Simultaneous and converse approximation theorems in weighted orlicz spaces

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Theorems in Weighted Orlicz Spaces

Ramazan Akg¨

un

Daniyal M. Israfilov

∗

Abstract

In the present work, we investigate the simultaneous and converse ap-proximation by trigonometric polynomials of the functions in the Orlicz spaces with weights satisfying so called Muckenhoupt’s Apcondition.

1 Introduction

A function Φ is called Young function if Φ is even, continuous, nonnegative in R, increasing on(0, ∞)such that

Φ₍₀_{) =}_{0, lim}

x→∞Φ(x) = ∞.

A nonnegative function M : [0, ∞)→ [0, ∞)is said to be quasiconvex if there exist a convex Young function Φ and a constant c₁≥1 such that

Φ₍_x_{) ≤} _M₍_x_{) ≤}Φ₍_c₁_x₎_, _∀_x _≥_0.

A Young function Φ is said to be satisfy ∆₂condition (Φ ∈ ∆₂) if there is a constant

c2 >0 such that

Φ₍_2x_{) ≤} _c₂Φ₍_x₎ for all x∈ R_.

∗_{Author for correspondence}

Received by the editors March 2008. Communicated by F. Brackx.

2000 Mathematics Subject Classification : Primary 46E30; Secondary 41A10, 41A25, 41A27, 42A10.

Key words and phrases : weighted Orlicz space, inverse theorems, weighted fractional

modu-lus of smoothness.

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Two Young functions Φ and Φ1 are said to be equivalent (we shall write

Φ _∼Φ₁_{) if there are c}₃_{, c}₄>0 such that

Φ₁₍c₃x) ≤ Φ₍x) ≤Φ₁₍c₄x), ∀x>0.

Let T := [−π, π]. A function ω : T → [0, ∞] will be called weight if ω is measurable and almost everywhere (a.e.) positive.

A 2π-periodic weight function ω belongs to the Muckenhoupt class Ap, p >1,

if sup J   1 |J| Z J ω(x)dx     1 |J| Z J ω−1/(p−1)(x)dx   p−1 ≤c5

with a finite constant c5independent of J, where J is any subinterval of T.

Let M be a quasiconvex Young function. We denote by ˜LM,ω(T)the class of

Lebesgue measurable functions f : T→C_{satisfying the condition} Z

T

M(|f(x)|)ω(x)dx <∞.

The linear span of the weighted Orlicz class ˜L_M,ω(T₎_{, denoted by L}

M,ω(T),

be-comes a normed space with the Orlicz norm

kfk_M,ω :=sup    Z T |f (x)g(x)|ω(x)dx : Z T ˜ M(|g|)ω(x)dx≤1    ,

where ˜M(y) :=sup_x_≥₀(xy−M(x)), y ≥0, is the complementary function of M. 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}_}_.

If ω ∈ A_p₍_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 L_M,ω(T₎ _is called weighted Orlicz space.

Detailed information about the classical Orlicz spaces, defined with respect to the convex Young function M, can be found in [24]. Since every convex function is quasiconvex, Orlicz spaces, considered in this work, are more general than the classical one and are investigated in the books [13] and [22].

For formulation of the new results we will begin with some required informa-tions. Let f(x) ∽ ∞

∑

k=−∞ c_keikx = a0 2 + ∞

∑

k=1 (a_kcos kx+b_ksin kx) (1.1) and ˜f(x) ∽ ∞

∑

k=1 (a_ksin kx−b_kcos kx)

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be the Fourier and the conjugate Fourier series of f ∈ L1(T₎_{, respectively. In} addi-tion, we put Sn(x, f) := n

∑

k=−n c_keikx = a0 2 + n

∑

k=1 (a_kcos kx+b_ksin kx), n =1, 2, . . . . By L1₀(T₎ _{we denote the class of L}1₍T₎ _{functions f for which the constant} term c0in (1.1) equals zero. If α >0, then α-th integral of f ∈ L1₀(T)is defined as

Iα(x, f) :=

_∑

k∈Z∗

c_k(ik)−αeikx,

where

(ik)−α := |k|−αe(−1/2)πiαsign k and Z∗ := {±1,±2,±3, . . .}. For α∈ (0, 1)let f(α)(x):= d dxI1−α(x, f), f(α+r)(x) :=f(α)(x)(r) = d r+1 dxr+1I1−α(x, f)

if the right hand sides exist, where r∈ Z+ _:_{= {}_{1, 2, 3, . . .}_}_.

