Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

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 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∞ .


Introduction
Let G 0 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 ∈ G 0 .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 ϕ 0 : D → G 0 of complex unit disc D : {w ∈ C : |w| 1} onto G 0 .Let γ r : ϕ 0 {w ∈ C : |w| r} for a given r ∈ 0, 1 .We denote by E p G 0 , 1 ≤ p ≤ ∞, Smirnov's classes of analytic functions f : where positive constant c is independent of r.

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 that A Young function Φ is said to satisfy Δ 2 condition Φ ∈ Δ 2 if there is a constant c > 0 such that for all x ∈ R. Two Young functions Φ and Φ 1 are said to be equivalent if there are c, C > 0 such that A function M : 0, ∞ → 0, ∞ is said to be quasiconvex if there exist a convex Young function Φ and a constant c ≥ 1 such that Φ 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 T M 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 : 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 holds xy ≤ M x M y , x,y ≥ 0.

2.7
For a quasiconvex function M we define the indice p M of M as 1 p M : inf p : p > 0, M p is quasiconvex , p 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 ⊂ L 1 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 : There exist 23, page 23 constants c, C > 0 such that

2.10
For a weight ω we denote by L p T, ω the class of measurable functions on T such that ω 1/p f belongs to Lebesgue space L p T on T. We set f p,ω : ω 1/p f p for f ∈ L p T, ω .A 2π-periodic weight function ω belongs to the Muckenhoupt class A p , 1 < p < ∞, if 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 θ 2 0, 1 a class of functions g satisfying Δ 2 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 θ 2 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 .
be the Fourier and the conjugate Fourier series of f ∈ L 1 T , respectively.Putting A k x : c k e ikx in 2.12 , we define for n 0, 1, 2, . . .
Let α ∈ R be given.We define fractional derivative of a function f ∈ L 1 T , satisfying 2.15 , as provided the right hand side exists.
Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from 2.20 that 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 and therefore Let M ∈ QC θ 2 0, 1 .For r ∈ R , we define the fractional modulus of smoothness of index r 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 θ 2 0, 1 , we have by 2.24 that where the constant c > 0, dependent only on r and M.
Remark 2.1.The modulus of smoothness Ω r M,ω f, δ , where For formulations of our results, we need several lemmas.
Lemma A see 36 .For α ∈ R , we suppose that

2.28
for some c > 0 if and only if there exists a R ∈ B such that where c and C are constants depending only on one another.
If M ∈ QC θ 2 0, 1 , ω ∈ A p M , and f ∈ L M,ω T , then from Theorem A ii and Abel's transformation we get and therefore from 2.14 and 2.30

2.31
From the property Proof.Without loss of generality one can assume that T n M,ω 1.Since we have by 2.30 and Theorem A iii that Hence from 2.33 and 2.31 , we find ≤ cn α T n M,ω .

2.38
General case follows immediately from this.
Let M ∈ QC θ 2 0, 1 .We denote by W α M T, ω , α > 0, ω ∈ A p M , the linear space of 2π-periodic real valued functions , 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 Proof.If α 0, then from boundedness see 2.21 of the operator S n we get that we have

2.42
From 2.21 we get the boundedness of W n in L M,ω T and we have

Journal of Function Spaces and Applications
From Lemma 2.2 we get

2.44
Now we have

2.45
Since we get Cn α E n f M,ω .

2.47
Now we show that

2.48
For this we set A k x, f : a k cos kx b k sin kx.

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 which by 2.7 this implies that and hence we obtain

2.52
In this case from 2.40 we have

2.58
Consequently, and 2.48 holds.Now 2.47 and 2.48 imply the result.
hold, where the constant c > 0 is dependent only on α and M.

2.62
Then, and hence 2.61 holds.We note that if , it can easily be obtained from the last inequality that the required inequality 2.61 holds.Now we will show

2.64
Putting Journal of Function Spaces and Applications we have g x s ds du dt.

2.69
Using 2.61 , 2.64 , and Lemma 2.2, we get 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 θ 2 0, 1 and ω ∈ A p M .

