Improved Inverse Theorems in Weighted Lebesgue and Smirnov Spaces
Article in Bulletin of the Belgian Mathematical Society, Simon Stevin · October 2007 DOI: 10.36045/bbms/1195157136 · Source: OAI
CITATIONS
11
READS
95 2 authors:
Some of the authors of this publication are also working on these related projects: Approximation in Variable Lebesgue and Smirnov spacesView project
Approximation in Variable Lebesgue and Smirnov SpacesView project Ali Guven Balikesir University 28PUBLICATIONS 196CITATIONS SEE PROFILE Daniyal M. Israfilov Balikesir University 84PUBLICATIONS 612CITATIONS SEE PROFILE
All content following this page was uploaded by Ali Guven on 17 February 2015.
Lebesgue and Smirnov Spaces
Ali Guven
Daniyal M. Israfilov
Abstract
The improvement of the inverse estimation of approximation theory by trigonometric polynomials in the weighted Lebesgue spaces was obtained and its application in the weighted Smirnov spaces was considered.
1
Introduction and the main results
Let T be the interval [−π, π] or the unit circle |z| = 1 of the complex plane C. A measurable 2π−periodic function ω : T → [0, ∞] is said to be a weight func-tion if ω−1({0, ∞}) has measure zero. With any given weight ω, we associate the ω−weighted Lebesgue space Lp(T, ω) , 1 ≤ p < ∞, consisting of all measurable
2π−periodic functions f on T such that kf kL p(T,ω):= Z T |f (x)|pω (x) dx 1/p < ∞.
Let 1 < p < ∞. A weight function ω belongs to the Muckenhoupt class Ap(T) if 1 |J| Z J ω (x) dx 1 |J| Z J [ω (x)]−1/(p−1)dx p−1 ≤ C
with a finite constant C independent of J, where J is any subinterval of [−π, π] and |J| denotes the length of J.
Received by the editors March 2006. Communicated by F. Brackx.
2000 Mathematics Subject Classification : 41A10, 41A25, 41A27, 42A10, 30E10.
Key words and phrases : Best approximation, Carleson curve, Muckenhoupt weight, weighted Lebesgue space, weighted Smirnov space.
The detailed information about the classes Ap(T)can be found in [22] and [9].
Let 1 < p < ∞ and ω ∈ Ap(T) . Since Lp(T, ω) is noninvariant with respect
to the usual shift, for the definition of the modulus of smoothness we consider the following mean value function as a shift for g ∈ Lp(T, ω):
σh(g) (x) := 1 2h h Z −h g (x + t) dt, 0 < h < π, x ∈ T.
It is known (see, [23]) that the operator σhis a bounded linear operator on Lp(T, ω) ,
1 < p < ∞, i. e.,
kσh(g)kLp(T,ω)≤ c kgkLp(T,ω), g ∈ Lp(T, ω) ,
holds with a constant c > 0 independent of g and h. The kth modulus of smoothness Ωk(g, ·)p,ω of the function g ∈ Lp(T, ω) is defined by
Ωk(g, δ)p,ω = sup 0<h≤δ T k hg L p(T,ω) , δ > 0 (1) where Thg = Th1g := g − σh(g) , Thkg := Th Thk−1g, k = 1, 2, ... .
The modulus of smoothness Ωk(g, ·)p,ωis nondecreasing, nonnegative, continuous
function and lim
δ→0 Ωk(g, δ)p,ω = 0, Ωk(g1 + g2, ·)p,ω ≤ Ωk(g1, ·)p,ω+ Ω (g2, ·)p,ω.
Let Tn (n = 0, 1, 2, ...) be the set of trigonometric polynomials of order at most
n. The best approximation to g ∈ Lp(T, ω) in the class Tn is defined by
En(g)p,ω = inf Tn∈Tn
kg − TnkLp(T,ω)
for n = 0, 1, 2, ... .
