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Ukrainian Mathematical Journal, Vol. 63, No. 1, June, 2011 (Ukrainian Original Vol. 63, No. 1, January, 2011)

TRIGONOMETRIC APPROXIMATION OF FUNCTIONS IN GENERALIZED LEBESGUE SPACES WITH VARIABLE EXPONENT ¨ R. Akgun

UDC 517.938.5

p./

We investigate the approximation properties of the trigonometric system in L2 : We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

1. Introduction Generalized Lebesgue spaces Lp.x/ with variable exponent and the corresponding Sobolev-type spaces are extensively applied in elasticity theory, fluid mechanics, differential operators [31, 10], nonlinear Dirichlet boundary-value problems [24], problems of nonstandard growth, and variational calculus [33]. These spaces appeared for the first time in [28] as an example of modular spaces [14, 26]. Sharapudinov [36] established the topological properties of Lp.x/ : Furthermore, if p WD ess sup p.x/ < 1; x2T

then Lp.x/ is a special case of the Musielak–Orlicz spaces [26]. Later, many mathematicians studied the principal properties of these spaces [36, 24, 32, 12]. There is a rich theory of boundedness of integral transforms of various types in Lp.x/ [22, 33, 9, 37]. For p.x/ WD p; 1 < p < 1; Lp.x/ coincides with the Lebesgue space Lp I the basic problems of trigonometric approximation in Lp were investigated by numerous mathematicians (among others, see [39, 19, 30, 40, 6, 4], etc.). The problems of approximation by algebraic polynomials and rational functions in Lebesgue spaces, Orlicz spaces, symmetric spaces, and their weighted versions on sufficiently smooth complex domains and curves were studied in [1–3, 15, 18, 16]. For a complete treatise on polynomial approximation, we refer the reader to the books [5, 8, 41, 29, 35, 23]. In the harmonic and Fourier analyses, some operators (e.g., the operator of partial sum of Fourier series, conjugate operator, operator of differentiation, and operator of shift f ! f . C h/ ; h 2 R/ are extensively used to prove approximation inequalities of direct and inverse types. Unfortunately, the space Lp.x/ is not p./continuous and not translation invariant [24]. Under various assumptions (including translation invariance) imposed on the modular space, Musielak [27] established some approximation theorems in modular spaces with respect to the ordinary moduli of smoothness. Since Lp.x/ is not translation invariant, by using Butzer–Wehrenstype moduli of smoothness (see [7, 13]) Israfilov et al. [17] obtained direct and inverse trigonometric approximation theorems in Lp.x/ : Balıkesir University, Balıkesir, Turkey. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 1, pp. 3–23, January, 2011. Original article submitted March 23, 2009; revision submitted October 22, 2010. 0041–5995/11/6301–0001

c 2011

Springer Science+Business Media, Inc.

1

¨ R. A KG UN

2 p./

In the present paper, we study the approximation properties of the trigonometric system in L2 : We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class. Let T WD Œ ; and let P be the class of 2-periodic Lebesgue measurable functions p D p.x/W T ! p./ p./ .1; 1/ such that p < 1: We introduce the class L2 WD L2 .T / of 2-periodic measurable functions f defined on T and satisfying the inequality Z

jf .x/jp.x/ dx < 1:

T p./

The class L2 is a Banach space [24] with the norms

kf .x/kp; WD kf .x/kp;;T

8
0W 1 jdxj ˇ ˛ ˇ : ; T

and WD sup kf .x/kp;

8 0 such that cY X CY: Throughout the paper, c; C; c1 ; c2 ; : : : denote constants different in different cases. The relation Xn D O .Yn / ; n D 1; 2; : : : ; means that there exists a constant C > 0 such that Xn CYn for n D 1; 2; : : : :

T RIGONOMETRIC A PPROXIMATION OF F UNCTIONS IN G ENERALIZED L EBESGUE S PACES WITH VARIABLE E XPONENT

3

p./

where f 2 L2 ; ˛ .˛ ˛ WD k

1/ : : : .˛ kŠ

k C 1/

for

˛ WD ˛; 1

k > 1;

