# Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Authors: Ramazan Akgün
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

LetG0 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 EM,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 EM,ω(G0) and EM,ω(G∞).


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## Body

## 1. Introduction

LetG0 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 γr:=φ0({w∈ℂ:|w|=r}) for a given r∈(0,1). We denote by Ep(G0), 1≤p≤∞, Smirnov’s classes of analytic functions f:G0→ℂ satisfying(1.1)sup⁡r∈(0,1)∫γr|f(z)|p|dz|<c,if1≤p<∞,max⁡z∈G0|f(z)|<C,ifp=∞,
where positive constant c is independent of r.It is well known thatEp(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 Ep(G0) is a Banach space with the norm(1.2)‖f‖p,Γ:={12π∫Γ|f(z)|p|dz|}1/p.
Smirnov classes Ep(G∞), 1≤p≤∞, of analytic functions f:G∞→ℂ can be defined similarly and Ep(G∞) are fulfilling the same above properties to that of Ep(G0).A smooth Jordan curveΓ 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(1.3)∫0δΩ(θ,s)sds<∞,δ>0.
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 spacesEp(G0), 1≤p≤∞, 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,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 EM,ω(G0) and EM,ω(G∞).Throughout the work, we will denote byc,C, the constants that are different in different places.


## 2. Approximation Theorems in Weighted Orlicz Space

A functionΦ is called Young function if Φ is even, continuous, nonnegative in ℝ, increasing on (0,∞) such that(2.1)Φ(0)=0,lim⁡x→∞Φ(x)=∞.
A Young function Φ is said to satisfy Δ2  condition (Φ∈Δ2) if there is a constant c>0 such that(2.2)Φ(2x)≤cΦ(x)
for all x∈ℝ.Two Young functionsΦ and Φ1 are said to be equivalent if there are c,C>0 such that(2.3)Φ1(cx)≤Φ(x)≤Φ1(Cx),∀x>0.A functionM:[0,∞)→[0,∞) is said to be quasiconvex if there exist a convex Young function Φ and a constant c≥1 such that(2.4)Φ(x)≤M(x)≤Φ(cx),∀x≥0,
holds.A nonnegative functionω 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 L̃M,ω(T) the class of Lebesgue measurable functions f:T→ℝ satisfying the condition(2.5)∫TM(|f(x)|)ω(x)dx<∞.
The linear span of the weighted Orlicz classL̃M,ω(T), denoted by LM,ω(T), becomes a normed space with the Orlicznorm(2.6)‖f‖M,ω:=sup⁡{∫T|f(x)g(x)|ω(x)dx:∫TM̃(|g|)ω(x)dx≤1},
where M̃(y):=supx≥0(xy-M(x)),y≥0, is the complementary function of M.IfM is quasiconvex and M̃ is its complementary function, then Young’s inequality holds(2.7)xy≤M(x)+M̃(y),x,y≥0.
For a quasiconvex function M we define the indice p(M) of M as(2.8)1p(M)∶=inf⁡{p:p>0,Mpisquasiconvex},p′(M)∶=p(M)p(M)-1.
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∈Ap(M), then it can be easily seen that LM,ω(T)⊂L1(T) and LM,ω(T) becomes a Banach space with the Orlicz norm. The Banach space LM,ω(T) is called weighted Orlicz space.We define theLuxemburgfunctional as(2.9)‖f‖(M),ω:=inf⁡{τ>0:∫TM(|f(x)|τ)ω(x)dx≤1}.
There exist [23, page 23] constants c,C>0 such that(2.10)c‖f‖(M),ω≤‖f‖M,ω≤C‖f‖(M),ω.
For a weight ω we denote by Lp(T,ω) the class of measurable functions on T such that ω1/pf belongs to Lebesgue space Lp(T) on T. We set ∥f∥p,ω:=∥ω1/pf∥p for f∈Lp(T,ω).A 2π-periodic weight function ω belongs to the Muckenhoupt class Ap, 1<p<∞, if(2.11)(1|J|∫Jω(x)dx)(1|J|∫Jω-1/(p-1)(x)dx)p-1≤c
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 byQC2θ(0,1) a class of functions g satisfying Δ2 condition such that gθ 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 spacesLM,ω(T), ω∈Ap(M), where M∈QC2θ(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].Letb0=0, ak,bk∈ℝ, ck=(ak-ibk)/2, c-k=(ak+ibk)/2, c0=a0/2(2.12)f(x)~∑k=-∞∞ckeikx=∑k=0∞(akcos⁡kx+bksin⁡kx),(2.13)f̃(x)~∑k=1∞(aksin⁡kx-bkcos⁡kx)be theFourier and the conjugate Fourier series of f∈L1(T), respectively. Putting Ak(x):=ckeikx in (2.12), we define for n=0,1,2,…(2.14)Sn(f):=Sn(x,f):=∑k=0n(Ak(x)+A-k(x))=a02+∑k=1n(akcos⁡kx+bksin⁡kx),Rn〈α〉(f,x):=∑k=0n(1-(kn+1)α)(Ak(x)+A-k(x)),α∈R+,Θm〈α〉:=11-((m+1)/(2m+1))rR2m〈α〉-1((2m+1)/(m+1))r-1Rm〈α〉,m=1,2,3,….For a givenf∈L1(T), assuming(2.15)∫Tf(x)dx=0,
we define αthfractional(α∈ℝ+)integral of f as [34, v.2, page 134](2.16)Iα(x,f):=∑k∈Z*ck(ik)-αeikx,
where(2.17)(ik)-α:=|k|-αe(-1/2)πiαsign⁡k
as principal value.Letα∈ℝ+ be given. We define fractional derivative of a function f∈L1(T), satisfying (2.15), as(2.18)f(α)(x):=d[α]+1dx[α]+1I1+α-[α](x,f)
provided the right hand side exists.Settingx,t∈T, r∈ℝ+, M∈QC2θ(0,1), ω∈Ap(M), and f∈LM,ω(T), we define(2.19)σtrf(x)∶=(I-σt)rf(x)=∑k=0∞(-1)k(rk)1(2t)k∫-tt⋯∫-ttf(x+u1+⋯uk)du1⋯duk,
where (rk):=r(r-1)⋯(r-k+1)/k! for k≥1 and (r0):=1 are Binomial coefficients, σtf(x):=(1/2t)∫-ttf(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 thatL is anyone of the operators Sn,  σh, and f̃.  If M∈QC2θ(0,1),  ω∈Ap(M), and f∈LM,ω(T), then there exists a constant c>0 such that
(2.20)∫TM(|Lf(t)|)ω(t)dt≤c∫TM(|f(t)|)ω(t)dt
holds.Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from (2.20) that(2.21)‖Lf‖M,ω≤c‖f‖M,ω
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(2.22)|(rk)|≤ckr+1,k=1,2,…,
we have(2.23)∑k=0∞|(rk)|<∞
and therefore(2.24)‖σtrf‖M,ω≤c‖f‖M,ω<∞
provided f∈LM,ω(T), ω∈Ap(M), where M∈QC2θ(0,1).LetM∈QC2θ(0,1). For r∈ℝ+, we define the fractional modulus of smoothness of indexr for f∈LM,ω(T), ω∈Ap(M) as(2.25)ΩM,ωr(f,δ):=sup⁡0<hi,t<δ‖∏i=1[r](I-σhi)(I-σt)r-[r]f‖M,ω,
where [x] denotes the integer part of a real number x.Since the operatorσt is bounded in LM,ω(T), ω∈Ap(M), where M∈QC2θ(0,1), we have by (2.24) that(2.26)ΩM,ωr(f,δ)≤c‖f‖M,ω,
where the constant c>0, dependent only on r and M.Remark 2.1.
The modulus of smoothnessΩM,ωr(f,δ), wherer∈ℝ+,M∈QC2θ(0,1),ω∈Ap(M), f∈LM,ω(T) has the following properties:(i)
ΩM,ωr(f,δ)is nonnegative, nondecreasing function ofδ≥0and subadditive,(ii)
limδ→0ΩM,ωr(f,δ)=0.For formulations of our results, we need several lemmas.Lemma A (see [36]).
