||𝑇𝑔_𝑟−𝑇(𝑔)(𝑧)||||𝑑𝑧||=𝑐2𝜋_Γ≤𝑐||𝑇𝑔_𝑟−𝑇(𝑔)(𝑧)||𝜔⁻¹(𝑧)𝜔(𝑧)||𝑑𝑧||2𝜋‖‖𝑇𝑔^𝑟−𝑇(𝑔)‖‖_𝑀,Γ,𝜔‖‖𝜔⁻¹‖‖_{𝑀,Γ,𝜔}≤𝑐2𝜋‖‖𝑇𝑔^𝑟−𝑇(𝑔)‖‖_𝑀,Γ,𝜔,(3.97) because ‖𝜔⁻¹‖_{𝑀,Γ,𝜔}≤𝑀(1)mes(Γ)+1≤𝑐<∞.
Using here the relation (3.92), we get𝑎_𝑘𝑇𝑔_𝑟⟶𝑎_𝑘(𝑇(𝑔)),as𝑟⟶1⁻,(3.98) and then by (3.95), 𝑎_𝑘(𝑇(𝑔))=𝛼_𝑘 for 𝑘=0,1,2,…. If 𝑇(𝑔)=0, then 𝛼_𝑘=𝑎_𝑘(𝑇(𝑔))=0,𝑘=0,1,2,…, and therefore 𝑔=0. This means that the operator 𝑇 is one-to-one.
Now we take a function 𝑓∈𝐸_𝑀,𝜔(𝐺₀) and consider the function 𝑓₀=𝑓∘𝜓∈𝐿_𝑀,𝜔0(𝕋). The Cauchy type integral12𝜋𝑖_𝕋𝑓₀(𝜏)𝜏−𝑤𝑑𝜏(3.99) represents analytic functions 𝑓⁺⁰ and 𝑓⁻⁰ in 𝔻 and 𝔻_∞, respectively. Since 𝜔₀∈𝐴₁(𝕋), by Lemma 3.18, we have 𝑓⁺⁰∈𝐸_𝑀,𝜔0(𝔻),𝑓⁻⁰∈𝐸_𝑀,𝜔0𝔻_∞,(3.100) and moreover 𝑓₀(𝑤)=𝑓⁺⁰(𝑤)−𝑓⁻⁰(𝑤)(3.101) a.e. on 𝕋. Since 𝑓⁻⁰∈𝐸¹(𝔻_∞) and 𝑓⁻⁰(∞)=0, we have 𝑎_𝑘=12𝜋𝑖_𝕋𝑓₀(𝑤)𝑤^𝑘+1𝑑𝑤=12𝜋𝑖_𝕋𝑓⁺⁰(𝑤)𝑤^𝑘+1𝑑𝑤−12𝜋𝑖_𝕋𝑓⁻⁰(𝑤)𝑤^𝑘+1𝑑𝑤=12𝜋𝑖_𝕋𝑓⁺⁰(𝑤)𝑤^𝑘+1𝑑𝑤,(3.102) which proves that the coefficients 𝑎_𝑘,𝑘=0,1,2,…, also become the Taylor coefficients of the function 𝑓⁺⁰ at the origin, that is, 𝑓⁺⁰(𝑤)=^∞_𝑘=0𝑎_𝑘𝑤^𝑘,𝑤∈𝔻,(3.103) and also 𝑇𝑓⁺⁰∼^∞_𝑘=0𝑎_𝑘𝐹_𝑘.(3.104) Hence the functions 𝑇(𝑓⁺⁰) and 𝑓 have the same Faber coefficients 𝑎_𝑘,𝑘=0,1,2,…, and therefore 𝑇(𝑓⁺⁰)=𝑓. This proves that the operator 𝑇 is onto.

