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J. Numer. Anal. Approx. Theory, vol. 50 (2021) no. 1, pp. 60–72 ictp.acad.ro/jnaat

APPROXIMATION BY MATRIX TRANSFORM IN GENERALIZED GRAND LEBESGUE SPACES WITH VARIABLE EXPONENT

AHMET TESTICIand DANIYAL M. ISRAFILOV

Abstract. In this work, the Lipschitz subclass of the generalized grand Lebesgue space with variable exponent is defined and the error of approximation by matrix transforms in this subclass is estimated.

MSC 2020. 41A10, 42A10.

Keywords. trigonometric approximation, matrix transforms, grand variable exponent, Lebesgue spaces, Lipschitz classes, rate of approximation, Fourier series.

1. INTRODUCTION

Let T := [0,2π] and let p(·) : T → [0,∞) be a Lebesgue measurable 2π periodic function satisfying the conditions

1≤p:= ess infx∈T p(x)≤ess supx∈

T p(x) :=p+<∞,

|p(x)−p(y)|ln|x−y| c, x, y∈[0,2π] and |x−y| ≤1/2, x6=y with some positive constantc=c(p). From now on, the class of such functions p(·) we denote by P(T). We also denote P0(T) := {p(·)∈ P(T) :p >1}.

The Lebesgue space Lp(·)(T) with variable exponent is defined as the set of all Lebesgue measurable 2π periodic functionsf such that

ρp(·)(f) :=

Z

0

|f(x)|p(x)dx <∞.

Ifp(·)∈ P0(T), then equipped with the norm

kfkp(·)= infnλ >0 :ρp(·)(f /λ)≤1o

Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail:

[email protected].

Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail:

[email protected].

Institute of Mathematics and Mechanics, National Academy of Science of Azerbaijan, Az.-1141 Baku, Azerbaijan.

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Lp(·)(T) becomes a Banach space. Let θ > 0 and p ∈ P0(T). The general- ized grand Lebesgue space with variable exponent Lp(·),θ(T) is the set of all measurable 2π periodic functions f :T→R such that

kfkp(·),θ = sup

0<ε<p−1ε

θ

p−−εkfkp(·)−ε<∞.

It is easily seen that

Lp(·)(T)⊂Lp(·),θ(T)⊂Lp(·)−ε(T) , 0< ε < p−1.

When θ = 0 and p(·) = p =const, the spaces Lp,0(T) reduce to classical Lebesgue spacesLp(T) and whenθ= 0 andp(·)6=const, the spacesLp(·),0(T) reduce to the variable exponent Lebesgue spacesLp(·)(T), investigated in detail in the monograph [8, 4]. When p = const and θ > 0 it was introduced by Iwaniec and Sbordone in [19] (for θ= 1) and by Greco, Iwaniec and Sbordone in [9] (forθ >1).

The grand and generalized grand Lebesgue spaces have been applied in the various fields; in particular in theory PDE [20, 32,33]. The fundamental problems of the spaces Lp(·),θ(T) in view of potential theory, maximal and singular operator theory were investigated in the monograph [26].

Regarding to grand spaces, it will be observed that these spaces in gen- eral are not separable; in particular, Lebesgue space is not dense in grand Lebesgue space. Similar situations are also valid in the case of variable expo- nents. The closure of Lp(·)(T) in Lp(·),θ(T) doesn’t coincide with Lp(·),θ(T) [27]. Henceforth, the closure ofLp(·)(T) inLp(·),θ(T) we denote byLp(·),θ0 (T).

Then Lp(·),θ0 (T) comprises the set of functions f such that

ε→0limε

θ

p−−ε kfkp(·)−ε = 0.

Note that in the generalizations of classical Lebesgue spaces different problems of approximation theory are also investigated. In particular, in [14,34,35,12, 13, 15, 23] the approximation properties of different summation methods in the variable exponent Lebesgue spaces were studied. Similar investigations have been done also in the grand and generalized grand spaces, with constant p and variable exponentp(·) in [5,6,16, 17,22].

Let fLp(·),θ(T), p(·) ∈ P0(T) and θ > 0. For α ∈ (0,1] we define the Lipschitz class

Lip (α, p(·), θ) :=nfLp(·),θ0 (T) : Ω (f, δ)p(·),θ =O(δα), δ >0o, where Ω (f, δ)p(·),θ is the modulus of smoothness for fLp(·),θ0 (T), defined as

Ω (f, δ)p(·),θ := sup

|h|≤δ

1 h

Z h 0

|f(x+t)f(x)|dt p(·),θ

, δ >0.