Forα∈ℝ+, we suppose that(i)
a1+a2+⋯+an+⋯,(ii)
a1+2αa2+⋯+nαan+⋯,
be two series in a Banach space(B,∥·∥). Let(2.27)Rn〈α〉:=∑k=0n(1-(kn+1)α)ak,Rn〈α〉*:=∑k=0n(1-(kn+1)α)kαak
for n=1,2,…. Then,
(2.28)‖Rn〈α〉*‖≤c,n=1,2,…
for some c>0 if and only if there exists a  R∈B such that
(2.29)‖Rn〈α〉-R‖≤Cnα,
where c and C are constants depending only on one another.IfM∈QC2θ(0,1), ω∈Ap(M), and f∈LM,ω(T), then from Theorem A(ii) and Abel’s transformation we get(2.30)‖Rn〈α〉(f,⋅)‖M,ω≤c‖f‖M,ω,n=1,2,3,…,x∈T
and therefore from (2.14) and(2.30)(2.31)‖Θn〈α〉(f,⋅)‖M,ω≤c‖f‖M,ω,n=1,2,3,…,x∈T.
From the property(2.32)Θm〈α〉(f)(x)=1∑k=m+12m[(k+1)α-kα]∑k=m+12m[(k+1)α-kα]Sk(x,f),x∈T,f∈L1(T)
it is known that(2.33)Θm〈α〉(Tm)=Tm
for Tm∈𝒯m, m=1,2,3,….Lemma 2.2.
LetTn∈𝒯n,n=1,2,3,…,M∈QC2θ(0,1), and ω∈Ap(M). If  α∈ℝ+,  then there exists a constant c>0 independent of n such that
(2.34)‖Tn(α)‖M,ω≤cnα‖Tn‖M,ω
holds.Proof.
Without loss of generality one can assume that∥Tn∥M,ω=1. Since
(2.35)Tn=∑k=0n(Ak(x)+A-k(x)),T̃nnα=∑k=1n[(Ak(x)-A-k(x))nα],Tn(α)(in)α=∑k=1nkα[(Ak(x)-A-k(x))nα]
we have by (2.30) and Theorem A(iii) that
(2.36)‖Rm〈α〉(T̃nnα)‖M,ω≤cnα‖T̃n‖M,ω≤cnα‖Tn‖M,ω=cnα
and from Lemma A
(2.37)‖Rm〈α〉(Tn(α)(in)α)‖M,ω≤c.
Hence from (2.33) and (2.31), we find
(2.38)‖Tn(α)‖M,ω=nα‖Θm〈α〉(Tn(α)(in)α)‖M,ω≤cnα‖Tn‖M,ω.
General case follows immediately from this.LetM∈QC2θ(0,1). We denote by WMα(T,ω), α>0, ω∈Ap(M), the linear space of 2π-periodic real valued functions f∈LM,ω(T) such that f(α)∈LM,ω(T).Lemma 2.3.
LetM∈QC2θ(0,1). If  f∈WMα(T,ω) with ω∈Ap(M) and α≥0,  then for n=0,1,2,…,  there is a constant c>0 dependent only on α and M such that
(2.39)‖f(α)(⋅)-Sn(α)(⋅,f)‖M,ω≤cEn(f(α))M,ω
holds.Proof.
Ifα=0, then from boundedness (see (2.21)) of the operator Sn we get that
(2.40)‖f-Snf‖M,ω≤cEn(f)M,ω.
Let Wn(f):=Wn(x,f):=(1/(n+1))∑ν=n2nSν(x,f), n=0,1,2,…. Since
(2.41)Wn(⋅,f(α))=Wn(α)(⋅,f),
we have
(2.42)‖f(α)(⋅)-Sn(α)(⋅,f)‖M,ω≤‖f(α)(⋅)-Wn(⋅,f(α))‖M,ω+‖Sn(α)(⋅,Wn(f))-Sn(α)(⋅,f)‖M,ω+‖Wn(α)(⋅,f)-Sn(α)(⋅,Wn(f))‖M,ω:=I1+I2+I3.
From (2.21) we get the boundedness of Wn in LM,ω(T) and we have
(2.43)I1≤‖f(α)(⋅)-Sn(⋅,f(α))‖M,ω+‖Sn(⋅,f(α))-Wn(⋅,f(α))‖M,ω≤cEn(f(α))M,ω+‖Wn(⋅,Sn(f(α))-f(α))‖M,ω≤cEn(f(α))M,ω.
From Lemma 2.2 we get
(2.44)I2≤cnα‖Sn(⋅,Wn(f))-Sn(⋅,f)‖M,ω,I3≤c(2n)α‖Wn(⋅,f)-Sn(⋅,Wn(f))‖M,ω≤c(2n)αEn(Wn(f))M,ω.
Now we have
(2.45)‖Sn(⋅,Wn(f))-Sn(⋅,f)‖M,ω≤‖Sn(⋅,Wn(f))-Wn(⋅,f)‖M,ω+‖Wn(⋅,f)-f(⋅)‖M,ω+‖f(⋅)-Sn(⋅,f)‖M,ω≤cEn(Wn(f))M,ω+cEn(f)M,ω.
Since
(2.46)En(Wn(f))M,ω≤cEn(f)M,ω,
we get
(2.47)‖f(α)(⋅)-Tn(α)(⋅,f)‖M,ω≤cEn(f(α))M,ω+cnαEn(Wn(f))M,ω+cnαEn(f)M,ω+c(2n)αEn(Wn(f))M,ω≤cEn(f(α))M,ω+CnαEn(f)M,ω.
Now we show that
(2.48)En(f)M,ω≤c(n+1)αEn(f(α))M,ω.
For this we set
(2.49)Ak(x,f):=akcos⁡kx+bksin⁡kx.
For given f∈LM,ω(T) and ε>0, by Lemma 3 of [37], there exists a trigonometric polynomial T such that
(2.50)∫TM(|f(x)-T(x)|)ω(x)dx<ε
which by (2.7) this implies that
(2.51)‖f-T‖M,ω<ε,
and hence we obtain
(2.52)En(f)M,ω⟶0asn⟶∞.
In this case from (2.40) we have
(2.53)f(x)=∑k=0∞Ak(x,f)
in ∥·∥M,ω norm. If k=1,2,3,…, then
(2.54)Ak(x,f)=Ak(x+απ2k,f)cos⁡απ2+Ak(x+απ2k,f̃)sin⁡απ2,Ak(x,f(α))=kαAk(x+απ2k,f).
Hence,
(2.55)∑k=0∞Ak(x,f)=A0(x,f)+cos⁡απ2∑k=1∞Ak(x+απ2k,f)+sin⁡απ2∑k=1∞Ak(x+απ2k,f̃)=A0(x,f)+cos⁡απ2∑k=1∞k-αAk(x,f(α))+sin⁡απ2∑k=1∞k-αAk(x,f̃(α)).
Therefore,
(2.56)f(x)-Sn(x,f)=cos⁡απ2∑k=n+1∞1kαAk(x,f(α))+sin⁡απ2∑k=n+1∞1kαAk(x,f̃(α)).
Since
(2.57)∑k=n+1∞k-αAk(x,f(α))=∑k=n+1∞k-α[(Sk(⋅,f(α))-f(α)(⋅))-(Sk-1(⋅,f(α))-f(α)(⋅))]=∑k=n+1∞(k-α-(k+1)-α)(Sk(⋅,f(α))-f(α)(⋅))-(n+1)-α(Sn(⋅,f(α))-f(α)(⋅)),∑k=n+1∞k-αAk(x,f̃(α))=∑k=n+1∞(k-α-(k+1)-α)(Sk(⋅,f̃(α))-f̃(α)(⋅))-(n+1)-α(Sn(⋅,f̃(α))-f̃(α)(⋅)),
we obtain
(2.58)‖f(⋅)-Sn(⋅,f)‖M,ω≤∑k=n+1∞(k-α-(k+1)-α)‖Sk(⋅,f(α))-f(α)(⋅)‖M,ω+(n+1)-α‖Sn(⋅,f(α))-f(α)(⋅)‖M,ω+∑k=n+1∞(k-α-(k+1)-α)‖Sk(⋅,f̃(α))-f̃(α)(⋅)‖M,ω+(n+1)-α‖Sn(⋅,f̃(α))-f̃(α)(⋅)‖M,ω≤c[∑k=n+1∞(k-α-(k+1)-α)Ek(f)M,ω+(n+1)-αEn(f(α))M,ω]+C[∑k=n+1∞(k-α-(k+1)-α)Ek(f)M,ω+(n+1)-αEn(f̃(α))M,ω].
Consequently,
(2.59)‖f(x)-Sn(x,f)‖M,ω≤cEk(f(α))M,ω[∑k=n+1∞(k-α-(k+1)-α)+(n+1)-α]+cEn(f̃(α))M,ω[∑k=n+1∞(k-α-(k+1)-α)+(n+1)-α]≤cEn(f(α))M,ω[∑k=n+1∞(k-α-(k+1)-α)+(n+1)-α]≤c(n+1)αEn(f(α))M,ω,
and (2.48) holds. Now (2.47) and (2.48) imply the result.Lemma 2.4.
