Simultaneous and Converse Approximation

Theorems in Weighted Orlicz Spaces

Ramazan Akgun 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 spaceswith weights satisfying so called Muckenhoupt’s Ap condition.

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, limx→∞

Φ (x) = ∞.

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

Φ (x) ≤ M (x) ≤ Φ (c1x) , ∀x ≥ 0.

A Young function Φ is said to be satisfy ∆2 condition (Φ ∈ ∆2) if there is a constantc2 > 0 such that

Φ (2x) ≤ c2Φ (x)

for all x ∈ R.

∗Author for correspondenceReceived 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.

Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 1–16

2 R. Akgun – D. M. Israfilov

Two Young functions Φ and Φ1 are said to be equivalent (we shall writeΦ ∼ Φ1) if there are c3, c4 > 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 ismeasurable and almost everywhere (a.e.) positive.

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

supJ

1

|J|

∫

J

ω (x) dx

1

|J|

∫

J

ω−1/(p−1) (x) dx

p−1

≤ c5

with a finite constant c5 independent 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

∫

T

M (| f (x)|) ω (x) dx < ∞.

The linear span of the weighted Orlicz class LM,ω (T), denoted by LM,ω (T), be-comes a normed space with the Orlicz norm

‖ f‖M,ω := sup

∫

T

| f (x) g (x)|ω (x) dx :∫

T

M (|g|) ω (x) dx ≤ 1

,

where M (y) := supx≥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, Mp is quasiconvex .

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) iscalled weighted Orlicz space.

Detailed information about the classical Orlicz spaces, defined with respect tothe convex Young function M, can be found in [24]. Since every convex functionis quasiconvex, Orlicz spaces, considered in this work, are more general than theclassical 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=−∞

ckeikx =a0

2+

∞

∑k=1

(ak cos kx + bk sin kx) (1.1)

and

f (x) ∽

∞

∑k=1

(ak sin kx − bk cos kx)

Approximation Theorems in Weighted Orlicz Spaces 3

be the Fourier and the conjugate Fourier series of f ∈ L1 (T), respectively. In addi-tion, we put

Sn (x, f ) :=n

∑k=−n

ckeikx =a0

2+

n

∑k=1

(ak cos kx + bk sin kx) , n = 1, 2, . . . .

By L10 (T) we denote the class of L1 (T) functions f for which the constant

term c0 in (1.1) equals zero. If α > 0, then α-th integral of f ∈ L10 (T) is defined as

Iα (x, f ) := ∑k∈Z∗

ck (ik)−α eikx,

where

(ik)−α := |k|−α e(−1/2)πiα sign 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)

=dr+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 areabsolute or depend only on the parameters given in their brackets.

Let x, t ∈ R, r ∈ R+ := (0, ∞) and let

∆rt f (x) :=

∞

∑k=0

(−1)k [Crk] f (x + (r − k) t) , f ∈ L1 (T) , (1.2)

where[

Crk

]

:=r(r−1)...(r−k+1)

k! for k > 1,[

Crk

]

:= r for k = 1 and[

Crk

]

:= 1 fork = 0.

Since [34, p. 14]

|[Crk]| =

∣

∣

∣

∣

r (r − 1) . . . (r − k + 1)

k!

∣

∣

∣

∣

≤c6 (r)

kr+1, k ∈ Z

+

we have that

C (r) :=∞

∑k=0

|[Crk]| < ∞,

and therefore ∆rt f (x) is defined a.e. on R. Furthermore, the series in (1.2) con-

verges absolutely a.e. and ∆rt f (x) is measurable [37].

If r ∈ Z+, then the fractional difference ∆rt f (x) coincides with usual forward

difference. Now we define

σrδ f (x) :=

1

δ

δ∫

0

|∆rt f (x)| dt, f ∈ LM,ω (T) , ω ∈ Ap(M).