The problems of the approximation theory by trigonometric polynomials in the space Lp(T, ω), when the weight function satisfies the Muckenhoupt condition, were
investigated by E. A. Haciyeva in [11]. Haciyeva obtained the direct and inverse estimates in terms of the modulus of smoothness (1). N. X. Ky, using a relevant modulus of smoothness, investigated the approximation problems in the weighted Lebesgue spaces with Muckenhoupt weights (see [19], [20]). For more general class of weights, namely for doubling weights, similar problems were investigated by G. Mastroianni and V. Totik in [21]. Also, M. C. De Bonis, G. Mastroianni and M. G. Russo gave results for some special weight functions in [6].
The following inverse theorem was proved in [11].
Theorem A. Let 1 < p < ∞ and ω ∈ Ap(T) . Then, for g ∈ Lp(T, ω) the inequality
Ωk g,1 n p,ω ≤ c n2k ( E0(g)p,ω+ n X ν=1 ν2k−1Eν(g)p,ω ) , n = 1, 2, ..., (2) holds with a constant c > 0 independent of n.
In this work we improve the estimate (2). We shall denote by c, the constants (in general, different in different relations) depending only on numbers that are not important for the questions of interest.
Our main result is the following.
Theorem 1. Let 1 < p < ∞ and ω ∈ Ap(T) . Then, for g ∈ Lp(T, ω) the estimate
Ωk g, 1 n p,ω ≤ c n2k ( n X ν=1 ν2βk−1Eνβ(g)p,ω )1/β , n = 1, 2, ..., (3) holds with a constant c > 0 independent of n, where β = min (p, 2) .
The estimate (3) is better than (2). Indeed, let
x : = 1 2 ν X µ=1 µ2k−1Eµ(g)p,ω+ (ν − 1) ν 2k−1E ν(g)p,ω = 1 2 ν−1 X µ=1 µ2k−1Eµ(g)p,ω+ νν 2k−1 Eν(g)p,ω and x − h : = (ν − 1) ν2k−1Eν(g)p,ω, x + h := ν X µ=1 µ2k−1Eµ(g)p,ω x − δ : = νν2k−1Eν(g)p,ω, x + δ := ν−1 X µ=1 µ2k−1Eµ(g)p,ω
for ν = 1, 2, ... . Since the function xβ is convex for β = min (p, 2), we obtain
h νν2k−1Eν(g)p,ω iβ −h(ν − 1) ν2k−1Eν(g)p,ω iβ ≤ ν X µ=1 µ2k−1Eµ(g)p,ω β − ν−1 X µ=1 µ2k−1Eµ(g)p,ω β .
After summation with respect to ν we have
n X ν=1 h νν2k−1Eν(g)p,ω iβ −h(ν − 1) ν2k−1Eν(g)p,ω iβ ≤ n X ν=1 ν X µ=1 µ2k−1Eµ(g)p,ω β − ν−1 X µ=1 µ2k−1Eµ(g)p,ω β ,
and after simple computations we obtain
( n X ν=1 ν2βk−1Eνβ(g)p,ω )1/β ≤ 2 n X ν=1 ν2k−1Eν(g)p,ω.
Consequently the estimate (3) is never worse than (2). In addition, in some cases, it leads to a more precise result. For example, if
En(g)p,ω = O
1
n2k
, n = 1, 2, ..., then we obtain from (2)
Ωk(g, δ)p,ω = O δ2k log1 δ (4) and from (3) Ωk(g, δ)p,ω = O δ 2k log 1 δ 1/β! , which is better than (4).
The analogue of Theorem 1 in nonweighted Lebesgue spaces, in terms of the usual modulus of smoothness, was proved by M. F. Timan in [26] (see also [25, p. 338]).
We also give an improvement of the appropriate inverse theorem in the weighted Smirnov spaces, obtained in [16]. For its formulation we have to give some auxiliary definitions and notations.
Let G be a finite domain in the complex plane, bounded by a rectifiable Jordan curve Γ, and let G−:= ExtΓ be the exterior of Γ. Further let
D := {z ∈ C : |z| < 1} , D−:= C\D.
We denote by ϕ the conformal mapping of G− onto D− normalized by ϕ (∞) = ∞, lim
z→∞
ϕ (z) z > 0.