LetTn∈𝒯n, n=0,1,2,…, M∈QC2θ(0,1), and ω∈Ap(M).  If α∈ℝ+,  then
(2.60)ΩM,ωα(Tn,πn+1)≤c(n+1)α‖Tn(α)‖M,ω
hold, where the constant c>0 is dependent only on α and M.Proof.
First we prove that if0<α<β, then
(2.61)ΩM,ωβ(Tn,⋅)≤cΩM,ωα(Tn,⋅).
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):=σtαTn(x), we have
(2.62)σtβ-αK(x)=∑j=0∞(-1)j(β-αj)1(2t)j∫-tt⋯∫-ttK(x+u1+⋯uj)du1⋯duj=∑j=0∞(-1)j(β-αj)1(2t)j×∫-tt⋯∫-tt[∑k=0∞(-1)k(αk)1(2t)k×∫-tt⋯∫-ttTn(x+u1+⋯uj+uj+1+⋯uj+k)duj+1⋯duj+k[∑k=0∞(-1)k(αk)1(2t)k]du1⋯duj=∑j=0∞∑k=0∞(-1)j+k(β-αj)(αk)[1(2t)j+k∫-tt⋯∫-ttTn(x+u1+⋯uj+k)du1⋯duj+k]=∑υ=0∞{∑μ=0υ(-1)υ-μ(β-αυ-μ)(αμ)1(2t)υ∫-tt⋯∫-ttTn{x+u1+⋯uυ}du1⋯duυ}=∑υ=0∞(-1)υ1(2t)υ∫-tt⋯∫-ttTn(x+u1+…uυ)du1⋯duυ∑μ=0υ(β-αυ-μ)(αμ)=∑υ=0∞(-1)υ(βυ)1(2t)υ∫-tt⋯∫-ttTn(x+u1+…uυ)du1⋯duυ=σtβTn(x).
Then,
(2.63)‖σtβTn‖M,ω=‖σtβ-αK‖M,ω≤c‖σtαTn‖M,ω,
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∈WM(2r)(T,ω), r=1,2,3,…, then
(2.64)ΩM,ωr(ϱ,δ)≤cδ2r‖ϱ(2r)‖M,ω.
Putting
(2.65)g(x):=∏i=2r(I-σhi)ϱ(x),
we have
(2.66)(I-σh1)g(x)=∏i=1r(I-σhi)ϱ(x),∏i=1r(I-σhi)ϱ(x)=12h1∫-h1h1(g(x)-g(x+t))dt=-18h1∫0h1∫0t∫-uug′′(x+s)dsdudt.
Therefore,
(2.67)‖∏i=1r(I-σhi)ϱ(x)‖M,ω=18h1sup⁡⁡{∫T|∫0h1∫0t∫-uug′′(x+s)dsdudt||v(x)|ω(x)dx:∫TM̃(|v(x)|)ω(x)dx≤1}≤18h1∫0h1∫0t2u‖12u∫-uug′′(x+s)ds‖M,ωdudt≤c8h1∫0h1∫0t2u‖g′′‖M,ωdudt=ch12‖g′′‖M,ω.
Since
(2.68)g′′(x)=∏i=2r(I-σhi)ϱ′′(x),
we obtain that
(2.69)ΩM,ωr(ϱ,δ)≤sup⁡0<hi≤δi=1,2,…,rch12‖g′′‖M,ω=cδ2‖∏i=2r(I-σhi)ϱ′′(x)‖M,ω=cδ2sup⁡0<hi≤δi=2,…,r‖∏i=2r(I-σhi)ϱ′′(x)‖M,ω=cδ2ΩM,ωr-1(ϱ′′,δ)≤cδ4ΩM,ωr-2(ϱ(4),δ)≤⋯≤Cδ2r‖ϱ(2r)‖M,ω.
Using (2.61), (2.64), and Lemma 2.2, we get
(2.70)ΩM,ωα(Tn,πn+1)≤cΩM,ω[α](Tn,πn+1)≤c(πn+1)2[α]‖Tn(2[α])‖M,ω≤c(n+1)2[α](n+1)[α]-(α-[α])‖Tn(α)‖M,ω=c(n+1)α‖Tn(α)‖M,ω
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 LM,ω(T) where M∈QC2θ(0,1) and ω∈Ap(M).Definition 2.5.
Forf∈LM,ω(T),δ>0, and r=1,2,3,…, the PeetreK-functional is defined as
(2.71)K(δ,f;LM,ω(T),WMα(T,ω)):=inf⁡g∈WMα(T,ω){‖f-g‖M,ω+δ‖g(r)‖M,ω}.Proposition 2.6.
LetM∈QC2θ(0,1),  ω∈Ap(M), and f∈LM,ω(T).  Then the K-functional K(δ2r,f;LM,ω(T),WM2r(T,ω)) in (2.71) and the modulus ΩM,ωr(f,δ),r=1,2,3,…,  are equivalent.Proof.
Ifh∈WM2r(T,ω), then we have
(2.72)ΩM,ωr(f,δ)≤c‖f-h‖M,ω+cδ2r‖h(2r)‖M,ω≤cK(δ2r,f;LM,ω(T),WM2r(T,ω)).
Putting(2.73)(Lδf)(x):=3δ-3∫0δ∫0u∫-ttf(x+s)dsdtdu,x∈T,
we have
(2.74)d2dx2Lδf=cδ2(I-σδ)f,
and hence
(2.75)d2rdx2rLδrf=cδ2r(I-σδ)r,r=1,2,3,….
On the other hand, we find
(2.76)‖Lδf‖M,ω≤3δ-3∫0δ∫0u2t‖σtf‖M,ωdtdu≤c‖f‖M,ω.
Now, let Aδr:=I-(I-Lδr)r. Then Aδrf∈WM2r(T,ω) and
(2.77)‖d2rdx2rAδrf‖M,ω≤c‖d2rdx2rLδrf‖M,ω=cδ2r‖(I-σδ)r‖M,ω≤cδ2rΩM,ωr(f,δ).
Since
(2.78)I-Lδr=(I-Lδ)∑j=0r-1Lδj,
we get
(2.79)‖(I-Lδr)g‖M,ω≤c‖(I-Lδ)g‖M,ω≤3cδ-3∫0δ∫0u2t‖(I-σt)g‖M,ωdtdu≤csup⁡0<t≤δ‖(I-σt)g‖M,ω.
Taking into account
(2.80)‖f-Aδrf‖M,ω=‖(I-Lδr)rf‖M,ω
by a recursive procedure, we obtain
(2.81)‖f-Aδrf‖M,ω≤csup⁡0<t1≤δ‖(I-σt1)(I-Lδr)r-1f‖M,ω≤csup⁡0<t1≤δsup⁡0<t2≤δ‖(I-σt1)(I-σt2)(I-Lδr)r-2f‖M,ω≤⋯≤csup⁡0<ti≤δi=1,2,…,r‖∏i=1r(I-σti)f(x)‖M,ω=cΩM,ωr(f,δ).
Now we can formulate the results.Theorem 2.7.
LetM∈QC2θ(0,1) and r∈ℝ+.  If f∈LM,ω(T) with ω∈Ap(M),  then there is a constant c>0 dependent only on r and M such that for n=0,1,2,3,…(2.82)En(f)M,ω≤cΩM,ωr(f,1n+1)
holds.Proof.
We putk-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∈WM,ω2k(𝕋) and n=0,1,2,3,…(2.83)En(f)M,ω≤En(f-g)M,ω+En(g)M,ω≤c[‖f-g‖M,ω+(n+1)-2k‖g(2k)‖M,ω]≤cK((n+1)-2k,f;LM,ω(T),WM2k(T,ω))≤cΩM,ωk(f,1(n+1))≤cΩM,ωr(f,1(n+1)).Theorem 2.8.
LetM∈QC2θ(0,1) and r∈ℝ+. If f∈LM,ω(T) with ω∈Ap(M),  then there is a constant c>0 dependent only on r and M such that for n=0,1,2,3,…(2.84)ΩM,ωr(f,πn+1)≤c(n+1)r∑ν=0n(ν+1)r-1Eν(f)M,ω
holds.Proof.
LetTn∈𝒯n be the best approximating polynomial of f∈LM,ω(T) and let m∈ℤ+. Then,
(2.85)ΩM,ωr(f,πn+1)≤ΩM,ωr(f-T2m,π(n+1))+ΩM,ωr(T2m,π(n+1))≤cE2m(f)M,ω+ΩM,ωr(T2m,π(n+1)).