2.72
Putting we have and hence

2.75
On the other hand, we find

2.77
Since Journal of Function Spaces and Applications we get

2.79
Taking into account by a recursive procedure, we obtain

2.81
Now we can formulate the results.
Proof.Let T n ∈ T n be the best approximating polynomial of f ∈ L M,ω T and let m ∈ Z .Then,

2.85
By Lemma 2.4 we have

2.88
Fractional Bernstein inequality of Lemma 2.2 gives T r

Journal of Function Spaces and Applications
It is easily seen that where

2.94
Last two inequalities complete the proof.
From Theorems 2.7 and 2.8 we have the following corollaries.
Corollary 2.12.Let 0 < σ < r and let f ∈ L M,ω T , ω ∈ A p M , where M ∈ QC θ 2 0, 1 .Then the following conditions are equivalent: 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

2.101
On the other hand, we find and Theorem 2.13 is proved.
As a corollary of Theorems 2.7, 2.8, and 2.13 we have the following.
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 hold.

Near Best Approximants in Weighted Smirnov-Orlicz Space
Let w ϕ z and w ϕ 1 z be the conformal mappings of G ∞ and G 0 onto the complement respectively.We denote by ψ and ψ 1 the inverse mappings of ϕ and ϕ 1 , respectively, and T : ∂D.These mappings ψ and ψ 1 have in some deleted neighborhood of ∞ the representations Therefore, the functions are analytic in D ∞ and have, respectively, simple zero and zero of order 2 at ∞. Hence they have expansions where F k z and F k 1/z are, respectively, Faber Polynomials of degree k for continuums G 0 and C \ G 0 , with the integral representations 38, pp.35, 255 We put

Journal of Function Spaces and Applications
This series is called the Faber-Laurent series of the function f and the coefficients a k and a k 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 G 0 and show that interpolating polynomials and best approximating polynomial in E p G 0 , 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 G 0 , 1 < p < ∞ norm.
The space L M,ω Γ becomes a Banach space with the Orlicz norm where N is the complementary function of M and ρ g; N : Γ N g z ω z |dz|.

Definition 3.1. Let ω be a weight on Γ and let E M,ω G
The classes of functions E M,ω G 0 and E M,ω G ∞ will be called weighted Smirnov-Orlicz classes with respect to domains G 0 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 G 0 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 G 0 .
In the case that all of the zeros of the nth Faber polynomial F n are in G 0 , we denote by L n f, • the n − 1 th interpolating polynomial for f ∈ E M,ω G 0 based on the zeros of F n .
Let f ∈ L 1 Γ .Then the functions f and f − defined by are analytic in G 0 and G ∞ , respectively, and f − ∞ 0. We denote by 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 ∈ L 1 Γ , and let be Cauchy's singular integral of f at the point t.The linear operator S Γ : f → S Γ f is called the Cauchy singular operator.
If one of the functions f or f − has the nontangential limits a.e. on Γ, then S Γ f z exists a.e. on Γ and also the other one has the nontangential limits a.e. on Γ.Conversely, if S Γ f 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 where the constant c depends only on Γ and M.
Proof.Assertion 3.20 immediately follows from modular inequality given in 7.5.13 of 23 .
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 G 0 .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 G 0 .We suppose that P n−1 is the n − 1 th best approximating algebraic polynomial for f in E M,ω G 0 .In this case we have

3.23
Since we assumed the interpolation nodes as the zeros of the Faber polynomials F n , using 39, page 59 , we have and consequently

3.25
By Lemma 3.2, we get

3.26
We set κ : max z∈Γ |ϑ z − 1|, where ϑ z π 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 0, 5 − 0, 5 From the last inequality we obtain we obtain that L n f, • is uniformly bounded in E M,ω G 0 , namely, L n ≤ c.

3.31
Therefore, we conclude that 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 that

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 where the constant c depends only on Γ and M.
Proof.From 3.8 and 3.5 , we have where z ∈ G 0 and

3.38
Using we find

3.40
Taking in the last inequality, the nontangential boundary values from inside of Γ, z → z 0 ∈ Γ and using 3.18 , we have and taking nontangential limit in 3.42 we get and hence by transformation z 0 ψ w 0 we obtain

3.44
Since one has

Journal of Function Spaces and Applications
From equality we have On the other hand,

3.49
We denote by A a subarc of T with the center w 0 such that it has arc lenght O 1/n .In this case and, by 1.3 ,

3.53
For every w ∈ T, one has and therefore we get the required inequality of Theorem 3.4.
Theorem 3.4 signifies that Poisson polynomial is near best approximant for f.For g ∈ L M,ω T , we set