˛ WD 1; 0

and I is the identity operator. ˛ Since the binomial coefficients satisfy the relation [34, p. 14] k ˇ ˇ ˇ ˛ ˇ c.˛/ ˇ ˇ ˇ k ˇ k ˛C1 ;

k 2 ZC ;

we get 1 ˇ ˇ X ˇ ˛ ˇ ˇ ˇ C.˛/ WD ˇ k ˇ 0 depends only on ˛; r; and p: Remark 1. The modulus of smoothness ˛ .f; ı/p./ ; ˛ 2 RC ; has the following properties for p 2 P possessing the property DL1 : (i) ˛ .f; ı/p./ is a nonnegative nondecreasing function of ı 0I (ii) ˛ .f1 C f2 ; /p./ ˛ .f1 ; /p./ C ˛ .f2 ; /p./ I (iii) lim ˛ .f; ı/p./ D 0: ı!0

¨ R. A KG UN

4

Let En .f /p./ WD inf kf

T kp; ;

T 2Tn

n D 0; 1; 2; : : : ;

p./

be the error of approximation of a function f 2 L2 ; where Tn is the class of trigonometric polynomials of degree not greater than n: For a given f 2 L1 ; assuming that Z f .x/dx D 0;

(4)

T

we define the ˛th fractional .˛ 2 RC / integral of f as follows [42, Vol. 2, p. 134]: I˛ .x; f / WD

X

ck .ik/

˛

e i kx ;

k2Z

where Z ck WD

f .x/e

ikx

dx

k 2 Z WD f˙1; ˙2; ˙3; : : :g

for

T

and .i k/

˛

WD jkj

˛

e.

1=2/ i˛ sign k

as the principal value. Let ˛ 2 RC be given. We define the fractional derivative of a function f 2 L1 satisfying (4) as f .˛/ .x/ WD

d Œ˛C1 I1CŒ˛ dx Œ˛C1

˛ .x; f

/;

provided that the right-hand side exists; here, Œx denotes the integer part of a real number x: p./ p./ ˛ ˛ Let Wp./ ; p 2 P; ˛ > 0; be the class of functions f 2 L2 such that f .˛/ 2 L2 : The class Wp./ becomes a Banach space with the norm

kf kW ˛ WD kf kp; C f .˛/ p; : p./

The main results of this work are the following: ˛ Theorem 1. Let f 2 Wp./ ; ˛ 2 RC ; and let p 2 P possess the property DL with 1: Then, for every natural n; there exists a constant c > 0 independent of n and such that

En .f /p./

c En .f .˛/ /p./ : .n C 1/˛


5

Corollary 1. Under the conditions of Theorem 1, the following relation is true: En .f /p./

.˛/ c

; ˛ f p; .n C 1/

where c > 0 is a constant independent of n D 0; 1; 2; 3; : : : : p./

Theorem 2. If ˛ 2 RC ; p 2 P possesses the property DL with 1; and f 2 L2 ; then there exists a constant c > 0 dependent only on ˛ and p and such that the following relation holds for n D 0; 1; 2; 3; : : : W 2 En .f /p./ c ˛ f; : n C 1 p./ The following inverse theorem of trigonometric approximation is true: p./

Theorem 3. If ˛ 2 RC ; p 2 P possesses the property DL with 1; and f 2 L2 ; then the following relation holds for n D 0; 1; 2; 3; : : : W ˛

f; nC1

p./

n X c . C 1/˛ .n C 1/˛

1

E .f /p./ ;

D0

where the constant c > 0 depends only on ˛ and p: p./

Corollary 2. Let ˛ 2 RC ; let p 2 P possess the property DL with 1; and let f 2 L2 : If En .f /p./ D O n

;

> 0;

n D 1; 2; : : : ;

then

˛ .f; ı/p./

8 ˆ O.ı /; ˛ > ; ˆ ˆ ˆ < D O .ı jlog .1=ı/j/ ; ˛ D ; ˆ ˆ ˆ ˆ :O.ı ˛ /; ˛ < :

Definition 1. For 0 < < ˛; we set ¸ ¹ p./ Lip .˛; p.// WD f 2 L2 W ˛ .f; ı/p./ D O ı ; ı > 0 : p./

Corollary 3. Let 0 < < ˛; let p 2 P possess the property DL with 1; and let f 2 L2 : Then the following conditions are equivalent: (a) f 2 Lip .˛; p.// I (b) En .f /p./ D O .n

/ ;

n D 1; 2; : : : :

¨ R. A KG UN

6 p./

Theorem 4. Let p 2 P possess the property DL with 1 and let f 2 L2 : If ˇ 2 .0; 1/ and 1 X

ˇ

1

E .f /p; < 1;

D1 ˇ

then f 2 Wp./ and

En .f .ˇ / /p./ c .n C 1/ˇ En .f /p./ C

1 X

! ˇ

1

E .f /p./ ;