By Lemma 2.4 we have
(2.86)ΩM,ωr(T2m,π(n+1))≤c(1n+1)r‖T2m(r)‖M,ω.
Since
(2.87)T2m(r)(x)=T1(r)(x)+∑ν=0m-1{T2ν+1(r)(x)-T2ν(r)(x)},
we get
(2.88)ΩM,ωr(T2m,π(n+1))≤c(n+1)r{‖T1(r)‖M,ω+∑ν=0m-1‖T2ν+1(r)-T2ν(r)‖M,ω}.
Fractional Bernstein inequality of Lemma 2.2 gives
(2.89)‖T2ν+1(r)-T2ν(r)‖M,ω≤c2νr‖T2ν+1-T2ν‖M,ω≤c2νr+1E2ν(f)M,ω,‖T1(r)‖M,ω=‖T1(r)-T0(r)‖M,ω≤cE0(f)M,ω.
Hence,
(2.90)ΩM,ωr(T2m,π(n+1))≤c(n+1)r{E0(f)M,ω+∑ν=0m-12(ν+1)rE2ν(f)M,ω}.
It is easily seen that
(2.91)2(ν+1)rE2ν(f)M,ω≤c*∑μ=2ν-1+12νμr-1Eμ(f)M,ω,ν=1,2,3,…,
where
(2.92)c*={2r+1,0<r<1,22r,r≥1.
Therefore,
(2.93)ΩM,ωr(T2m,π(n+1))≤c(n+1)r{E0(f)M,ω+2rE1(f)M,ω+C∑ν=1m∑μ=2ν-1+12νμr-1Eμ(f)M,ω}≤c(n+1)r{E0(f)M,ω+∑μ=12mμr-1Eμ(f)M,ω}≤c(n+1)r∑ν=02m-1(ν+1)r-1Eν(f)M,ω.
If we choose 2m≤n+1≤2m+1, then
(2.94)ΩM,ωr(T2m,π(n+1))≤c(n+1)r∑ν=0n(ν+1)r-1Eν(f)M,ω,E2m(f)M,ω≤E2m-1(f)M,ω≤c(n+1)r∑ν=0n(ν+1)r-1Eν(f)M,ω.
Last two inequalities complete the proof.From Theorems2.7 and 2.8 we have the following corollaries.Corollary 2.9.
LetM∈QC2θ(0,1) and r∈ℝ+.  If f∈LM,ω(T) with ω∈Ap(M) and
(2.95)En(f)M,ω=O(n-σ),σ>0,n=1,2,…,
then
(2.96)ΩM,ωr(f,δ)={O(δσ);r>σ,O(δσ|log⁡(1δ)|);r=σ,O(δα);r<σ,
hold.Definition 2.10.
LetM∈QC2θ(0,1) and r∈ℝ+. Iff∈LM,ω(T) and ω∈Ap(M) then for 0<σ<r we set Lipσ(r,M,ω):={f∈LM,ω(T):ΩM,ωr(f,δ)=𝒪(δσ),δ>0}.Corollary 2.11.
LetM∈QC2θ(0,1) and r∈ℝ+.  If f∈LM,ω(T),  ω∈Ap(M),  0<σ<r and En(f)M,ω=𝒪(n-σ),n=1,2,…, then  f∈Lipσ(r,M,ω).Corollary 2.12.
Let0<σ<r and let f∈LM,ω(T),  ω∈Ap(M),  where M∈QC2θ(0,1). Then the following conditions are equivalent:
(2.97)(a)f∈Lipσ(r,M,ω),(b)En(f)M,ω=O(n-σ),n=1,2,….Theorem 2.13.
Letf∈LM,ω(T),  ω∈Ap(M), where M∈QC2θ(0,1). If  α∈(0,∞) and
(2.98)∑ν=1∞να-1Eν(f)M,ω<∞,
then
(2.99)En(f(α))M,ω≤c((n+1)αEn(f)M,ω+∑ν=n+1∞να-1Eν(f)M,ω)
hold where the constant c>0 is dependent only on α and M.Proof of Theorem2.13.
The condition (2.98) and Lemma 2.3 implies that f(α) exist and f(α)∈LM,ω(T). Since
(2.100)‖f(α)-Sn(f(α))‖M,ω≤‖S2m+2(f(α))-Sn(f(α))‖M,ω+∑k=m+2∞‖S2k+1(f(α))-S2k(f(α))‖M,ω,
we have for 2m<n<2m+1(2.101)‖S2m+2(f(α))-Sn(f(α))‖M,ω≤c2(m+2)αEn(f)M,ω≤C(n+1)αEn(f)M,ω.
On the other hand, we find
(2.102)∑k=m+2∞‖S2k+1(f(α))-S2k(f(α))‖M,ω≤c∑k=m+2∞2(k+1)αE2k(f)M,ω≤C∑k=m+2∞∑μ=2k-1+12kμα-1Eμ(f)M,ω=c∑ν=2m+1+1∞να-1Eν(f)M,ω≤c∑ν=n+1∞να-1Eν(f)M,ω,
and Theorem 2.13 is proved.As a corollary of Theorems2.7, 2.8, and 2.13 we have the following.Corollary 2.14.
Letf∈WMα(T,ω),  ω∈Ap(M),  r∈(0,∞), and
(2.103)∑ν=1∞να-1Eν(f)M,ω<∞
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
(2.104)ΩM,ωr(f(α),πn+1)≤c(n+1)r∑ν=0n(ν+1)α+r-1Eν(f)M,ω+c∑ν=n+1∞να-1Eν(f)M,ω
hold.


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

Letw=φ(z) and w=φ1(z) be the conformal mappings of G∞ and G0 onto the complement 𝔻∞ of 𝔻, normalized by the conditions(3.1)φ(∞)=∞,lim⁡z→∞φ(z)/z>0,φ1(0)=∞,lim⁡z→0zφ1(z)>0,
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 representations(3.2)ψ(w)=αw+α0+∑k=1∞αkwk,α>0,ψ1(w)=∑l=1∞βlwl,β1>0.
Therefore, the functions(3.3)ψ′(w)ψ(w)-z,z∈G0,ψ1′(w)ψ1(w)-z,z∈G∞
are analytic in 𝔻∞ and have, respectively, simple zero and zero of order 2 at ∞. Hence they have expansions(3.4)ψ′(w)ψ(w)-z=∑k=0∞Fk(z)wk+1,z∈G0,w∈D∞,ψ1′(w)ψ1(w)-z=∑k=1∞F̃k(1/z)wk+1,z∈G∞,w∈D∞,
where Fk(z) and F̃k(1/z) are, respectively, Faber Polynomials of degree k for continuums G0¯ and ℂ¯∖G0, with the integral representations [38, pp. 35, 255](3.5)Fk(z)=12πi∫Twkψ′(w)ψ(w)-zdw,z∈G0,F̃k(1z)=12πi∫Twkψ1′(w)ψ1(w)-zdw,z∈G∞,(3.6)Fk(z)=φk(z)+12πi∫Γφk(ς)ς-zdς,z∈G∞,k=0,1,2,…,(3.7)F̃k(1z)=φ1k(z)-12πi∫Γφ1k(ς)ς-zdς,z∈G0∖{0}.
We put(3.8)ak:=ak(f):=12πi∫Tf0(w)wk+1dw,k=0,1,2,…,ãk:=ãk(f):=12πi∫Tf1(w)wk+1dw,k=1,2,…
and correspond the series(3.9)∑k=0∞akFk(z)+∑k=1∞ãkF̃k(1z)
with the function f∈L1(Γ), that is,(3.10)f(z)~∑k=0∞akFk(z)+∑k=1∞ãkF̃k(1z).
This series is called the Faber-Laurent series of the function f and the coefficients ak and ãk 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 Ep(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 Ep(G0), 1<p<∞ norm.A functionω:Γ→[0,∞] is called a weight on Γ, if ω is measurable and ω-1({0,∞}) has measure zero. We denote by LM,ω(Γ) the linear space of Lebesgue measurable functions f:Γ→ℂ satisfying the condition(3.11)∫ΓM[α|f(z)|]ω(z)|dz|<∞
for some α>0.The spaceLM,ω(Γ) becomes a Banach space with the Orlicz norm(3.12)‖f‖M,Γ,ω:=sup⁡{∫Γ|f(z)g(z)|ω(z)|dz|:g∈LN,ω(Γ);ρ(g;N)≤1},
where N is the complementary function of M and(3.13)ρ(g;N):=∫ΓN[|g(z)|]ω(z)|dz|.The Banach spaceLM,ω(Γ) is called weighted Orlicz space on Γ.Forz∈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 relation(3.14)sup⁡z∈Γsup⁡ϵ>0(1ϵ∫Γ(z,ϵ)ω(τ)|dτ|)(1ϵ∫Γ(z,ϵ)ω(τ)-1/(p-1)|dτ|)p-1<∞,ifp>1,sup⁡ϵ>01ϵ∫Γ(z,ϵ)ω(τ)|dτ|≤cω(z),∀z∈Γ,ifp=1
is denoted by Ap(Γ).We denote byLp(Γ,ω) the set of all measurable functions f:Γ→ℂ such that |f|ω1/p belongs to Lebesgue space Lp(Γ), 1≤p<∞, on Γ.Definition 3.1.