Throughout this work by C(r), c, c₁, c2, . . ., ci(α, . . .), cj(β, . . .), . . . we

de-note the constants, which can be different in different places, such that they are absolute or depend only on the parameters given in their brackets.

Let x, t ∈R_{, r} _∈R+ _:_{= (}_{0, ∞}₎_{and let} ∆r_t_f ₍_x₎ _:₌ ∞

∑

k=0 (−1)k[Cr_k] f (x+ (r−k)t), f ∈ L1(T₎_, _(1.2) where Cr_k := r(r−1)..._k!(r−k+1) for k > 1, Cr k

:= r for k = 1 and C_kr := 1 for

k=0. Since [34, p. 14] |[Cr_k]| = r(r−1). . .(r−k+1) k! ≤ c6(r) kr+1 , k ∈ Z + we have that C(r) := ∞

∑

k=0 |[C_kr]| < ∞,

and therefore ∆r_tf (x) is defined a.e. on R. Furthermore, the series in (1.2) con-verges absolutely a.e. and ∆r_tf (x)is measurable [37].

If r ∈ Z+_{, then the fractional difference ∆}r

tf(x)coincides with usual forward

difference. Now we define

σr_δf(x) := 1 δ δ Z 0 |∆r_t_f₍_x_)|_{dt, f} _∈ _L_M,ω₍T₎_{, ω} _∈ _A p(M).

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Let M ∈ △2, Mθ is quasiconvex for some θ ∈ (0, 1) and ω ∈ Ap(M). Since the

series in (1.2) converges absolutely a.e., we have σr_δf (x) < ∞ a.e. and using the boundedness of the Hardy-Littlewood Maximal function [13, Th. 6.4.4, p.250] in

LM,ω(T), ω ∈ Ap(M), we get

kσr_δf (x)k_M,ω ≤c₇(M, r) kfk_M,ω <∞. (1.3)

Hence, if r ∈ R+ _{and ω} _∈ _A

p(M) we can define the r-th mean modulus of

smoothness of a function f ∈ LM,ω(T)as Ω_r₍_{f , h}₎ M,ω := sup |δ|≤h kσr_δf (x)k_M,ω. (1.4) If r ∈ Z+_{, M}₍_x₎ _:₌ _xp_{/p, 1} < p < ∞ and ω ∈ A_p then Ω_r(f , h) M,ω coincides

with Ky’s mean modulus of smoothness, defined in [26].

Remark 1. Let L_M,ω(T₎_{be a weighted Orlicz space with M} _{∈ △}₂ _{and ω} _∈ _A

p(M). If

Mθ is quasiconvex for some θ ∈ (0, 1), then r-th mean modulus of smoothness

Ω_r₍_{f , h}₎_M,ω_{, r}_∈ R+_{, has the following properties:}

(i) Ω_r₍_{f , h}₎_M,ω_{is non-negative and non-decreasing function of h}_≥_0.

(ii)Ω_r₍_f₁₊_f₂_,_·) M,ω ≤Ωr(f1,·)M,ω+Ωr(f2,·)M,ω. (iii)lim h→0 Ω_r₍_{f , h}₎ M,ω =0. Let En(f)_M,ω := inf T∈Tn kf −Tk_M,ω, f ∈ L_M,ω(T₎_{, n} ₌_{0, 1, 2, . . . ,}

whereTn is the class of trigonometric polynomials of degree not greater than n.

A polynomial Tn(x, f) := Tn(x) of degree n is said to be a near best approxi-mant of f if

kf −TnkM,ω ≤c8(M)En(f)M,ω, n=0, 1, 2, . . . .

Let Wα

M,ω(T), α > 0, be the class of functions f ∈ LM,ω(T) such that f(α) ∈ LM,ω(T). W_M,ωα (T), α>0, becomes a Banach space with the norm

kfk_Wα M,ω(T) := kfkM,ω+ f (α) M,ω.

In this work we investigate the simultaneous and inverse theorems of approx-imation theory in the weighted Orlicz spaces L_M,ω(T₎_.

Simultaneous approximation problems in nonweighted Orlicz spaces, defined with respect to the convex Young function M, was studied in [12]. In the weighted case, where the weighted Orlicz spaces are defined as the subclass of the measur-able functions on T satisfying the condition

Z

T

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some direct and inverse theorems of approximation theory were obtained in [17]. Some generalizations of these results to the weighted Lebesgue and Orlicz spaces defined on the curves of complex plane, were proved in [19], [21], [14], [15], [18], [16], [2] and [1].