Proof of Theorem 3.6. We prove that the rational function 𝑅_𝑛(𝑧,𝑓)∶=^𝑛_𝑘=0𝑎_𝑘𝐹_𝑘(𝑧)+^𝑛_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧(3.105) satisfies the required inequality of Theorem 3.6. This inequality is true if we can show that ‖‖‖‖𝑓₋(𝑧)+^𝑛_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧‖‖‖‖_𝑀,Γ,𝜔≤𝑐Ω^{𝑟𝑀,Γ,𝜔}‖‖‖‖𝑓𝑓,(𝑛+11),(3.106)₊(𝑧)−^𝑛_𝑘=0𝑎_𝑘𝐹_𝑘(𝑧)‖‖‖‖_𝑀,Γ,𝜔≤𝑐Ω^{𝑟𝑀,Γ,𝜔}𝑓,(𝑛+11),(3.107) because 𝑓(𝑧)=𝑓⁺(𝑧)−𝑓⁻(𝑧) a.e. on Γ.
First we prove (3.106). Let 𝑓∈𝐿_𝑀,𝜔(Γ). Then 𝑓₁∈𝐿_𝑀,𝜔1(𝕋) and 𝑓₀∈𝐿_𝑀,𝜔0(𝕋). According to (3.101),𝑓(𝜍)=𝑓⁺⁰(𝜑(𝜍))−𝑓⁻⁰(𝜑(𝜍))(3.108) a.e. on Γ. On the other hand, 𝑓₁(𝑤)=𝑓⁺¹(𝑤)−𝑓⁻¹(𝑤),(3.109) which implies the inequality 𝑓(𝜍)=𝑓⁺¹𝜑₁(𝜍)−𝑓⁻¹𝜑₁(𝜍)(3.110) a.e. on Γ.
Let 𝑧^′∈𝐺₀⧵{0}. Using (3.7) and (3.110), we have^𝑛_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧^=^𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1𝑧^−12𝜋𝑖_Γ∑^𝑛𝑘=1𝑎_𝑘̃𝜑^𝑘1(𝜍)𝜍−𝑧^=𝑑𝜍_𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1𝑧^−12𝜋𝑖_Γ∑^𝑛𝑘=1𝑎_𝑘̃𝜑^𝑘1(𝜍)−𝑓⁺¹𝜑₁(𝜍)𝜍−𝑧^𝑑𝜍−12𝜋𝑖_Γ𝑓⁻¹𝜑₁(𝜍)𝜍−𝑧^−1𝑑𝜍2𝜋𝑖_Γ𝑓(𝜍)𝜍−𝑧^=𝑑𝜍_𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1𝑧^−12𝜋𝑖_Γ∑^𝑛𝑘=1𝑎_𝑘̃𝜑^𝑘1(𝜍)−𝑓⁺¹𝜑₁(𝜍)𝜍−𝑧^𝑑𝜍−𝑓⁻¹𝜑₁𝑧^−𝑓⁻𝑧^.(3.111) Hence, taking the nontangential limit 𝑧^→𝑧∈Γ, inside of Γ, we obtain ^𝑛_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧=^𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1(𝑧)−12^𝑛_𝑘=1−𝑆_Γ𝑎_𝑘̃𝜑^𝑘1(𝑧)−𝑓⁺¹𝜑₁(𝑧)^𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1−𝑓⁺¹∘𝜑₁−𝑓⁻¹𝜑₁(𝑧)−𝑓⁺(𝑧)(3.112) a.e. on Γ.