Let ψ be the inverse of ϕ. The functions ϕ and ψ have continuous extensions to Γ and T, their derivatives ϕ0(z) and ψ0(w) have definite nontangential limit values on Γ and T a. e., and they are integrable with respect to Lebesgue measure on Γ and T, respectively [10, pp. 419-438].
We denote by Ep(G) , 1 ≤ p < ∞, the Smirnov class of analytic functions in G.
Each function f ∈ Ep(G) has a nontangential limit almost everywhere (a. e.) on Γ,
and the nontangential limit of f , belongs to the Lebesgue space Lp(Γ) . The general
information about Ep(G) can be found in [8, pp. 168-185] and [10, pp. 438-453].
Let ω be a weight function on Γ and let Lp(Γ, ω) be the ω−weighted Lebesgue
space on Γ. The ω−weighted Smirnov space Ep(G, ω) defined as
Ep(G, ω) := {f ∈ E1(G) : f ∈ Lp(Γ, ω)} .
The approximation problems in Ep(G, ω) and Lp(Γ, ω) , 1 < p < ∞, was studied
in [14] , [15] and [16]. The nonweighted case was considered in [1], [17], [2], [13] and [4].
Definition 1. A rectifiable Jordan curve Γ is called a Carleson curve if the condition sup z∈Γ sup ε>0 1 ε |Γ (z, ε)| < ∞
holds, where Γ (z, ε) is the portion of Γ in the open disk of radius ε centered at z, and |Γ (z, ε)| its length.
The Muckenhoupt class on the rectifiable Jordan curve Γ seems as follows: Definition 2. Let 1 < p < ∞. A weight function ω belongs to the Muckenhoupt class Ap(Γ) if the condition
sup z∈Γ sup ε>0 1 ε Z Γ(z,ε) ω (τ ) |dτ | 1 ε Z Γ(z,ε) [ω (τ )]−1/(p−1)|dτ | p−1 < ∞ holds.
The Carleson curves and Muckenhoupt classes Ap(Γ) were studied in details in
[3].
Let Γ be a rectifiable Jordan curve and f ∈ L1(Γ). Then the function f+ defined
by f+(z) := 1 2πi Z Γ f (ς) ς − zdς, z ∈ G (5)
is analytic in G. Furthermore, if Γ is a Carleson curve and ω ∈ Ap(Γ) , then f+ ∈
Ep(G, ω) for f ∈ Lp(Γ, ω) , 1 < p < ∞ (see [14, Lemma 3]).
With every weight function ω on the rectifiable Jordan curve Γ, we associate another weight ω0 on T defined by ω0 := ω ◦ ψ.
Let ω ∈ Ap(Γ) and ω0 ∈ Ap(T) , where 1 < p < ∞. If f ∈ Lp(Γ, ω) , then
f0 := (f ◦ ψ)
ψ01/p ∈ Lp(T, ω0) .
We define the kth modulus of smoothness of the function f ∈ Lp(Γ, ω) by
Ωk(f, δ)Γ,p,ω := Ωk
f0+, δ
p,ω0
, δ > 0. (6)
The following theorem was proved in [16].
Theorem B. Let Γ be a Carleson curve, 1 < p < ∞, ω ∈ Ap(Γ) and ω0 ∈ Ap(T) .
Then, for f ∈ Ep(G, ω) the estimate
Ωk f, 1 n Γ,p,ω ≤ c n2k ( E0(f )Γ,p,ω+ n X ν=1 ν2k−1Eν(f )Γ,p,ω ) , n = 1, 2, ..., (7) holds with a constant c > 0 independent of n.
This theorem can be improved by the aim of Theorem 1 as follows:
Theorem 2. Let Γ be a Carleson curve, 1 < p < ∞, ω ∈ Ap(Γ) and ω0 ∈ Ap(T) .
Then, for f ∈ Ep(G, ω) the estimate
Ωk f, 1 n Γ,p,ω ≤ c n2k ( n X ν=1 ν2βk−1Eνβ(f )Γ,p,ω )1/β , n = 1, 2, ..., (8) holds with a constant c > 0 independent of n, where β = min (p, 2) .