Since Orlicz spaces considered by us in this work are more general than the Orlicz space studied in the above mentioned works, the results obtained in this paper are new also in the nonweighted cases.

The similar problems in the weighted Lebesgue spaces Lp(T, ω), under

dif-ferent conditions on the weight function ω, were investigated in the works [11], [25], [6], [30], [29], [31], [8], [10] and also in the books [39], [7], [9], [32].

Our new results are the following.

Theorem 1. Let Mθ be quasiconvex for some θ ∈ (0, 1), M ∈ ∆₂_{, ω} _∈ _A_p₍_M₎ _and

f ∈W_M,ωα (T₎_{, α} _∈ R+

0 := [0, ∞). If Tn ∈ Tn is a near best approximant of f , then

f (α)₋_T(α) n M,ω ≤cEn f(α) M,ω, n=0, 1, 2, . . . (1.5) with a constant c=c(M, α) >0.

This simultaneous approximation theorem in case of α ∈ Z+ _{for Lebesgue} spaces Lp(T₎_{, 1} _≤ _p_≤_{∞, was proved in [5]. In the classical Orlicz spaces L}_M₍T₎ some results about simultaneous trigonometric and algebraic approximation of type (1.5), where En

f(α)

M,1is replaced by the modulus of smoothness of f

(α)_,

α ∈Z+_{, were obtained in [33] and [12].}

Theorem 2. If Mθ is quasiconvex for some θ ∈ (0, 1), M ∈ ∆₂_{, ω} _∈ _A_p₍_M₎ _and

f ∈W_M,ωr (T₎_{, r} _∈ R+_{, then} Ω_r₍_{f , h}₎ M,ω ≤chr f (r) M,ω, 0 <h≤π with a constant c=c(M, r) > 0.

In the case of r ∈ Z+_{, for the usual non weighted modulus of smoothness} defined in the Lebesgue spaces Lp(T₎_{, 1}_≤ _p _≤_{∞, this inequality was proved in} [28] and for the general case r ∈ R+ _{was obtained by Butzer, Dyckhoff, G ¨orlich} and Stens in [4] (See also Taberski [37]). In case of r ∈ Z+_{, ω} _∈ _A_p_{, 1} < p < ∞,

this inequality in the weighted Lebesgue spaces Lp(T_,ω₎_{was proved in [26]. For} the classical Orlicz spaces similar result in nonweighted and weighted cases were obtained in [33] and [17] (see also [3]).

The following converse theorem holds:

Theorem 3. Let LM,ω(T)be a weighted Orlicz space with M ∈ △2and ω ∈ Ap(M).

If Mθis quasiconvex for some θ ∈ (0, 1)and f ∈ LM,ω(T), then for a given r∈ R+

Ω_r₍_{f , π/}₍_n₊₁₎₎ M,ω ≤ c (n+1)r n

∑

ν=0 (ν+1)r−1Eν(f)_M,ω, n=0, 1, 2, . . . with a constant c=c(M, r) > 0.

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In the space Lp(T₎_{, 1} _≤ _p _≤ _{∞, this inequality was proved in [37]. In case of}

r ∈ Z+ _{this theorem in the spaces L}p₍T_,ω₎_{, 1}< p <∞, ω ∈ A_p, was proved by

Ky in [26]. For the positive and even integer r this theorem in the spaces Lp(T_,ω₎_, 1< p<∞, ω ∈ A_p, by using Butzer-Wehrens’s type modulus of smoothness was

obtained in [11]. In case of r ∈Z+ _{for weighted Orlicz spaces L}_M₍T_,ω₎_{, ω} _∈ _Ap_, similar results were obtained in [17] and [3].

Theorem 4. Let Mθ be quasiconvex for some θ ∈ (0, 1), M∈ ∆₂_{and ω} _∈ _A_p₍_M₎_{. If}

∞

∑

ν=1

να−1Eν(f)_M,ω<∞

for some α ∈ (0, ∞), then f ∈W_M,ωα (T₎_and

En f(α) M,ω ≤c ( (n+1)αEn(f)_M,ω+ ∞

∑

ν=n+1 να−1Eν(f)_M,ω ) (1.6) with a constant c=c(M, α) > 0.