Using (3.19), (3.110), Minkowski’s inequality, and the boundedness of 𝑆_Γ, we get‖‖‖‖𝑓₋(𝑧)+^𝑛_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧‖‖‖‖_𝑀,Γ,𝜔=‖‖‖‖12^𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1(𝑧)−𝑓⁺¹𝜑₁(𝑧)−𝑆_Γ^𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1−𝑓⁺¹∘𝜑₁(𝑧)‖‖‖‖_𝑀,Γ,𝜔≤𝑐‖‖‖‖^𝑛_𝑘=1𝑎_𝑘̃𝜑^𝑘1(𝑧)−𝑓⁺¹𝜑₁(𝑧)‖‖‖‖_𝑀,Γ,𝜔≤𝑐‖‖‖‖𝑓⁺¹(𝑤)−^𝑛_𝑘=1𝑎_𝑘̃𝑤^𝑘‖‖‖‖_{𝑀,𝕋,𝜔1}.(3.113) On the other hand, from the proof of Theorem 3.22 we know that the Faber-Laurent coefficients 𝑎_𝑘̃ of the function 𝑓 and the Taylor coefficients of the function 𝑓⁺¹ at the origin are the same. Then taking Lemma 3.19 into account, we conclude that ‖‖‖‖𝑓₋+^𝑛_𝑘=1𝑎_𝑘̃𝐹_𝑘1𝑧‖‖‖‖_𝑀,Γ,𝜔≤𝑐Ω^{𝑟𝑀,𝕋,𝜔}₁𝑓⁺¹,1(𝑛+1)=𝑐Ω^{𝑟𝑀,Γ,𝜔}𝑓,(𝑛+11),(3.114) 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 𝑓∈𝐸_𝑀,𝜔(𝐺₀). Then we have 𝑇(𝑓⁺⁰)=𝑓. Since by Theorem 3.22 the operator 𝑇∶𝐸_𝑀,𝜔0(𝔻)→𝐸_𝑀,𝜔(𝐺₀) is linear, bounded, one-to-one and onto, the operator 𝑇⁻¹∶𝐸_𝑀,𝜔(𝐺₀)→𝐸_𝑀,𝜔0(𝔻) is also linear and bounded. We take a 𝑝^∗𝑛∈𝒫_𝑛 as the best approximating algebraic polynomial to 𝑓 in 𝐸_𝑀,𝜔(𝐺₀), that is, 𝐸_𝑛(𝑓)_𝑀,Γ,𝜔=‖‖𝑓−𝑝^∗𝑛‖‖_𝑀,Γ,𝜔.(3.115) Then, 𝑇⁻¹(𝑝^∗𝑛)∈𝒫_𝑛(𝔻), and therefore 𝐸_𝑛𝑓⁺⁰_{𝑀,𝕋,𝜔0}≤‖‖𝑓⁺⁰−𝑇⁻¹𝑝^∗𝑛‖‖_{𝑀,𝕋,𝜔0}=‖‖𝑇⁻¹(𝑓)−𝑇⁻¹𝑝^∗𝑛‖‖_{𝑀,𝕋,𝜔0}=‖‖𝑇⁻¹𝑓−𝑝^∗𝑛‖‖_{𝑀,𝕋,𝜔0}≤‖‖𝑇⁻¹‖‖‖‖𝑓−𝑝^∗𝑛‖‖_𝑀,Γ,𝜔=‖‖𝑇⁻¹‖‖𝐸_𝑛(𝑓)_𝑀,Γ,𝜔,(3.116) because the operator 𝑇⁻¹ is bounded.
Theorem 2.8 and (3.116) imply thatΩ^{𝑟𝑀,Γ,𝜔}𝑓,1𝑛=Ω^{𝑟𝑀,𝕋,𝜔0}𝑓⁺⁰,1𝑛≤𝑛𝑐^𝑟𝐸⁰𝑓⁺⁰_{𝑀,𝕋,𝜔0}+^𝑛_𝑘=1𝑘^𝑟−1𝐸_𝑘𝑓⁺⁰_{𝑀,𝕋,𝜔0}≤𝑐‖‖𝑇₋₁‖‖𝑛^𝑟𝐸⁰(𝑓)_𝑀,Γ,𝜔+^𝑛_𝑘=1𝑘^𝑟−1𝐸_𝑘(𝑓)_𝑀,Γ,𝜔,𝑟>0.(3.117)