4 R. Akgun – D. M. Israfilov

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 σrδ f (x) < ∞ a.e. and using the

boundedness of the Hardy-Littlewood Maximal function [13, Th. 6.4.4, p.250] inLM,ω (T), ω ∈ Ap(M), we get

‖σrδ f (x)‖M,ω ≤ c7 (M, r) ‖ f‖M,ω < ∞. (1.3)

Hence, if r ∈ R+ and ω ∈ Ap(M) we can define the r-th mean modulus of

smoothness of a function f ∈ LM,ω (T) as

Ωr ( f , h)M,ω := sup|δ|≤h

‖σrδ f (x)‖M,ω . (1.4)

If r ∈ Z+, M(x) := xp/p, 1 < p < ∞ and ω ∈ Ap then Ωr ( f , h)M,ω coincideswith Ky’s mean modulus of smoothness, defined in [26].

Remark 1. Let LM,ω (T) be a weighted Orlicz space with M ∈ 2 and ω ∈ Ap(M). If

Mθ is quasiconvex for some θ ∈ (0, 1), then r-th mean modulus of smoothnessΩr ( f , h)M,ω, r ∈ R+, has the following properties:

(i) Ωr ( f , h)M,ω is non-negative and non-decreasing function of h ≥ 0.(ii) Ωr ( f1 + f2, ·)M,ω ≤ Ωr ( f1, ·)M,ω + Ωr ( f2, ·)M,ω.(iii) lim

h→0Ωr ( f , h)M,ω = 0.

Let

En ( f )M,ω := infT∈Tn

‖ f − T‖M,ω , f ∈ LM,ω (T) , n = 0, 1, 2, . . . ,

where Tn 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

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

Let WαM,ω (T), α > 0, be the class of functions f ∈ LM,ω (T) such that f (α) ∈

LM,ω (T). WαM,ω (T), α > 0, becomes a Banach space with the norm

‖ f‖WαM,ω(T) := ‖ f‖M,ω +

∥

∥

∥f (α)

∥

∥

∥

M,ω.

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

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

∫

T

M (| f (x)|ω (x)) dx < ∞,

Approximation Theorems in Weighted Orlicz Spaces 5

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 theOrlicz space studied in the above mentioned works, the results obtained in thispaper 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 ∈ Ap(M) and

f ∈ WαM,ω (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 Lebesguespaces Lp (T), 1 ≤ p ≤ ∞, was proved in [5]. In the classical Orlicz spaces LM (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 ∈ Ap(M) and

f ∈ WrM,ω (T), r ∈ R+, then

Ωr ( 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, Gorlichand Stens in [4] (See also Taberski [37]). In case of r ∈ Z+, ω ∈ Ap, 1 < p < ∞,this inequality in the weighted Lebesgue spaces Lp (T,ω) was proved in [26]. Forthe classical Orlicz spaces similar result in nonweighted and weighted cases wereobtained 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 ∈ 2 and ω ∈ Ap(M).

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

Ωr ( f , π/ (n + 1))M,ω ≤c

(n + 1)r

n

∑ν=0

(ν + 1)r−1 Eν ( f )M,ω , n = 0, 1, 2, . . .

with a constant c = c(M, r) > 0.

6 R. Akgun – D. M. Israfilov

In the space Lp (T), 1 ≤ p ≤ ∞, this inequality was proved in [37]. In case ofr ∈ Z+ this theorem in the spaces Lp (T,ω), 1 < p < ∞, ω ∈ Ap, was proved by

Ky in [26]. For the positive and even integer r this theorem in the spaces Lp (T,ω),1 < p < ∞, ω ∈ Ap, by using Butzer-Wehrens’s type modulus of smoothness wasobtained in [11]. In case of r ∈ Z+ for weighted Orlicz spaces LM (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 ∈ Ap(M). If

∞

∑ν=1

να−1Eν ( f )M,ω < ∞

for some α ∈ (0, ∞), then f ∈ WαM,ω (T) and

En

(

f (α))

M,ω≤ c

(n + 1)α En ( 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 LM (T), similar inequality wasproved in [20]. In case of α ∈ Z+, in Lp (T,ω), 1 < p < ∞, ω ∈ Ap, an inequalityof type (1.6) was recently proved in [23].