2
Auxiliary results
Let Γ be a rectifiable Jordan curve and f ∈ L1(Γ) . Then the limit
SΓ(f ) (z) := lim ε→0 1 2πi Z Γ\Γ(z,ε) f (ς) ς − z dς (9)
is exists and is finite for almost all z ∈ Γ (see [3, pp. 117-144]). SΓ(f ) (z) is called
the Cauchy singular integral of f at z ∈ Γ.
For f ∈ L1(Γ) the function f+ (defined in (5)) has nontangential limits and the
formula
f+(z) = SΓ(f ) (z) +
1
2f (z) (10)
holds a. e. on Γ [10, p. 431].
For f ∈ L1(Γ), we associate the function SΓ(f ) taking the value SΓ(f ) (z) a.
e. on Γ. The linear operator SΓ defined in such way is called the Cauchy singular
operator. The following theorem, which is analogously deduced from David’s theo-rem (see [5]), states the necessary and sufficient condition for boundedness of SΓ in
Lp(Γ, ω) (see also [3, pp. 117-144]).
Theorem 3. Let Γ be a Carleson curve, 1 < p < ∞, and let ω be a weight function on Γ. The inequality
kSΓ(f )kLp(Γ,ω) ≤ c kf kLp(Γ,ω)
holds for every f ∈ Lp(Γ, ω) if and only if ω ∈ Ap(Γ) .
For k = 0, 1, 2, ..., and R > 1 let
Fk,p(z) := 1 2πi Z |t|=R tkψ0(t)1−1/p ψ (t) − z dt, z ∈ G.
Obviously, Fk,p is a polynomial of degree k. The polynomials Fk,p are called the
p−Faber polynomials for G (see [17] and [2]).
For detailed information about Faber polynomials and Faber series see [24, pp. 33-116].
Let Pn (n = 0, 1, 2, ...) be the set of the complex polynomials of degree at most
n, P be the set of all polynomials (with no restrictions on degrees), and let P (D) be the set of restrictions of the polynomials to D. If we define an operator Tp on P (D)
as Tp(P ) (z) := 1 2πi Z T P (t)ψ0(t)1−1/p ψ (t) − z dt, z ∈ G, then it is clear that
Tp n X k=0 aktk ! = n X k=0 akFk,p(z) . From (5), we have Tp(P ) z0= 1 2πi Z Γ P (ϕ (ς))ϕ0(ς)1/p ς − z0 dς = (P ◦ ϕ)ϕ01/p + z0
for z0 ∈ G. Taking the limit z0 → z ∈ Γ, over all nontangential paths inside Γ, we obtain by (10) Tp(P ) (z) = SΓ (P ◦ ϕ)ϕ01/p (z) + 1 2 (P ◦ ϕ)ϕ01/p (z) for almost all z ∈ Γ.
We can state the following theorem as a corollary of Theorem 3.
Theorem 4. Let Γ be a Carleson curve, 1 < p < ∞, and let w be a weight function on Γ. If ω ∈ Ap(Γ) and ω0 ∈ Ap(T) , then the linear operator Tp : P (D) →
Ep(G, ω) is bounded.
Hence if ω ∈ Ap(Γ) and ω0 ∈ Ap(T) , then the operator Tp can be extended to
the whole of Ep(D, ω0) as a bounded linear operator and we have the representation
Tp(g) (z) = 1 2πi Z T g (t)ψ0(t)1−1/p ψ (t) − z dt, z ∈ G, for all g ∈ Ep(D, ω0) .
Theorem 5 ([16]). Let Γ be a Carleson curve, 1 < p < ∞, and let ω be a weight function on Γ such that ω ∈ Ap(Γ) and ω0 ∈ Ap(T) . Then the operator Tp :
Ep(D, ω0) → Ep(G, ω) is one-to-one and onto. In fact, we have Tp
f0+ = f for f ∈ Ep(G, ω) .