DnC1

where the constant c > 0 depends only on ˇ and p: p./

Corollary 4. Suppose that p 2 P possesses the property DL with 1; f 2 L2 ; ˇ 2 .0; 1/ ; and 1 X

˛

1

E .f /p./ < 1

D1

for some ˛ > 0: In this case, for n D 0; 1; 2; : : : ; there exists a constant c > 0 dependent only on ˛; ˇ; and p and such that ˇ f

.˛/

; nC1

p./

n X c . C 1/˛Cˇ .n C 1/ˇ D0

1

E .f /p./ C c

1 X

˛

1

E .f /p./ :

DnC1

The following theorem on simultaneous approximation is true: p./

Theorem 5. Let ˇ 2 Œ0; 1/; let p 2 P possess the property DL with 1; and let f 2 L2 : Then there exist T 2 Tn and a constant c > 0 dependent only on ˛ and p and such that

.ˇ /

f

T .ˇ / p; cEn f .ˇ / p./ :

Definition 2 (Hardy space of variable exponent H p./ on a unit disc D with boundary T WD @D/ [21]. Let p.z/W T !.1; 1/ be a measurable function. We say that a complex-valued analytic function ˆ in D belongs to the Hardy space H p./ if Z2ˇ ˇp.#/ ˇ i# ˇ sup d# < C1; ˇˆ re ˇ

0 0; then nontangential p./ boundary values f e i exist a.e. on T and f e i 2 L2 .T / : Under the conditions 1 < p and p < 1; H p./ becomes a Banach space with the norm

kf kH p./

WD f e i

p;;T

8 Z < D inf > 0W : T

9 ˇ ˇ ˇ f e i ˇp. / = ˇ ˇ d 1 : ˇ ˇ ˇ ˇ ;

Theorem 6. If p 2 P possesses the property DL with 1; f belongs to the Hardy space H p./ on D; and r 2 RC ; then there exists a constant c > 0 independent of n and such that

f .z/

n X kD0

ak .f /z k

c r

i

f e ;

H p./

1 nC1

;

n D 0; 1; 2; : : : ;

p./

where ak .f /; k D 0; 1; 2; 3; : : : ; are the Taylor coefficients of f at the origin. 2. Some Auxiliary Results We begin with the following lemma: Lemma A [20]. For r 2 RC ; let (i) a1 C a2 C : : : C an C : : : and (ii) a1 C 2r a2 C : : : C nr an C : : : be two series in a Banach space .B; kk/ : Let Rnhri

WD

n X

1

kD1

k nC1

r ak

and Rnhri

WD

n X

1

kD1

k nC1

r

k r ak

for n D 1; 2; : : : : Then

hri R

n c;

n D 1; 2; : : : ;

for some c > 0 if and only if there exists R 2 B such that

hri

Rn

C

R r ; n

where c and C are constants that depend only on one another.

¨ R. A KG UN

8 p./

Lemma B [38]. If p 2 P possesses the property DL with 1 and f 2 L2 ; then there are constants c; C > 0 such that

fQ c kf kp; (5) p; and

Sn .; f /

p;

C kf kp;

(6)

for n D 1; 2; : : : : Remark 2. Under the conditions of Lemma B, the following conclusions can be made: (i) it readily follows from (5) and (6) that there exists a constant c > 0 such that

Sn .; f / p; cEn .f /p./ En fQ p./ I

f

(ii) it follows from the generalized H¨older inequality [24] (Theorem 2.1) that p./

L2 L1 : For a given f 2 L1 ; let f .x/ v

1 1 X X a0 .ak cos kx C bk sin kx/ D C ck e i kx 2 kD1

(7)

kD 1

and fQ.x/ v

1 X

.ak sin kx

bk cos kx/

kD1

be the Fourier series and the conjugate Fourier series of f; respectively. Setting Ak .x/ WD ck e i kx in (7), we define Sn .f / WD Sn .x; f / WD

n X

.Ak .x/ C A

k .x// D

kD0

Rnh˛i .f; x/ WD

n X a0 .ak cos kx C bk sin kx/ ; C 2

n D 0; 1; 2; : : : ;

kD1

n X 1 kD0

k nC1

˛ .Ak .x/ C A

k .x// ;

and ‚hri m WD

1

1 hri R m C 1 r 2m 2m C 1

1 2m C 1 r mC1

hri Rm

1

for

m D 1; 2; 3; : : : :