Corollary 1. Let Mθ be quasiconvex for some θ ∈ (0, 1), M ∈ ∆2, ω ∈ Ap(M) andr > 0. If

∞

∑ν=1

να−1Eν ( f )M,ω < ∞

for some α ∈ (0, ∞), then f ∈ WαM,ω (T) and for n = 0, 1, 2, . . .

Ωr

(

f (α),π

n + 1

)

M,ω

≤ c

1

(n + 1)r

n

∑ν=0

(ν + 1)α+r−1 Eν ( 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 Lp (T), 1 ≤p ≤ ∞, was proved in [38] (See also [35]) and in [36], respectively. In the case ofα ∈ R+ and r ∈ Z+, in the classical Orlicz spaces LM (T) the similar result wasobtained [20]. For the weighted Lebesgue spaces Lp (T,ω), 1 < p < ∞, whenω ∈ Ap and α, r ∈ Z+, similar type inequality was obtained in [23].

Approximation Theorems in Weighted Orlicz Spaces 7

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 andM such that

Ωr (Tn, h)M,ω ≤ chr∥

∥

∥T

(r)n

∥

∥

∥

M,ω, 0 < h ≤ π/n.

Proof. Since

∆rt Tn

(

x −r

2t)

= ∑ν∈Z∗

n

(

2i sint

2ν

)r

cνeiνx,

∆[r]t T

(r−[r])n

(

x −[r]

2t

)

= ∑ν∈Z∗

n

(

2i sint

2ν

)[r]

(iν)r−[r] cνeiνx

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

ϕ (z) :=

(

2i sint

2z

)[r]

(iz)r−[r] , g (z) :=

(

2

zsin

t

2z

)r−[r]

, − n ≤ z ≤ n,

g (0) := tr−[r],

we get

∆[r]t T

(r−[r])n

(

x −[r]

2t

)

= ∑ν∈Z∗

n

ϕ (ν) cνeiνx, ∆rt Tn

(

x −r

2t)

= ∑ν∈Z∗

n

ϕ (ν) g (ν) cνeiνx.

Taking into account the fact that [37]

g (z) =∞

∑k=−∞

dkeikπz/n

uniformly in [−n, n], with d0 > 0, (−1)k+1 dk ≥ 0, d−k = dk (k = 1, 2, . . .), wehave

∆rt Tn (·) =

∞

∑k=−∞

dk∆[r]t T

(r−[r])n

(

· +kπ

n+

r − [r]

2t

)

.

Consequently we get

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

|∆rt Tn (·)| dt

∥

∥

∥

∥

∥

∥

M,ω

=

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

∣

∣

∣

∣

∣

∞

∑k=−∞

dk∆[r]t T

(r−[r])n

(

· +kπ

n+

r − [r]

2t

)

∣

∣

∣

∣

∣

dt

∥

∥

∥

∥

∥

∥

M,ω

≤∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

∣

∣

∣

∣

∆[r]t T

(r−[r])n

(

·+kπ

n+

r − [r]

2t

)∣

∣

∣

∣

dt

∥

∥

∥

∥

∥

∥

M,ω

8 R. Akgun – D. M. Israfilov

and since [39, p.103]

∆[r]t T

(r−[r])n (·) =

t∫

0

· · ·

t∫

0

T(r)n

(

·+ t1 + . . . + t[r]

)

dt1 . . . dt[r]

we find

Ωr (Tn, h)M,ω ≤ sup|δ|≤h

∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

∣

∣

∣

∣

∆[r]t T

(r−[r])n

(

·+kπ

n+

r − [r]

2t

)∣

∣

∣

∣

dt

∥

∥

∥

∥

∥

∥

M,ω

= sup|δ|≤h

∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

∣

∣

∣

∣

∣

∣

t∫

0

· · ·

t∫

0

T(r)n

(

·+kπ

n+

r − [r]

2t + t1 + . . . + t[r]