3
Proofs of the main results
Let g ∈ Lp(T, ω) has the Fourier series
g (x) ∼ a0 2 + ∞ X ν=1 (aνcos νx + bνsin νx) .
We denote the nth partial sum of this series by Sn(g, x) . Let also
Aν(g, x) := aνcos νx + bνsin νx, ν = 1, 2, ..., and ∆µ(g, x) := 2µ−1 X ν=2µ−1 Aν(g, x) .
By a simple calculation, one can show that the kth difference Tk
h has the Fourier
series Thkg (x) ∼ ∞ X ν=1 1 −sin νh νh !k Aν(g, x) .
It is also known that (see [12]) the partial sums of the Fourier series are bounded in the space Lp(T, ω) and hence
Proof of Theorem 1. Let h > 0 and let m be any natural number. It is clear that Thk(g) (x) = Thk(g) (x) − Thk(S2m−1(g, ·)) (x) + Thk(S2m−1(g, ·)) (x) = Thk(g − S2m−1(g, ·)) (x) + Thk(S2m−1(g, ·)) (x) . Using (11) yields T k h (g − S2m−1(g, ·)) p,ω ≤ c kg − S2m−1(g, ·)kp,ω ≤ c E2m−1(g)p,ω.
On the other hand, by Theorem 1 of [18],
T k h (S2m−1(g, ·)) p,w = 2m−1 X ν=1 1 −sin νh νh !k Aν(g, ·) p,ω ≤ c m X µ=1 ∆2µ(g, ·; k, h) 2 p,ω , where ∆µ(g, x; k, h) := 2µ−1 X ν=2µ−1 1 − sin νh νh !k Aν(g, x) .
By simple calculations, we obtain
m X µ=1 ∆2µ(g, ·; k, h) 2 p,ω ≤ m X µ=1 k∆µ(g, ·; k, h)k2p,ω 1/2 if p > 2, and m X µ=1 ∆2µ(g, ·; k, h) 2 p,ω ≤ m X µ=1 k∆µ(g, ·; k, h)kpp,ω 1/p
if p ≤ 2. Hence we have to estimate k∆µ(g, ·; k, h)kp,ω.
Let’s assume that k = 1. By Abel’s transformation, we get ∆µ(g, x; 1, h) = 2µ−1 X ν=2µ−1 1 −sin νh νh ! Aν(g, x) = 2µ−2 X ν=2µ−1 " 1 −sin νh νh ! − 1 −sin (ν + 1) h (ν + 1) h !# ν X j=2µ−1 Aj(g, x) + 1 − sin (2 µ− 1) h (2µ− 1) h ! 2µ−1 X ν=2µ−1 Aν(g, x) = 2µ−2 X ν=2µ−1 sin (ν + 1) h (ν + 1) h − sin νh νh ! ν X j=2µ−1 Aj(g, x) + 1 − sin (2 µ− 1) h (2µ− 1) h ! 2µ−1 X ν=2µ−1 Aν(g, x) .
If we take the norm, we obtain k∆µ(g, ·; 1, h)kp,ω ≤ 2µ−2 X ν=2µ−1 sin νh νh − sin (ν + 1) h (ν + 1) h ! ν X j=2µ−1 Aj(g, ·) p,ω + 1 −sin (2 µ− 1) h (2µ− 1) h 2µ−1 X ν=2µ−1 Aν(g, ·) p,ω . Using (11) we get ν X j=2µ−1 Aj(g, ·) p,ω = ∞ X j=2µ−1 Aj(g, ·) − ∞ X j=ν+1 Aj(g, ·) p,ω ≤ ∞ X j=2µ−1 Aj(g, ·) p,ω + ∞ X j=ν+1 Aj(g, ·) p,ω = kg − S2µ−1−1(g, ·)k p,ω+ kg − Sν(g, ·)kp,ω ≤ c E2µ−1−1(g) p,ω, and similarly 2µ−1 X ν=2µ−1 Aν(g, ·) p,ω ≤ c E2µ−1−1(g) p,ω. Hence we have k∆µ(g, ·; 1, h)kp,ω ≤ c E2µ−1−1(g)p,ω 2µ−2 X ν=2µ−1 sin νh νh − sin (ν + 1) h (ν + 1) h ! +c E2µ−1−1(g) p,ω 1 − sin (2 µ− 1) h (2µ− 1) h ≤ c E2µ−1−1(g) p,ω2 2µ h2. By the same way, for k > 1 we can obtain
k∆µ(g, ·; k, h)kp,ω ≤ c E2µ−1−1(g) p,ω2 2kµh2k. Thus we have T k h (S2m−1(g, ·)) p,ω ≤ c m X µ=1 k∆µ(g, ·; k, h)k β p,ω 1/β ≤ c m X µ=1 E2βµ−1−1(g)p,ω22kµβh2βk 1/β , and hence T k h (g) p,ω ≤ c E2m−1(g)p,ω+ h2k m X µ=1 22kµβE2βµ−1−1(g)p,ω 1/β .