In the space Lp(T₎_{, 1} _≤ _p _≤ _{∞, this inequality for α} _∈ Z+ _{was proved in} [35]. When α ∈ R+ _{in the classical Orlicz spaces L}_M₍T₎_{, similar inequality was} proved in [20]. In case of α∈ Z+_{, in L}p₍T_,ω₎_{, 1} < p< ∞, ω ∈ A_p, an inequality

of type (1.6) was recently proved in [23].

Corollary 1. Let Mθ be quasiconvex for some θ ∈ (0, 1), M ∈ ∆₂, ω ∈ A_p₍_M₎ and r>0. If

∞

∑

ν=1

να−1Eν(f)_M,ω<∞

for some α ∈ (0, ∞), then f ∈W_M,ωα (T₎_{and for n} ₌_{0, 1, 2, . . .}

Ω_r f(α), π n+1 M,ω ≤c ( 1 (n+1)r n

∑

ν=0 (ν+1)α+r−1Eν(f)_M,ω+ ∞

∑

ν=n+1 να−1Eν(f)_M,ω ) with a constant c=c(M, α, r) >0.

In cases of α, r ∈ Z+ _{and α, r} _∈ R+_{, this corollary in the spaces L}p₍T₎_{, 1} _≤

p ≤ ∞, was proved in [38] (See also [35]) and in [36], respectively. In the case of

α ∈ R+ _{and r} _∈ Z+_{, in the classical Orlicz spaces L}_M₍T₎ _{the similar result was} obtained [20]. For the weighted Lebesgue spaces Lp(T_,ω₎_{, 1} < p < ∞, when ω ∈ Apand α, r ∈ Z+, similar type inequality was obtained in [23].

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2 Auxiliary Facts

We begin with

Lemma 1. Let M ∈ △2, ω ∈ Ap(M) and r ∈ R+. If Mθ is quasiconvex for some

θ ∈ (0, 1)and Tn ∈ Tn, n≥1, then there exists a constant c>0 depends only on r and M such that Ω_r₍_T_n_{, h}₎ M,ω ≤chr T (r) n M,ω, 0 <h≤π/n. Proof. Since ∆r_t_T_n_x₋ r 2t =

∑

ν∈Z∗_n 2i sin t 2ν r cνeiνx, ∆[r] t T (r−[r]) n x− [r] 2 t =

∑

ν∈Z∗_n 2i sin t 2ν [r]

(iν)r−[r]cνeiνx

with Z∗_n := {∓1,∓2, . . . ,∓n},[r] ≡integer part of r, putting

ϕ(z):= 2i sin t 2z [r] (iz)r−[r], g(z):= 2 zsin t 2z r−[r] , −n≤z≤n, g(0):=tr−[r], we get ∆[_tr]T_n(r−[r]) x− [r] 2 t =

_∑

ν∈Z∗_n ϕ(ν)cνeiνx, ∆rtTn x− r 2t =

_∑

ν∈Z∗_n ϕ(ν)g(ν)cνeiνx.

Taking into account the fact that [37]

g(z) = ∞

∑

k=−∞ d_keikπz/n uniformly in [−n, n], with d0 > 0, (−1)k+1dk ≥ 0, d−k = dk (k =1, 2, . . .), we have ∆r_tTn(·) = ∞

∑

k=−∞ d_k∆[_tr]T_n(r−[r]) · +kπ n + r− [r] 2 t . Consequently we get 1 δ δ Z 0 |∆r_t_T_n_(·)|_dt M,ω = 1 δ δ Z 0 ∞

∑

k=−∞ d_k∆[r] t T (r−[r]) n · +kπ n + r− [r] 2 t dt M,ω ≤ ∞

∑

k=−∞ |d_k| 1 δ δ Z 0 ∆[_tr]_T_n(r−[r]) · + kπ n + r− [r] 2 t dt M,ω

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and since [39, p.103] ∆[r]_t _T_n(r−[r])_{(·) =} t Z 0 · · · t Z 0 Tn(r) · +t1+. . .+t[r] dt1. . . dt[r] we find Ω_r₍_T_n_{, h}₎ M,ω ≤ sup |δ|≤h ∞

∑

k=−∞ |dk| 1 δ δ Z 0 ∆[r]_t _T_n(r−[r]) · +kπ n + r− [r] 2 t dt M,ω = sup |δ|≤h ∞

∑

k=−∞ |dk| 1 δ δ Z 0 t Z 0 · · · t Z 0 Tn(r) · +kπ n + r− [r] 2 t+t1+. . .+t[r] dt1. . . dt[r] dt M,ω ≤h[r]sup |δ|≤h ∞