Letω be a weight on Γ and let EM,ω(G0):={f∈E1(G0):f∈LM,ω(Γ)}, EM,ω(G∞):={f∈E1(G∞):f∈LM,ω(Γ)}, ẼM,ω(G∞):={f∈EM,ω(G∞):f(∞)=0}. The classes of functions EM,ω(G0) and EM,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 theFn is the same with the best approximating algebraic polynomials in the weighted Smirnov-Orlicz class EM,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 EM,ω(G0).In the case that all of the zeros of thenth Faber polynomial Fn are in G0, we denote by Ln(f,·) the (n-1)th interpolating polynomial for f∈EM,ω(G0) based on the zeros of Fn.Letf∈L1(Γ). Then the functions f+ and f- defined by(3.15)f+(z)=12πi∫Γf(ς)ς-zdς,z∈G0,f-(z)=12πi∫Γf(ς)ς-zdς,z∈G∞
are analytic in G0 and G∞, respectively, and f-(∞)=0.We denote by(3.16)En(f)M,Γ,ω:=inf⁡{‖f-p‖M,Γ,ω:p∈Pn}
the minimal error of approximation by polynomials of f, where 𝒫n is the set of algebraic polynomials of degree not greater than n.LetΓ be a rectifiable Jordan curve, f∈L1(Γ), and let(3.17)(SΓf)(t):=lim⁡ε→012πi∫Γ∖Γ(t,ε)f(ς)ς-tdς,t∈Γ
be Cauchy’s singular integral of f at the point t. The linear operator SΓ:f→SΓf is called the Cauchy singular operator.If one of the functionsf+ or f- has the nontangential limits a.e. on Γ, then SΓf(z) exists a.e. on Γ and also the other one has the nontangential limits a.e. on Γ. Conversely, if SΓf(z) exists a.e. on Γ, then both functions f+ and f- have the nontangential limits a.e. on Γ. In both cases, the formulae(3.18)f+(z)=(SΓf)(z)+f(z)2,f-(z)=(SΓf)(z)-f(z)2
hold, and hence(3.19)f=f+-f-
holds a.e. on Γ (see, e.g., [1, page 431]).Lemma 3.2.
fΓ is a regular curve,  M∈QC2θ(0,1) and ω∈Ap(M)(Γ),  then for every f∈EM,ω(G0) one has
(3.20)‖SΓ(f,⋅)‖M,Γ,ω≤c‖f‖M,Γ,ω,
where the constant c depends only on Γ and M.Proof.
Assertion (3.20) immediately follows from modular inequality
(3.21)∫ΓM(|SΓ(f,t)|)ω(t)dt≤c∫ΓM(|f(t)|)ω(t)dt
given in (7.5.13) of [23].Theorem 3.3.
IfΓ is a closed Radon curve, M∈QC2θ(0,1) and ω∈Ap(M)(Γ),  then for every f∈EM,ω(G0) one has
(3.22)‖f-Ln(f,⋅)‖M,Γ,ω≤cEn(f)M,Γ,ω,
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 Ln(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, Ln(f,·) is uniformly bounded in weighted Smirnov-Orlicz class EM,ω(G0). We suppose that Pn-1 is the (n-1)th best approximating algebraic polynomial for f in EM,ω(G0). In this case we have
(3.23)‖f-Ln(f,⋅)‖M,Γ,ω=‖f-Pn-1-Ln(f-Pn-1,⋅)‖M,Γ,ω≤(1+‖Ln‖)‖f-Pn-1‖M,Γ,ω.
Since we assumed the interpolation nodes as the zeros of the Faber polynomials Fn, using [39, page 59], we have
(3.24)f(z)-Ln(f,z)=Fn(z)2πi∫Γf(ς)Fn(ς)(ς-z)dς=Fn(z)(SΓ(fFn))(z),z∈G0
and consequently
(3.25)‖f(z)-Ln(f,z)‖M,Γ,ω=‖Fn(⋅)(SΓ(fFn))(⋅)‖M,Γ,ω≤{max⁡z∈Γ|Fn(z)|}‖SΓ(fFn)‖M,Γ,ω.
By Lemma 3.2, we get
(3.26)‖f-Ln(f,⋅)‖M,Γ,ω≤c{max⁡z∈Γ|Fn(z)|}‖fFn‖M,Γ,ω≤c{max⁡z,ς∈Γ|Fn(z)Fn(ς)|}‖f‖M,Γ,ω.
We set κ∶=maxz∈Γ|ϑz-1|, where ϑzπ 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∈Γ(3.27)0,5-0,5⋅κ<|Fn(z)|<1,5+0,5⋅κ,
and therefore
(3.28)max⁡z,ς∈Γ|Fn(z)Fn(ς)|≤3+κ1-κ.
From the last inequality we obtain
(3.29)‖f-Ln(f,⋅)‖M,Γ,ω≤c3+κ1-κ‖f‖M,Γ,ω.
Since
(3.30)‖Ln(f,⋅)‖M,Γ,ω≤‖f‖M,Γ,ω+‖f-Ln(f,⋅)‖M,Γ,ω≤(1+c3+κ1-κ)‖f‖M,Γ,ω,
we obtain that Ln(f,·) is uniformly bounded in EM,ω(G0), namely,
(3.31)‖Ln‖≤c.
Therefore, we conclude that
(3.32)‖f-Ln(f,⋅)‖M,Γ,ω≤c‖f-Pn-1‖M,Γ,ω=cEn(f)M,Γ,ω
and interpolating polynomial Ln(f,·) is near best approximant for f.IfΓ is Dini-smooth, then [44] there exist constants c and C such that(3.33)0<c<|ψ′(w)|<C<∞,|w|≥1.
Similar inequalities hold also for ψ1′ and φ1′, in case of |w|=1 and z∈Γ, respectively.We define Poisson polynomial for functionf∈EM,ω(G0)(3.34)Vn(f,z):=∑k=0nckFk(z)+∑k=n+12n-1(2-kn)ckFk(z),z∈G0.Theorem 3.4.
IfΓ is a Dini-smooth curve,  M∈QC2θ(0,1) and ω∈Ap(M)(Γ),  then for every f∈EM,ω(G0) one has
(3.35)‖f-Vn(f,⋅)‖M,Γ,ω≤cEn(f)M,Γ,ω,
where the constant c depends only on Γ and M.Proof.
From (3.8) and (3.5), we have
(3.36)Vn(f,z)=12π∫02πf(ψ(eit))dt2πi∫Γ∑k=-(2n-1)2n-1λ|k|φk(ς)e-iktς-zdς,
where z∈G0 and
(3.37)λ|k|:={1,0≤k≤n,2-kn,n+1≤k≤2n-1.
If Pn∈𝒫n is near best approximant for f∈EM,ω(G0), we get
(3.38)‖f-Vn(f,⋅)‖M,Γ,ω≤En(f)M,Γ,ω+‖Pn-Vn(f,⋅)‖M,Γ,ω.
Using
(3.39)Pn(z)=12π∫02πPn(ψ(eit))dt2πi∫Γ∑k=-(2n-1)2n-1λ|k|kφ(ς)e-iktς-zdς,z∈G0,
we find
(3.40)Pn(z)-Vn(f,z)=12π∫02π{Pn(ψ(eit))-f(ψ(eit))}dt2πi∫Γ∑k=-(2n-1)2n-1λ|k|kφ(ς)e-iktς-zdς,z∈G0.
Taking in the last inequality, the nontangential boundary values from inside of Γ, z→z0∈Γ and using (3.18), we have
(3.41)Pn(z0)-Vn(f,z0)=12π∫02π{Pn(ψ(eit))-f(ψ(eit))}dt×[12∑k=-(2n-1)2n-1λ|k|kφ(z0)e-ikt+12πi∫Γ∑k=-(2n-1)2n-1λ|k|kφ(ς)e-iktς-z0dς].