Choosing h = 1/n for a given n, by the definition of the modulus of smoothness we have Ωk g, 1 n p,ω ≤ c E2m−1(g)p,ω+ 1 n2k m X µ=1 22kµβE2βµ−1−1(g)p,ω 1/β . If we use the inequality
E2m−1(g)p,ω ≤ 24βk 22mβk 2m−1 X ν=2m−2+1 ν2βk−1Eνβ(g)p,ω and select m such that 2m ≤ n < 2m+1, we obtain
Ωk g, 1 n p,ω ≤ c 26k n2k 2m−1 X ν=2m−2+1 ν2βk−1Eνβ(g)p,ω 1/β + 2 6k n2k 2m−2 X ν=1 ν2βk−1Eνβ(g)p,ω 1/β ≤ c n2k " n X ν=1 ν2βk−1Eνβ(g)p,ω #1/β , and the theorem is proved.
Proof of Theorem 2. Let f ∈ Ep(G, ω) . Then by Theorem 5 we have Tp f0+= f , where f0(t) = f (ψ (t)) ψ0(t)1/p, t ∈ T.
Since Tp : Ep(D, ω0) → Ep(G, ω) is bounded, one to one and onto, the linear
operator Tp−1 : Ep(G, ω) → Ep(D, ω0) is also bounded.
Let Pn∗ ∈ Pn (n = 0, 1, 2, ...) be the polynomials of best approximation to f in
Ep(G, ω) , that is,
En(f )Γ,p,ω = kf − P
∗
nkLp(Γ,ω).
The existence of such polynomials follows, for example, from Theorem 1.1 in [7, p. 59]. Since Tp−1(Pn∗) is a polynomial of degree n, by the boundedness of Tp−1 we get
En f0+ p,ω0 ≤ f + 0 − T −1 p (P ∗ n) L p(T,w0) = T −1 p (f ) − T −1 p (P ∗ n) L p(T,w0) ≤ T −1 p kf − P ∗ nkLp(Γ,ω), and hence En f0+ p,ω0 ≤ T −1 p En(f )Γ,p,ω. (12)
By (6), Theorem 1 and (12) we obtain Ωk f, 1 n Γ,p,ω = Ωk f0+, 1 n p,ω0 ≤ c n2k ( n X ν=1 ν2βk−1Eνβf0+ p,ω0 )1/β ≤ c T −1 p n2k ( n X ν=1 ν2βk−1Eνβ(f )Γ,p,ω )1/β , which prove the theorem.
Acknowledgement. We would like to thank Professor V. Kokilashvili for useful discussions.
References
[1] S. Ja. Al’per, Approximation in the Mean of Analytic Functions of Class Ep
(Russian), Moscow: Gos. Izdat. Fiz.-Mat. Lit. (1960), 273-286.
[2] J. E. Andersson, On the Degree of Polynomial Approximation in Ep(D), J. Approx. Theory 19 (1977), 61-68.
[3] A. B¨ottcher, Yu I. Karlovich, Carleson Curves, Muckenhoupt Weights and Toeplitz Operators, Birkhauser-Verlag (1997).