∑

k=−∞ |dk| 1 δ δ Z 0 1 δ[r] δ Z 0 · · · δ Z 0 Tn(r) · +kπ n + r− [r] 2 t+t1+. . .+t[r] × ×dt1. . . dt[r]dt M,ω ≤h[r]sup |δ|≤h ∞

∑

k=−∞ |dk| 1 δ[r] δ Z 0 · · · δ Z 0    1 δ δ Z 0 T (r) n · +kπ n + r− [r] 2 t+t1+. . .+t[r] × ×dt}dt1. . . dt[r] M,ω ≤c9(M, r)h[r]sup |δ|≤h ∞

∑

k=−∞ |dk| 1 δ δ Z 0 T (r) n · +kπ n + r− [r] 2 t dt M,ω ≤c9(M, r)h[r]sup |δ|≤h ∞

∑

k=−∞ |dk| 1 r−[r] 2 δ ·+kπ n+ r−[r] 2 δ Z ·+kπ n T (r) n (u) du M,ω .

On the other hand [37]

∞

∑

k=−∞

|d_k| <2g(0) =2tr−[r], 0<t≤_π/n

and for 0<t<_δ <h≤_π_{/n we have}

∞

∑

k=−∞

|d_k| < 2hr−[r].

Therefore the boundedness of Hardy-Littlewood Maximal function in LM,ω(T)

implies that Ω_r₍_T_n_{, h}₎ M,ω ≤c10(M, r)hr T (r) n M,ω.

Further, by the similar way for 0 < −h ≤ π/n, the same inequality also holds

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Lemma 2. Let M ∈ △2, Mθ is quasiconvex for some θ ∈ (0, 1) and ω ∈ Ap(M). If

Tn ∈ Tn and α >0, then there exists a constant c>0 depending only on α and M such

that T (α) n M,ω ≤cn α_k_T nk_M,ω.

Proof. Since M ∈ △2, Mθ is quasiconvex for some θ ∈ (0, 1)and ω ∈ Ap(M) we

have [3] kSnfk_M,ω ≤c11(M) kfkM,ω, ˜f M,ω ≤c12(M) kfkM,ω.

Following the method given in [27] we obtain the required result.

Definition 1. For f ∈ LM,ω(T), δ > 0 and r =1, 2, 3, . . ., the Peetre K-functional is defined as K δ, f ; LM,ω(T), WM,ωr (T) := inf g∈W_M,ωr (T₎ kf −gk_M,ω+δ g (r) M,ω . (2.1) Lemma 3. Let M ∈ △₂, Mθ is quasiconvex for some θ ∈ (0, 1) and ω ∈ A_p₍_M₎. If

f ∈ LM,ω(T)and r=1, 2, 3, . . ., then

(i) the K-functional (2.1) and the modulus (1.4) are equivalent and (ii) there exists a constant c >0 depending only on r and M such that

En(f)M,ω ≤cΩr f , 1 n M,ω .

Proof. (i) can be proved by the similar way to that of Theorem 1 in [26] and later

(ii) is proved by standard way (see for example, [17]).

3 Proof of the results

Proof of Theorem 1. We set

Wn(f) :=Wn(x, f):= 1 n+1 2n

∑

ν=n Sν(x, f), n=0, 1, 2, . . . . Since Wn(·, f(α)) =Wn(α)(·, f) we have f (α)_{(·) −}_T(α) n (·, f) M,ω ≤ f (α)_{(·) −}_W n(·, f(α)) M,ω+ T (α) n (·, Wn(f)) −Tn(α)(·, f) M,ω+ W (α) n (·, f) −Tn(α)(·, Wn(f)) M,ω =: I1+I2+I3.

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We denote by T_n∗(x, f)the best approximating polynomial of degree at most n to f in LM,ω(T, ω). In this case, from the boundedness of Wn in LM,ω(T, ω)we

have I₁ ≤ f (α)_{(·) −}_T∗ n(·, f(α)) M,ω+ T ∗ n(·, f(α)) −Wn(·, f(α)) M,ω ≤c13(M)En f(α) M,ω+ Wn(·, T ∗ n(f(α)) − f(α)) M,ω ≤c14(M, α)En f(α) M,ω.