Since φ-2n(ς)∑k=-(2n-1)2n-1λ|k|φk(ς)e-ikt is analytic in G∞, we have
(3.42)12πi∫Γ∑k=-(2n-1)2n-1λ|k|φk(ς)e-ikt(ς-z)φ2n(ς)dς=0,z∈G0,
and taking nontangential limit in (3.42) we get
(3.43)12∑k=-(2n-1)2n-1λ|k|φk(z0)e-ikt=-12πi∫Γφ2n(z0)φ2n(ς)∑k=-(2n-1)2n-1λ|k|φk(ς)e-iktς-z0dς,
and hence by transformation z0=ψ(w0) we obtain
(3.44)Pn(z0)-Vn(f,z0)=12π∫02π{Pn(ψ(eit))-f(ψ(eit))}dt2πi∫T(1-w02nw2n)∑k=-(2n-1)2n-1λ|k|wke-iktψ(w)-ψ(w0)ψ′(w)dw.
Since one has
(3.45)12πi∫T(1-w02nw2n)∑k=-(2n-1)2n-1λ|k|wke-iktψ(w)-ψ(w0)ψ′(w)dw=12πi∫T(1-w02nw2n)∑k=-(2n-1)2n-1λ|k|wke-ikt[ψ′(w)ψ(w)-ψ(w0)-1w-w0]dw+12πi∫T(1-w02nw2n)1w-w0∑k=-(2n-1)2n-1λ|k|wke-ikt,
we can write
(3.46)Pn(z0)-Vn(f,z0)=12π∫02π{Pn(ψ(eit))-f(ψ(eit))}dt×12πi∫T(1-w02nw2n)∑k=-(2n-1)2n-1λ|k|wke-ikt[ψ′(w)ψ(w)-ψ(w0)-1w-w0]dw+12π∫02π{Pn(ψ(eit))-f(ψ(eit))}dt2πi∫T(1-w02nw2n)1w-w0×∑k=-(2n-1)2n-1λ|k|wke-ikt:=I1+I2.
From equality
(3.47)12πi∫T(1-w02nw2n)1w-w0∑k=-(2n-1)2n-1λ|k|wke-iktdw=∑k=-(2n-1)2n-1λ|k|w0ke-ikt,
we have
(3.48)‖I2‖M,Γ,ω≤En(f)M,Γ,ω12π∫02π|∑k=-(2n-1)2n-1λ|k|w0ke-ikt|dt.
On the other hand,
(3.49)‖I1‖M,Γ,ω≤En(f)M,Γ,ω14π2∫02π|∑k=-(2n-1)2n-1λ|k|wke-ikt|∫T|1-w02nw2n||ψ′(w)ψ(w)-ψ(w0)-1w-w0||dw|.
We denote by A a subarc of 𝕋 with the center w0 such that it has arc lenght 𝒪(1/n). In this case
(3.50)∫A|1-w02nw2n||ψ′(w)ψ(w)-ψ(w0)-1w-w0||dw|≤∫A|w2n-w02n|||ψ′(w)||ψ(w)-ψ(w0)|-1|w-w0|||dw|≤c
and, by (1.3),
(3.51)∫T∖A|1-w02nw2n||ψ′(w)ψ(w)-ψ(w0)-1w-w0||dw|≤2∫T∖A|ψ′(w)(w-w0)-[ψ(w)-ψ(w0)]||ψ(w)-ψ(w0)||w-w0||dw|≤c∫1/n1ω(ψ′,t)tdt≤c.
Hence,
(3.52)‖I1‖M,Γ,ω≤En(f)M,Γ,ω∫02π|∑k=-(2n-1)2n-1λ|k|wke-ikt|dt.
Inequalities (3.46), (3.48), and (3.52) imply that
(3.53)‖Pn-Vn(f,⋅)‖M,Γ,ω≤En(f)M,Γ,ω{∫02π|∑k=-(2n-1)2n-1λ|k|w0ke-ikt|+∫02π|∑k=-(2n-1)2n-1λ|k|wke-ikt|}.
For every w∈𝕋, one has
(3.54)∫02π|∑k=-(2n-1)2n-1λ|k|wke-ikt|≤c,
and therefore we get the required inequality of Theorem 3.4.Theorem3.4 signifies that Poisson polynomial is near best approximant for f.Forg∈LM,ω(𝕋), we set(3.55)σh(g)(w):=12h∫-hhg(weit)dt,0<h<π,w∈T.
If M∈QC2θ(0,1) and ω∈Ap(M)(𝕋), then by Theorem A(ii) we have(3.56)‖σh(g)‖M,T,ω≤c‖g‖M,T,ω,
and consequently σh(g)∈LM,ω(𝕋) for any g∈LM,ω(𝕋).Definition 3.5.
LetM∈QC2θ(0,1), ω∈Ap(M)(𝕋), and r>0. The function
(3.57)ΩM,T,ωr(g,δ):=sup⁡0<hi,t≤δ‖∏i=1[r](I-σhi)(I-σt)r-[r]g‖M,T,ω,δ>0
is called rth modulus of smoothness of g∈LM,ω(𝕋).It can easily be verified that the functionΩM,𝕋,ωr(g,·) is continuous, nonnegative, subadditive and satisfy limδ→0ΩM,𝕋,ωr(g,δ)=0 for g∈LM,ω(𝕋).LetΓ be a Dini-smooth curve and ω be a weight on Γ. We associate with ω the following two weights defined on 𝕋 by(3.58)ω0:=ω∘ψ,ω1:=ω∘ψ1
and let f0:=f∘ψ,f1:=f∘ψ1 for f∈LM,ω(Γ). Then from (3.33), we have f0∈LM,ω0(𝕋) and f1∈LM,ω1(𝕋) for f∈LM,ω(Γ). Using the nontangential boundary values of f0+ and f1+ on 𝕋, we define(3.59)ΩM,Γ,ωr(f,δ):=ΩM,T,ω0r(f0+,δ),Ω̃M,Γ,ωr(f,δ):=ΩM,T,ω1r(f1+,δ)
for r,δ>0.We set(3.60)En(f)M,T,ω:=inf⁡P∈Pn(D)‖f-P‖M,T,ω,Ẽn(g)M,Γ,ω:=inf⁡R∈Rn‖g-R‖M,Γ,ω,
where f∈EM,ω(𝔻), g∈EM,ω(G∞), and ℛn is the set of rational functions of the form ∑k=0nakz-k.Now we can give several applications of approximation theorems of Section2.Theorem 3.6.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈LM,ω(Γ) with ω∈A1(Γ).  Then there is a constant c>0 such that for any natural number n(3.61)‖f-Rn(⋅,f)‖M,Γ,ω≤c{ΩM,Γ,ωr(f,1(n+1))+Ω̃M,Γ,ωr(f,1(n+1))},
where r>0 and Rn(·,f) is the nth partial sum of the Faber-Laurent series of f.Corollary 3.7.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈EM,ω(G0) with ω∈A1(Γ).  Then there is a constant c>0 such that for every natural number n(3.62)‖f-Pn(⋅,f)‖M,Γ,ω≤cΩM,Γ,ωr(f,1(n+1)),r>0,
where Pn(·,f) is the nth partial sum of the Faber series of f.Corollary 3.8.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈ẼM,ω(G∞) with  ω∈A1(Γ). Then there is a constant c>0 such that for every natural number n(3.63)‖f-Rn(⋅,f)‖M,Γ,ω≤c8Ω̃M,Γ,ωr(f,1(n+1)),r>0,
where Rn(·,f) is as in Theorem 3.6.Theorem 3.9.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈EM,ω(G0) with ω∈A1(Γ). Then for r>0 there exists a constant c>0 such that
(3.64)ΩM,Γ,ωr(f,1n)≤cnr{E0(f)M,Γ,ω+∑k=1nkr-1Ek(f)M,Γ,ω}
hold.Corollary 3.10.
Under the conditions of Corollary3.7, if
(3.65)En(f)M,Γ,ω=O(n-α),α>0,n=1,2,3,…,
then for f∈EM,ω(G0) and r>0(3.66)ΩM,Γ,ωr(f,δ)={O(δα),r>α,O(δα|log⁡1δ|),r=α,O(δr),r<α.Definition 3.11.
LetM∈QC2θ(0,1) and α∈ℝ+.If f∈EM,ω(G0), then for 0<σ<α we set
(3.67)Lip̃σ(α,M,Γ,ω):={f∈ẼM,ω(G∞):Ω̃M,Γ,ωα(f,δ)=O(δα)},Lipσ(α,M,Γ,ω):={f∈EM,ω(G0):ΩM,Γ,ωα(f,δ)=O(δσ),δ>0}.Corollary 3.12.