)

dt1 . . . dt[r]

∣

∣

∣

∣

∣

∣

dt

∥

∥

∥

∥

∥

∥

M,ω

≤ h[r] sup|δ|≤h

∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

1

δ[r]

δ∫

0

· · ·

δ∫

0

∣

∣

∣

∣

T(r)n

(

·+kπ

n+

r − [r]

2t + t1 + . . . + t[r]

)∣

∣

∣

∣

×

×dt1 . . . dt[r]dt∥

∥

∥

M,ω

≤ h[r] sup|δ|≤h

∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

1

δ[r]

δ∫

0

· · ·

δ∫

0

1

δ

δ∫

0

∣

∣

∣

∣

T(r)n

(

·+kπ

n+

r − [r]

2t + t1 + . . . + t[r]

)∣

∣

∣

∣

×

×dt dt1 . . . dt[r]

∥

∥

∥

M,ω

≤ c9 (M, r) h[r] sup|δ|≤h

∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

1

δ

δ∫

0

∣

∣

∣

∣

T(r)n

(

·+kπ

n+

r − [r]

2t

)∣

∣

∣

∣

dt

∥

∥

∥

∥

∥

∥

M,ω

≤ c9 (M, r) h[r] sup|δ|≤h

∞

∑k=−∞

|dk|

∥

∥

∥

∥

∥

∥

∥

∥

1r−[r]

2 δ

·+ kπn +

r−[r]2 δ

∫

·+ kπn

∣

∣

∣T(r)n (u)

∣

∣

∣ du

∥

∥

∥

∥

∥

∥

∥

∥

M,ω

.

On the other hand [37]

∞

∑k=−∞

|dk| < 2g (0) = 2tr−[r], 0 < t ≤ π/n

and for 0 < t < δ < h ≤ π/n we have

∞

∑k=−∞

|dk| < 2hr−[r].

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

Ωr (Tn, 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 holdsand the proof of Lemma 1 is completed.

Approximation Theorems in Weighted Orlicz Spaces 9

Lemma 2. Let M ∈ 2, Mθ is quasiconvex for some θ ∈ (0, 1) and ω ∈ Ap(M). IfTn ∈ Tn and α > 0, then there exists a constant c > 0 depending only on α and M suchthat

∥

∥

∥T

(α)n

∥

∥

∥

M,ω≤ cnα ‖Tn‖M,ω .

Proof. Since M ∈ 2, Mθ is quasiconvex for some θ ∈ (0, 1) and ω ∈ Ap(M) wehave [3]

‖Sn f‖M,ω ≤ c11 (M) ‖ f‖M,ω ,∥

∥ f∥

∥

M,ω≤ c12 (M) ‖ f‖M,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 isdefined as

K(

δ, f ; LM,ω (T) , WrM,ω (T)

)

:= infg∈Wr

M,ω(T)

‖ f − g‖M,ω + δ∥

∥

∥g(r)

∥

∥

∥

M,ω

. (2.1)

Lemma 3. Let M ∈ 2, Mθ is quasiconvex for some θ ∈ (0, 1) and ω ∈ Ap(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, . . . .

SinceWn(·, f (α)) = W

(α)n (·, f )

we have

∥

∥

∥f (α)(·) − T

(α)n (·, f )

∥

∥

∥

M,ω≤

∥

∥

∥f (α)(·) − Wn(·, f (α))

∥

∥

∥

M,ω+

∥

∥

∥T(α)n (·, Wn( f )) − T

(α)n (·, f )

∥

∥

∥

M,ω+

∥

∥

∥W

(α)n (·, f ) − T

(α)n (·, Wn( f ))

∥

∥

∥

M,ω=: I1 + I2 + I3.

10 R. Akgun – D. M. Israfilov

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) wehave

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) nα ‖Tn(·, Wn( f )) − Tn(·, f )‖M,ω

andI3 ≤ c16 (M, α) (2n)α ‖Wn(·, f ) − Tn(·, Wn( f ))‖M,ω

≤ c17 (M, α) (2n)α En (Wn( f ))M,ω .