From Lemma 2 we get

I2 ≤c15(M, α)nαkTn(·, Wn(f)) −Tn(·, f)k_M,ω and I₃ ≤c₁₆(M, α) (2n)αkWn(·, f) −Tn(·, Wn(f))k_M,ω ≤c17(M, α) (2n)αEn(Wn(f))_M,ω. Now we have kTn(·, Wn(f)) −Tn(·, f)k_M,ω ≤ kTn(·, Wn(f)) −Wn(·, f)k_M,ω + kWn(·, f) − f (·)k_M,ω+ kf (·) −Tn(·, f)k_M,ω ≤c₁₈(M)En(Wn(f))_M,ω+c19(M)En(f)_M,ω+c20(M)En(f)_M,ω. Since En(Wn(f))_M,ω ≤c21(M)En(f)_M,ω, we get f (α)_{(·) −}_T(α) n (·, f) M,ω ≤c14(M, α)En f(α) M,ω+c22(M)n α_E n(Wn(f))_M,ω +c23(M)nαEn(f)M,ω+c17(M, α) (2n)αEn(Wn(f))M,ω ≤c24(M, α)En f(α) M,ω+c25(M)n α_E n(f)_M,ω. Since [3] En(f)_M,ω ≤ c26(M, α) (n+1)α En f(α) M,ω, (3.1) we obtain f (α)_{(·) −}_T(α) n (·, f) M,ω ≤c27(M, α)En f(α) M,ω

and the proof is completed.

Proof of Theorem 2. Let Tn ∈ Tn be the trigonometric polynomial of best

approxi-mation of f in LM,ω(T)metric. By Remark 1(ii), Lemma 1 and (1.3) we get

Ω_r₍_{f , h}₎ M,ω ≤Ωr(Tn, h)M,ω+Ωr(f −Tn, h)M,ω ≤c₁₀(M, r)hr T (r) n M,ω+c7(M, r)En(f)M,ω, 0 <h≤π/n.

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Using (3.1), Lemma 3 (ii) and Ω_l₍f , h)_M,ω ≤chl f (l) M,ω, f ∈W l M,ω(T), l =1, 2, 3, . . . ,

which can be showed using the judgements given in [26, Theorem 1], we have

En(f)_M,ω ≤ c26(M, r) (n+1)r−[r]En f(r−[r]) M,ω ≤ c28(M, r) (n+1)r−[r] Ω_[_r_] f(r−[r]), 2π n+1 M,ω ≤ c29(M, r) (n+1)r−[r] 2π n+1 [r] f (r) M,ω.

On the other hand, by Theorem 1 we find T (r) n M,ω ≤ T (r) n − f(r) M,ω+ f (r) M,ω ≤c27(M, r)En f(r) M,ω+ f (r) M,ω ≤c30(M, r) f (r) M,ω.

Then choosing h with π/(n+1) <h ≤π/n, (n=1, 2, 3, . . .), we obtain Ω_r₍_{f , h}₎ M,ω ≤c31(M, r)hr f (r) M,ω

and we are done.

Proof of Theorem 3. Let Tn ∈ Tn be the best approximating polynomial of f ∈

LM,ω(T), ω∈ Ap(M) and let m∈ Z+. Then by Remark 1(ii) and (1.3) we have

Ω_r₍f , π/(n+1))_M,ω ≤Ω_r₍f −T₂m, π/(n+1))_M,ω+Ω_r(T₂m, π/(n+1))_M,ω ≤c7(M, r)E2m(f)_M,ω+Ω_r(T₂m, π/(n+1))_M,ω. Since Ω_r₍_T₂m, π/(n+1))_M,ω ≤c₅₄(M, r) π n+1 r T (r) 2m M,ω, n+1≥2 m and T₂(mr)(x) = T (r) 1 (x) + m−1

∑

ν=0 n T(r) 2ν+1(x) −T (r) 2ν (x) o , we have Ω_r₍_T₂m, π/(n+1))_M,ω ≤ c₁₀(M, r) π n+1 r( T (r) 1 M,ω+ m−1

∑

ν=0 T (r) 2ν+1−T (r) 2ν M,ω ) . By Lemma 2 we find T (r) 2ν+1−T (r) 2ν M,ω ≤c32(M, r)2 νr_k_T 2ν+1−T2νk M,ω ≤c32(M, r)2νr+1E2ν(f)M,ω