LetM∈QC2θ(0,1) and α∈ℝ+.  If f∈EM,ω(G0),  ω∈A1(Γ),  0<σ<α and En(f)M,Γ,ω=𝒪(n-σ),n=1,2,…,  then f∈Lipσ(α,M,Γ,ω).By Corollaries3.7 and 3.10 we have the constructive characterization of the class Lipσ(α,M,Γ,ω).Corollary 3.13.
Let0<σ<α and f∈EM,ω(G0),ω∈A1(Γ),  where M∈QC2θ(0,1), be fulfilled. Then the following conditions are equivalent:(a)
f∈Lipσ(α,M,Γ,ω).(b)
En(f)M,Γ,ω=𝒪(n-σ),n=1,2,….The inverse theorem for unbounded domains has the following form.Theorem 3.14.
LetΓ be a Dini-smooth curve,  M∈QC2θ(0,1) and f∈ẼM,ω(G∞) with ω∈A1(Γ).  Then there is a constant c>0 such that for every natural number n(3.68)Ω̃M,Γ,ωr(f,1n)≤cnr{Ẽ0(f)M,Γ,ω+∑k=1nkr-1Ẽk(f)M,Γ,ω},r>0
holds.By the similar way to that ofEM,ω(G0), we obtain the following corollaries.Corollary 3.15.
Under the conditions of Corollary3.8, if
(3.69)Ẽn(f)M,Γ,ω=O(n-α),α>0,n=1,2,3,…,
then forf∈ẼM,ω(G∞) and r>0(3.70)Ω̃M,Γ,ωr(f,δ)={O(δα),r>α,O(δα|log⁡1δ|),r=α,O(δr),r<α.Corollary 3.16.
Under the conditions of Theorem3.14, if
(3.71)Ẽn(f)M,Γ,ω=O(n-α),α>0,n=1,2,3,…,
then f∈Lip̃σ(α,M,Γ,ω).By Corollaries3.8 and 3.15, we have the following.Corollary 3.17.
Letα>0 and the conditions of Theorem 3.14 be fulfilled. Then the following conditions are equivalent,(a)
f∈Lip̃σ(α,M,Γ,ω),(b)
Ẽn(f)M,Γ,ω=𝒪(n-α),n=1,2,3,….Before the proofs, we need some auxiliary lemmas.Lemma 3.18.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈LM,ω(Γ) with ω∈A1(Γ).  Then, f+∈EM,ω(G0) and f-∈ẼM,ω(G∞) for every f∈LM,ω(Γ).Proof.
UsingM∈Δ2, we can find a p∈(1,∞) such that LM,ω(Γ)⊂Lp(Γ,ω), where the inclusion maps being continuous (see, e.g., Lemma 2.13 of [20]). Since ω∈Ap(Γ) by [9], we get f+∈E1(G0) and f-∈E1(G∞). Using ω∈Ap(M)(Γ) and boundedness of operator SΓ in LM,ω(Γ), we obtain from (3.18) that
(3.72)f+∈LM,ω(Γ),f-∈LM,ω(Γ).Lemma 3.19.
LetM∈QC2θ(0,1) and ω∈Ap(M)(𝕋). Then there exists a constant c>0 such that for every natural number n(3.73)‖g-Tng‖M,T,ω≤cΩM,T,ωr(g,1(n+1)),g∈EM,ω(D),
where r>0 and Tng is nth partial sum of the Taylor series of g at the origin.Proof.
Using Theorem2.7 this lemma can be proved by the same method of Theorem 3 of [45].Let𝒫 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)→EM,ω(G0) and T̃:𝒫(𝔻)→ẼM,ω(G∞) defined on 𝒫(𝔻) as(3.74)T(P)(z):=12πi∫TP(w)ψ′(w)ψ(w)-zdw,z∈G0,T̃(P)(z):=12πi∫TP(w)ψ1′(w)ψ1(w)-zdw,z∈G∞.Then it is readily seen that(3.75)T(∑k=0nbkwk)=∑k=0nbkFk(z),T̃(∑k=0ndkwk)=∑k=0ndkF̃k(1z).Ifz′∈G0, then(3.76)T(P)(z′)=12πi∫TP(w)ψ′(w)ψ(w)-z′dw=12πi∫Γ(P∘φ)(ς)ς-z′dς=(P∘φ)+(z′),
which, by (3.18), implies that(3.77)T(P)(z)=SΓ(P∘φ)(z)+(12)(P∘φ)(z)a.e. onΓ.Similarly taking from outside ofΓ the nontangential limit z′′→z∈Γ in the relation(3.78)T̃(P)(z′′)=12πi∫ΓP(φ1(ς))ς-z′′dς=[(P∘φ1)]-(z′′),z′′∈G∞,
we get(3.79)T̃(P)(z)=-(12)(P∘φ1)(z)+SΓ(P∘φ1)(z)a.e. onΓ.SinceSΓ is bounded in LM,ω(Γ), we have the following result.Lemma 3.20.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈LM,ω(Γ) with ω∈Ap(M)(Γ).  Then the linear operators
(3.80)T:P(D)⟶EM,ω(G0),T̃:P(D)⟶ẼM,ω(G∞)
are bounded.The set of trigonometric polynomials is dense inLM,ω(T), which implies density of the algebraic polynomials in EM,ω(𝔻). Consequently, from Lemma 3.20, we can extend the operators T and T̃ from 𝒫(𝔻) to the spaces EM,ω0(𝔻) and EM,ω1(𝔻) as linear and bounded operators, respectively, and for the extensions T:EM,ω0(𝔻)→EM,ω(G0) and T̃:EM,ω1(𝔻)→ẼM,ω(G∞), we have the representations(3.81)T(g)(z)=12πi∫Tg(w)ψ′(w)ψ(w)-zdw,z∈G0,g∈EM,ω0(D),T̃(g)(z)=12πi∫Tg(w)ψ1′(w)ψ1(w)-zdw,z∈G∞,g∈EM,ω1(D).Lemma 3.21.
LetM∈QC2θ(0,1) and f∈LM,ω(𝕋) with ω∈Ap(M)(𝕋). Then,
(3.82)‖Pr(f)-f‖M,T,ω⟶0,asr⟶1-,
where
(3.83)Pr(f)(eiθ):=12π∫02πP(r,θ-t)f(eit)dt,0<r<1
and P(r,θ-t) is the Poisson kernel.Proof.
There are numbersp and q such that
(3.84)1<p<p(M)<q<∞,ω∈Ap(T).
Since [46, Theorem 10] Pr is a bounded operator in Lp(𝕋,ω) for every 1<p<∞, we have by Marcinkiewicz Interpolation Theorem
(3.85)‖Pr(f)‖M,T,ω≤c‖f‖M,T,ω.
From density of trigonometric polynomials in LM,ω(T), we have density of the set of continuous functions on 𝕋 in LM,ω(𝕋). Consequently, there is a continuous function f* on 𝕋 such that, for given ϵ>0 and f∈LM,ω(𝕋),
(3.86)‖f-f*‖M,T,ω<ϵ.
On the other hand, since the Poisson integral of a continuous function converges to it uniformly on𝕋 [47, page 239], we have by (2.7) and ω∈Ap(𝕋)(3.87)‖Pr(f*)-f*‖M,T,ω=sup⁡ρ(g,M̃)≤1∫T|Pr(f*)(w)-f*(w)||g(w)|ω(w)|dw|<ϵ(M(1)∫Tω(w)|dw|+1)≤Cϵ
for 0<1-r<δ(ϵ). Then, from (3.85), (3.86), and (3.87), we conclude that
(3.88)‖Pr(f)-f‖M,T,ω≤‖Pr(f)-Pr(f*)‖M,T,ω+‖Pr(f*)-f*‖M,T,ω+‖f*-f‖M,T,ω=‖Pr(f-f*)‖M,T,ω+‖Pr(f*)-f*‖M,T,ω+‖f*-f‖M,T,ω≤c‖f*-f‖M,T,ω+‖Pr(f*)-f*‖M,T,ω<{c+C}ϵ.
This completes the proof.Theorem 3.22.
LetΓ be a Dini-smooth curve, M∈QC2θ(0,1) and f∈LM,ω(Γ) with ω∈A1(Γ).  Then the linear operators
(3.89)T:EM,ω0(D)⟶EM,ω(G0),T̃:EM,ω1(D)⟶ẼM,ω(G∞)
are one-to-one and onto.Proof.
The proof we give, only for the operatorT. For the operator T̃ is the proof goes similarly. Let g∈EM,ω0(𝔻) with the Taylor expansion
(3.90)g(w):=∑k=0∞αkwk,w∈D.