3.55
If M ∈ QC θ 2 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 θ 2 0, 1 , ω ∈ A p M T , and r > 0. The function It can easily be verified that the function Ω r M,T,ω g, • is continuous, nonnegative, subadditive and satisfy lim δ → 0 Ω r M,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 by Using the nontangential boundary values of f 0 and f 1 on T, we define for r, δ > 0.
We set and R n is the set of rational functions of the form n k 0 a k z −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 θ 2 0, 1 and f ∈ L M,ω Γ with ω ∈ A 1 Γ .Then there is a constant c > 0 such that for any natural number 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 θ 2 0, 1 and f ∈ E M,ω G 0 with ω ∈ A 1 Γ .Then there is a constant c > 0 such that for every natural number n , r > 0, 3.62 where P n •, f is the nth partial sum of the Faber series of f.
where R n •, f is as in Theorem 3.6.
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 G 0 , ω ∈ A 1 Γ , where M ∈ QC θ 2 0, 1 , be fulfilled.Then the following conditions are equivalent: The inverse theorem for unbounded domains has the following form.Theorem 3.14.Let Γ be a Dini-smooth curve, M ∈ QC θ 2 0, 1 and f ∈ E M,ω G ∞ with ω ∈ A 1 Γ .Then there is a constant c > 0 such that for every natural number n Ω r M,Γ,ω f, By the similar way to that of E M,ω G 0 , we obtain the following corollaries.

3.70
Corollary 3.16.Under the conditions of Theorem 3.14, if 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, Before the proofs, we need some auxiliary lemmas.

3.72
Lemma 3.19.Let M ∈ QC θ 2 0, 1 and ω ∈ A p M T .Then there exists a constant c > 0 such that for every natural number n g − T n g M,T,ω ≤ c Ω r M,T,ω g, where r > 0 and T n g 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 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 G 0 and T : P D → E M,ω G ∞ defined on P D as

3.74
Then it is readily seen that
Similarly taking from outside of Γ the nontangential limit z → z ∈ Γ in the relation 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 θ 2 0, 1 and f ∈ L M,ω Γ with ω ∈ A p M Γ .Then the linear operators 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,ω 0 D and E M,ω 1 D as linear and bounded operators, respectively, and for the extensions 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 > 0 and f ∈ L M,ω T ,

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 for 0 < 1 − r < δ .Then, from 3.85 , 3.86 , and 3.87 , we conclude that

3.88
This completes the proof.
Proof.The proof we give, only for the operator T .For the operator T is the proof goes similarly.Let g ∈ E M,ω 0 D with the Taylor expansion g w : Since Γ is a Dini-smooth curve, the conditions ω ∈ A 1 Γ , ω 0 ∈ A 1 T , and ω 1 ∈ A 1 T are equivalent.
Let g r w : g rw , 0 < r < 1.Since g ∈ E 1 D is the Poisson integral of its boundary function 48, page 41 , we have g r − g M,T,ω 0 P r g − g M,T,ω 0 , 3.91 and using Lemma 3.21, we get g r − g M,T,ω 0 → 0, as r → 1 − .Therefore, the boundedness of the operator T implies that

3.93
From the last equality and Lemma 3 of 39, page 43 we have and therefore a k T g r −→ α k , as r −→ 1 − .

3.95
On the other hand, applying 3.33 , 2.7 , and weighted version of H ölder's inequality a k F 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 :

Journal of Function Spaces and Applications
Using 3.19 , 3.110 , Minkowski's inequality, and the boundedness of S Γ , we get a k w k M,T,ω 1 .

3.113
On the other hand, from the proof of Theorem 3.22 we know that the Faber-Laurent coefficients a k 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 Proof of Theorem 3.9.Let f ∈ E M,ω G 0 .Then we have T f 0 f.Since by Theorem 3.22 the operator T : E M,ω 0 D → E M,ω G 0 is linear, bounded, one-to-one and onto, the operator T −1 : E M,ω G 0 → E M,ω 0 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 G 0 , that is,

3.115
Then, T −1 p * n ∈ P n D , and therefore
−t f 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 θ 2 0, 1 , ω ∈ A p M , and f ∈ L M,ω T , then there exists a constant c > 0 such that