Now we have

‖Tn(·, Wn( f )) − Tn(·, f )‖M,ω ≤ ‖Tn(·, Wn( f )) − Wn(·, f )‖M,ω

+ ‖Wn(·, f ) − f (·)‖M,ω + ‖ f (·) − Tn(·, f )‖M,ω

≤ c18 (M) En (Wn( f ))M,ω + c19 (M) En ( f )M,ω + c20 (M) En ( f )M,ω .

SinceEn (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) nαEn (Wn( f ))M,ω

+c23 (M) nαEn ( f )M,ω + c17 (M, α) (2n)α En (Wn( f ))M,ω

≤ c24 (M, α) En

(

f (α))

M,ω+ c25 (M) nαEn ( 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

Ωr ( 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.

Approximation Theorems in Weighted Orlicz Spaces 11

Using (3.1), Lemma 3 (ii) and

Ωl ( 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

Ωr ( 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

Ωr ( f , π/ (n + 1))M,ω ≤ Ωr ( f − T2m , π/ (n + 1))M,ω + Ωr (T2m , π/ (n + 1))M,ω

≤ c7 (M, r) E2m ( f )M,ω + Ωr (T2m , π/ (n + 1))M,ω .

Since

Ωr (T2m , π/ (n + 1))M,ω ≤ c54 (M, r)

(

π

n + 1

)r ∥

∥

∥T

(r)2m

∥

∥

∥

M,ω, n + 1 ≥ 2m

and

T(r)2m (x) = T

(r)1 (x) +

m−1

∑ν=0

T(r)2ν+1 (x)− T

(r)2ν (x)

,

we have

Ωr (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) 2νr ‖T2ν+1 − T2ν‖M,ω ≤ c32 (M, r) 2νr+1E2ν ( f )M,ω

12 R. Akgun – D. M. Israfilov

and∥

∥

∥T

(r)1

∥

∥

∥

M,ω=

∥

∥

∥T

(r)1 − T

(r)0

∥

∥

∥

M,ω≤ c33 (M, r) E0 ( f )M,ω .

Hence

Ωr (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,

Ωr (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−1 Eν ( f )M,ω .

If we choose 2m ≤ n + 1 ≤ 2m+1, then

Ωr (T2m , π/ (n + 1))M,ω ≤c36 (M, r)

(n + 1)r

n

∑ν=0

(ν + 1)r−1 Eν ( f )M,ω ,

E2m ( f )M,ω ≤ E2m−1 ( f )M,ω ≤c37 (M, r)

(n + 1)r

n

∑ν=0

(ν + 1)r−1 Eν ( 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)αE2m ( f )M,ω

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

∞

∑m=1

‖T2m+1 − T2m‖WαM,ω(T) =

∞

∑m=1

‖T2m+1 − T2m‖M,ω +∞

∑m=1

∥

∥

∥T

(α)

2m+1 − T(α)2m

∥

∥

∥

M,ω

Approximation Theorems in Weighted Orlicz Spaces 13

= c39 (M, α)∞

∑m=1

2(m+1)αE2m ( f )M,ω ≤ c40 (M, α)∞

∑m=1

2m

∑j=2m−1+1

jα−1Ej ( f )M,ω

≤ c41 (M, α)∞

∑j=2

jα−1Ej ( f )M,ω < ∞.

Therefore,∞

∑m=1

‖T2m+1 − T2m‖WαM,ω(T) < ∞,

which implies that T2m is a Cauchy sequence in WαM,ω (T). Since T2m → f in

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

It is clear thatEn

(

f (α))

M,ω≤

∥

∥

∥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. (3.3)

By Lemma 2∥

∥

∥S2m+2 f (α) − Sn f (α)

∥

∥

∥

M,ω≤ c42 (M, α) 2(m+2)αEn ( f )M,ω

≤ c43 (M, α) (n + 1)α En ( 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+1 f (α) − S2k f (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 andvaluable comments related to this paper.

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