[4] A. C¸ avu¸s, D.M. Israfilov, Approximation by Faber-Laurent Rational Functions in the Mean of Functions of Class Lp(Γ) with 1<p<∞, Approx. Th. Appl. 11
(1995), 105-118.
[5] G. David, Op´erateurs Int´egraux Singuliers sur Certaines Courbes du Plan Com-plexe, Ann. Sci. Ecole. Norm. Sup. 17 (1984), 157-189.
[6] M. C. De Bonis, G. Mastroianni, M. G. Russo, Polynomial approximation with special doubling weights, Acta Sci. Math. (Szeged) 69 (2003), 1-2, 159-184. [7] R. A. Devore, G. G. Lorentz, Constructive Approximation, Springer-Verlag
(1993).
[8] P. L. Duren, Theory of Hp Spaces, Academic Press (1970).
[9] E. M. Dynkin, B.P. Osilenker, Weighted Norm Estimates for Singular Integrals and Their Applications, J. Sov. Math. 30 (1985), 2094-2154.
[10] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Trans-lation of Mathematical Monographs, Vol.26, Providence, RI: AMS (1968). [11] E. A. Haciyeva, Investigation the Properties of Functions with Quasimonotone
Fourier Coefficients in Generalized Nikolskii-Besov Spaces (Russian), Authors Summary of Candidates Dissertation, Tbilisi (1986).
[12] R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform, Trans. Amer. Math. Soc. 176 (1973), 227-251.
[13] D. M. Israfilov, Approximate Properties of Generalized Faber Series in an In-tegral Metric (Russian), Izv. Akad. Nauk. Az. SSR, Ser. Fiz.-Math. Nauk. 2 (1987),10-14.
[14] D. M. Israfilov, Approximation by p-Faber Polynomials in the Weighted Smirnov Class Ep(G, w) and the Bieberbach Polynomials, Constr. Approx. 17 (2001),
[15] D. M. Israfilov, Approximation by p−Faber −Laurent Rational Functions in the Weighted Lebesgue Spaces, Czechoslovak Mathematical Journal 54 (2004), 751-765.
[16] D. M. Israfilov, A. Guven, Approximation in Weighted Smirnov Classes, East J. Approx. 11 (2005), 91-102.
[17] V. Kokilashvili, A Direct Theorem on Mean Approximation of Analytic Func-tions by Polynomials, Soviet Math. Dokl. 10 (1969), 411-414.
[18] D. S. Kurtz, Littlewood-Paley and Multiplier Theorems on Weighted Lp Spaces, Trans. Amer. Math. Soc. 259 (1980), 235-254.
[19] N. X. Ky, On approximation by trigonometric polynomials in Lp
u−spaces, Studia
Sci. Math. Hungar., 28 (1993), 183-188.
[20] N. X. Ky, Moduli of Mean Smoothness and Approximation with Ap−weights,
Annales Univ. Sci. Budapest 40 (1997),37-48.
[21] G. Mastroianni, V. Totik, Jackson type inequalities for doubling and Ap weights,
Rend. Circ. Mat. Palermo Serie II Suppl. 52 (1998), 83-99.
[22] B. Muckenhoupt, Weighted Norm Inequalities for the Hardy Maximal Function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
[23] B. Muckenhoupt, Two weight Function norm Inequalities for the Poisson Inte-gral, Trans. Amer. Math. Soc. 210 (1975), 225-231.
[24] P. K. Suetin, Series of Faber Polynomials, Gordon and Breach Science Publish-ers (1998).
[25] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Perg-amon Press (1963).
[26] M. F. Timan, Converse Theorems of the Constructive Theory of Functions, Mat. Sborn. 46 (1958), 125-132.
[27] J. L. Walsh, H. G. Russel, Integrated continuity conditions and degree of approx-imation by polynomials or by bounded analytic functions, Trans. Amer. Math. Soc. 92 (1959), 355-370.
Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145, Balikesir, Turkey
E-mail :ag guven@yahoo.com, mdaniyal@balikesir.edu.tr
View publication stats View publication stats