Since Γ is a Dini-smooth curve, the conditions ω∈A1(Γ),ω0∈A1(𝕋), and ω1∈A1(𝕋) are equivalent.
Letgr(w):=g(rw),0<r<1. Since g∈E1(𝔻) is the Poisson integral of its boundary function [48, page 41], we have(3.91)‖gr-g‖M,T,ω0=‖Pr(g)-g‖M,T,ω0,
and using Lemma 3.21, we get ∥gr-g∥M,𝕋,ω0→0, as r→1-.
Therefore, the boundedness of the operatorT implies that(3.92)‖T(gr)-T(g)‖M,Γ,ω⟶0,asr⟶1-.
Since ∑k=0∞αkrkwk is uniformly convergent on 𝕋, one has
(3.93)T(gr)(z′)=12πi∫Tgr(w)ψ′(w)ψ(w)-z′dw=∑m=0∞αmrm12πi∫Twmψ′(w)ψ(w)-z′dw=∑m=0∞αmrmFm(z′),z′∈G0.
From the last equality and Lemma 3 of [39, page 43] we have
(3.94)ak(T(gr))=12πi∫TT(gr)(ψ(w))wk+1dw=12πi∫T∑m=0∞αmrmFm(ψ(w))wk+1dw=∑m=0∞αmrm12πi∫TFm(ψ(w))wk+1dw=αkrk,
and therefore
(3.95)ak(T(gr))⟶αk,asr⟶1-.
On the other hand, applying (3.33), (2.7), and weighted version of Hölder’s inequality
(3.96)∫Γ|u(z)v(z)|ω(z)|dz|≤c‖u‖M,Γ,ω‖v‖M̃,Γ,ω,u∈LM,ω(Γ),v∈LM̃,ω(Γ),
we obtain
(3.97)|ak(T(gr))-ak(T(g))|=|12πi∫T[T(gr)-T(g)](ψ(w))wk+1dw|≤12π∫T|[T(gr)-T(g)](ψ(w))||dw|=12π∫Γ|[T(gr)-T(g)](z)||φ′(z)||dz|≤c2π∫Γ|[T(gr)-T(g)](z)||dz|=c2π∫Γ|[T(gr)-T(g)](z)|ω-1(z)ω(z)|dz|≤c2π‖T(gr)-T(g)‖M,Γ,ω‖ω-1‖M̃,Γ,ω≤c2π‖T(gr)-T(g)‖M,Γ,ω,
because ∥ω-1∥M̃,Γ,ω≤M̃(1)mes(Γ)+1≤c<∞.
Using here the relation (3.92), we get(3.98)ak(T(gr))⟶ak(T(g)),asr⟶1-,
and then by (3.95), ak(T(g))=αk for k=0,1,2,…. If T(g)=0, then αk=ak(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 functionf∈EM,ω(G0) and consider the function f0=f∘ψ∈LM,ω0(𝕋). The Cauchy type integral(3.99)12πi∫Tf0(τ)τ-wdτ
represents analytic functions f0+ and f0- in 𝔻 and 𝔻∞, respectively. Since ω0∈A1(𝕋), by Lemma 3.18, we have
(3.100)f0+∈EM,ω0(D),f0-∈ẼM,ω0(D∞),
and moreover
(3.101)f0(w)=f0+(w)-f0-(w)
a.e. on 𝕋. Since f0-∈E1(𝔻∞) and f0-(∞)=0, we have
(3.102)ak=12πi∫Tf0(w)wk+1dw=12πi∫Tf0+(w)wk+1dw-12πi∫Tf0-(w)wk+1dw=12πi∫Tf0+(w)wk+1dw,
which proves that the coefficients ak,k=0,1,2,…, also become the Taylor coefficients of the function f0+ at the origin, that is,
(3.103)f0+(w)=∑k=0∞akwk,w∈D,
and also
(3.104)T(f0+)~∑k=0∞akFk.
Hence the functions T(f0+) and f have the same Faber coefficients ak,k=0,1,2,…, and therefore T(f0+)=f. This proves that the operator T is onto.Proof of Theorem3.6.
We prove that the rational function(3.105)Rn(z,f):=∑k=0nakFk(z)+∑k=1nãkF̃k(1z)
satisfies the required inequality of Theorem 3.6. This inequality is true if we can show that
(3.106)‖f-(z)+∑k=1nãkF̃k(1z)‖M,Γ,ω≤cΩ̃M,Γ,ωr(f,1(n+1)),(3.107)‖f+(z)-∑k=0nakFk(z)‖M,Γ,ω≤cΩM,Γ,ωr(f,1(n+1)),
because f(z)=f+(z)-f-(z) a.e. on Γ.
First we prove (3.106). Let f∈LM,ω(Γ). Then f1∈LM,ω1(𝕋) and f0∈LM,ω0(𝕋). According to (3.101),(3.108)f(ς)=f0+(φ(ς))-f0-(φ(ς))
a.e. on Γ. On the other hand,
(3.109)f1(w)=f1+(w)-f1-(w),
which implies the inequality
(3.110)f(ς)=f1+(φ1(ς))-f1-(φ1(ς))
a.e. on Γ.
Letz′∈G0∖{0}. Using (3.7) and (3.110), we have(3.111)∑k=1nãkF̃k(1z′)=∑k=1nãkφ1k(z′)-12πi∫Γ∑k=1nãkφ1k(ς)ς-z′dς=∑k=1nãkφ1k(z′)-12πi∫Γ(∑k=1nãkφ1k(ς)-f1+(φ1(ς)))ς-z′dς-12πi∫Γf1-(φ1(ς))ς-z′dς-12πi∫Γf(ς)ς-z′dς=∑k=1nãkφ1k(z′)-12πi∫Γ(∑k=1nãkφ1k(ς)-f1+(φ1(ς)))ς-z′dς-f1-(φ1(z′))-f-(z′).
Hence, taking the nontangential limit z′→z∈Γ, inside of Γ, we obtain
(3.112)∑k=1nãkF̃k(1z)=∑k=1nãkφ1k(z)-12(∑k=1nãkφ1k(z)-f1+(φ1(z)))-SΓ[∑k=1nãkφ1k-(f1+∘φ1)]-f1-(φ1(z))-f+(z)
a.e. on Γ.
Using (3.19), (3.110), Minkowski’s inequality, and the boundedness of SΓ, we get(3.113)‖f-(z)+∑k=1nãkF̃k(1z)‖M,Γ,ω=‖12(∑k=1nãkφ1k(z)-f1+(φ1(z)))-SΓ[∑k=1nãkφ1k-(f1+∘φ1)](z)‖M,Γ,ω≤c‖∑k=1nãkφ1k(z)-f1+(φ1(z))‖M,Γ,ω≤c‖f1+(w)-∑k=1nãkwk‖M,T,ω1.
On the other hand, from the proof of Theorem 3.22 we know that the Faber-Laurent coefficients ãk of the function f and the Taylor coefficients of the function f1+ at the origin are the same. Then taking Lemma 3.19 into account, we conclude that
(3.114)‖f-+∑k=1nãkF̃k(1z)‖M,Γ,ω≤cΩM,T,ω1r(f1+,1(n+1))=cΩ̃M,Γ,ωr(f,1(n+1)),
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 Theorem3.9.
Letf∈EM,ω(G0). Then we have T(f0+)=f. Since by Theorem 3.22 the operator T:EM,ω0(𝔻)→EM,ω(G0) is linear, bounded, one-to-one and onto, the operator T-1:EM,ω(G0)→EM,ω0(𝔻) is also linear and bounded. We take a pn*∈𝒫n as the best approximating algebraic polynomial to f in EM,ω(G0), that is,
(3.115)En(f)M,Γ,ω=‖f-pn*‖M,Γ,ω.
Then, T-1(pn*)∈𝒫n(𝔻), and therefore
(3.116)En(f0+)M,T,ω0≤‖f0+-T-1(pn*)‖M,T,ω0=‖T-1(f)-T-1(pn*)‖M,T,ω0=‖T-1(f-pn*)‖M,T,ω0≤‖T-1‖‖f-pn*‖M,Γ,ω=‖T-1‖En(f)M,Γ,ω,
because the operator T-1 is bounded.
Theorem2.8 and (3.116) imply that(3.117)ΩM,Γ,ωr(f,1n)=ΩM,T,ω0r(f0+,1n)≤cnr{E0(f0+)M,T,ω0+∑k=1nkr-1Ek(f0+)M,T,ω0}≤c‖T-1‖nr{E0(f)M,Γ,ω+∑k=1nkr-1Ek(f)M,Γ,ω},r>0.



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*Source: 982360-2012-02-08.xml*