(8)


9

Under the conditions of Lemma B, using (6) and the Abel transformation, we get

h˛i

R .f; x/ n

p;

c kf kp; ;

p./

n D 1; 2; 3; : : : ;

x 2 T;

f 2 L2 ;

m D 1; 2; 3; : : : ;

x 2 T;

f 2 L2 :

(9)

and, therefore, it follows from (8) and (9) that

hri

‚ .f; x/ c kf kp; ; m p;

p./

From the property (see (16) in [25]) 2m X .k C 1/r

1

‚hri m .f /.x/ D X2m kDmC1

.k C 1/r

k

r

k r Sk .x; f /;

x 2 T;

f 2 L1 ;

kDmC1

it is known (see (18) in [25]) that ‚hri m .Tm / D Tm

(10)

for Tm 2 Tm ; m D 1; 2; 3; : : : : Lemma 1. Let Tn 2 Tn ; let p 2 P possess the property DL with 1; and let r 2 RC : Then there exists a constant c > 0 independent of n and such that

.r/

T n

p;

cnr kTn kp; :

Proof. Without loss of generality, one can assume that kTn kp; D 1: Since

Tn D

n X

.Ak .x/ C A

k

.x// ;

kD0

we get n h X TQn .Ak .x/ D nr

A

k .x// =n

r

i

kD1

and n .r/ h X Tn r r .Ak .x/ D i k nr

A

i r .x// =n : k

kD1

In this case, by virtue of (9) and (5), we have

hri

Rn

TQn nr

!

p;

c c c

TQn r kTn kp; D r ; p; r n n n

¨ R. A KG UN

10

whence, applying Lemma A (with R D 0/ to the series n h X

.Ak .x/

A

i r .x// =n C 0 C 0 C ::: C 0 C :::; k

kD1

n X

h k r .Ak .x/

A

k

i .x// =nr C 0 C 0 C : : : C 0 C : : : ;

kD1

we obtain

n

X

1

kD1

k nC1

r

i

A k .x// =nr

h

k r .Ak .x/

c;

p;

namely,

hri

Rn

.r/

Tn nr

n

X

D i r 1

!

p;

kD1

n

X

D 1

kD1

hri

k nC1

k nC1

r k

r

h

.Ak .x/

i

A k .x// =n

r

p;

r k

r

h

.Ak .x/

i

A k .x// =n

r

c :

p;

hri

Since Rn .cf / D cRn .f / for every real c; it follows from relation (10) and the last inequality that

.r/

T n

p;

1 hri .r/

Dn

nr ‚n Tn

.r/ D ‚hri T

n n

r

p;

D nr ‚hri

n

.r/

Tn nr

p;

!

c nr D c nr kTn kp; :

p;

The general case follows immediately from this. 2 Lemma 2. If p 2 P possesses the property DL with 1; f 2 Wp./ ; and r D 1; 2; 3; : : : ; then

r .f; ı/p./ cı 2 r

1

f 00 ; ı

p./

;

with some constant c > 0: Proof. Setting g.x/ WD

r Y i D2

I

hi f .x/;

ı 0;


11

we get r Y h1 g.x/ D I

I

hi f .x/

i D1

and r Y

I

hi

i D1

hZ1 =2

1 f .x/ D h1

g .x C t // dt D

.g.x/

1 2h1

h1 =2

hZ1 =2Z2t Z u=2 0

0

g 00 .x C s/ dsdudt:

u=2

Therefore, it follows from (1) that

r

Y

I

hi

i D1

f .x/

p;

8 ˇ ˇ 9 ˇ ˆ > Z ˇˇ hZ1 =2Z2t Zu=2 Z < ˇ = c ˇ ˇ p 0 ./ p 0 .x/ 00 sup g .x C s/ dsdudt ˇ jg0 .x/j dxW g0 2 L2 and dx 1 jg0 .x/j ˇ ˆ ˇ > 2h1 : ˇˇ ; ˇ 0 0 u=2 T T

c 2h1

c 2h1

hZ1 =2Z2t 0

0

hZ1 =2Z2t 0

Zu=2

1

00 u g .x C s/ ds

u

u=2

dudt

p;

u g 00 p; dudt D ch21 g 00 p; :

0

Since g 00 .x/ D

r Y

I

hi f 00 .x/;

i D2

we obtain r .f; ı/p./

r

Y

sup ch21 g 00 p; D cı 2 sup I 0