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and T (r) 1 M,ω = T (r) 1 −T (r) 0 M,ω ≤c33(M, r)E0(f)M,ω. Hence Ω_r₍_T₂m, π/(n+1)) M,ω ≤ c34(M, r) π n+1 r( E0(f)_M,ω+ m−1

∑

ν=0 2(ν+1)rE2ν(f) M,ω ) . It is easily seen that

2(ν+1)rE2ν(f) M,ω ≤c35(r) 2ν

∑

µ=2ν−1₊₁ µr−1Eµ(f)_M,ω, ν=1, 2, 3, . . . . (3.2) Therefore, Ω_r₍_T₂m, π/(n+1)) M,ω ≤ c₃₄(M, r) π n+1 r    E₀(f)_M,ω+2rE₁(f)_M,ω+ c35(r) m

∑

ν=1 2ν

∑

µ=2ν−1₊₁ µr−1Eµ(f)_M,ω    ≤ c36(M, r) π n+1 r( E0(f)_M,ω+ 2m

∑

µ=1 µr−1Eµ(f)_M,ω ) ≤ c36(M, r) π n+1 r 2m−1

∑

ν=0 (ν+1)r−1Eν(f)_M,ω. If we choose 2m ≤n+1≤2m+1, then Ω_r₍_T₂m, π/(n+1))_M,ω ≤ c36(M, r) (n+1)r n

∑

ν=0 (ν+1)r−1Eν(f)_M,ω, E₂m(f)_M,ω ≤E₂m−1(f)_M,ω ≤ c₃₇(M, r) (n+1)r n

∑

ν=0 (ν+1)r−1Eν(f)_M,ω

and Theorem 3 is proved.

Proof of Theorem 4. If Tn is the best approximating trigonometric polynomial of f ,

then by Lemma 2 T (α) 2m+1 −T (α) 2m M,ω ≤c38(M, α)2 (m+1)α_E 2m(f) M,ω

and hence by this inequality, (3.2) and hypothesis of Theorem 4 we have

∞

∑

m=1 kT₂m+1 −T₂mk_Wα M,ω(T) = ∞

∑

m=1 kT₂m+1 −T₂mk_M,ω+ ∞

∑

m=1 T (α) 2m+1 −T (α) 2m M,ω

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=c39(M, α) ∞

∑

m=1 2(m+1)αE2m(f)_M,ω ≤c₄₀(M, α) ∞

∑

m=1 2m

∑

j=2m−1₊₁ jα−1E_j(f)_M,ω ≤c₄₁(M, α) ∞

∑

j=2 jα−1E_j(f)_M,ω <∞. Therefore, ∞

∑

m=1 kT₂m+1−T2mk_Wα M,ω(T) <∞,

which implies that {T₂m} is a Cauchy sequence in W_M,ωα (T). Since T₂m → f in

the Banach space LM,ω(T), we have f ∈W_M,ωα (T).

It is clear that En f(α) M,ω ≤ f (α)₋_S nf(α) M,ω ≤ S2m+2f (α)₋_S nf(α) M,ω+ ∞

∑

k=m+2 S2k+1f (α)₋_S 2kf(α) M,ω. (3.3) By Lemma 2 S2m+2f (α)₋_S nf(α) M,ω ≤c42(M, α)2 (m+2)α_E n(f)_M,ω ≤c43(M, α) (n+1)αEn(f)_M,ω (3.4) for 2m <n<2m+1.

On the other hand, by Lemma 2 and by (3.2)

∞

∑

k=m+2 S2k+1f (α)₋_S 2kf(α) M,ω≤c44(M, α) ∞

∑

k=m+2 2(k+1)αE₂k(f)_M,ω ≤c45(M, α) ∞

∑

k=m+2 2k

∑

µ=2k−1₊₁ µα−1Eµ(f)_M,ω =c46(M, α) ∞

∑

ν=2m+1₊₁ να−1Eν(f)_M,ω ≤c46(M, α) ∞

∑

ν=n+1 να−1Eν(f)_M,ω. (3.5) Now using the relations (3.4) and (3.5) in (3.3) we obtain the required inequality. Acknowledgement. The authors are indebted to the referees for efforts and valuable comments related to this paper.

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Balikesir University, Faculty of Science and Art, Department of Mathematics, 10145, Balikesir, Turkey email:rakgun@balikesir.edu.tr, mdaniyal@balikesir.edu.tr