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LetfL1(T) and

f(x)∼ a20 +

X

k=1

(akcoskx+bksinkx) be its Fourier series representation with the Fourier coefficients

ak:= 1

π

Z

−π

f(t) cos (kt) dtand bk:= 1

π

Z

−π

f(t) sin (kt) dt and let

Sn(f) (x) =

n

X

k=0

uk(f) (x), n= 1,2, . . . , be the nth partial sums of the Fourier series of f, where

uk(f) (x) :=a0/2 anduk(f) (x) := (akcoskx+bksinkx), k= 1,2, . . . Let A = (an,k) be a lower triangular infinite matrix of real elements an,k

such that

an,k≥0 forkn, and an,k= 0 for k > n, wherek= 0,1,2, . . . ,and let

s(A)n :=

n

X

k=0

an,k= 1, n= 0,1,2, . . .

Unless otherwise indicate, we assume that A = (an,k) is a matrix that the summation of row elements equal to one throughout this work. For a given matrixA= (an,k), the matrix transform of Fourier series of f is defined as

Tn(A)(f) (x) :=

n

X

k=0

an,n−kSk(f) (x), n= 0,1,2, . . .

Let Pn = Pnk=0pk and (pn) be a sequence of positive real numbers. If an,k=pk/Pn , thenTn(A)(f) coincides with the N¨orlund means

Nn(f) (x) = P1

n

n

X

k=0

pn−kSk(f) (x), n= 0,1,2, . . . ,

which in the case ofpn= 1, for alln= 0,1,2, . . . ,reduce to the Ces`aro means σn(f) (x) =n+11

n

X

k=0

Sk(f) (x) .

We consider the matrix transforms Tn(A)(f) as approximating trigonomet- ric polynomials to f and study approximation properties of Tn(A)(f) in the generalized grand Lebesgue space with variable exponent, exactly in the Lip- schitz classes defined above. The required conditions on the sequences (an,k) are crucial points to arrive better approach when matrix transform Tn(A)(f)

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is constructed with respect to a given matrixA= (an,k). We mention the no- tations and definitions of some classes of sequences consisting of nonnegative numbers to explain the amongst important relations.

A nonnegative sequence (an,k) is called almost monotone increasing (de- creasing) sequenceif there exists a constantK1:=K1(an,k) (K2 :=K2(an,k)) depending only the sequence (an,k) such that

an,kK1an,m (an,mK2an,k)

for all 0≤kmn. If (an,k) isalmost monotone decreasing sequence, then we write (an,k)∈AM DS and if (an,k) isalmost monotone increasing sequence then we write (an,k)∈AM IS.

Let

An,k= k+11

n

X

j=n−k

an,j.

If (An,k) ∈ AM DS, then (An,k) is called almost monotone decreasing upper mean sequence and we write (An,k) ∈AM DU M S. If (An,k) ∈ AM IS, then (An,k) is calledalmost monotone increasing upper mean sequence and then we write (An,k)∈AM IU M S.

There exist following embedding relations between these sequence classes N ISAM DSAM IU M S

and

N DSAM ISAM DU M S,

whereN IS is the class of nonnegative and nondecreasing sequences,N DS is the class of nonnegative and nonincreasing sequences [36].

We use also the notations

∆ak=akak+1, ∆kan,k=an,kan,k+1.

We writem=O(n) if there exists a positive constantK3 such thatmK3n.

Let nowLp(T) be the classical Lebesgue space and let ω(f, δ) := sup

0<h≤δ

kf(·+h)fkp, δ >0

is the modulus of continuity for fLp(T), wherek·kp := ρ1/pp (·) forp(·) :=

p isconst.

In this case Lipschitz classes can be defined as

Lip (α, p) :={f ∈Lp(T) :ω(f, δ) =O(δα), δ >0}, whereα∈(0,1].

The rate of trigonometric approximation in the Lipschitz classes Lip (α, p), α ∈ (0,1] and 1 < p < ∞, were investigated by a great number of authors.

Initially, degree of approximation by σn(f), when f ∈ Lip (α, p) was studied by Quade. He proved [31] that if f ∈Lip (α, p) forα∈(0,1] and 1< p <∞, then kf −σn(f)kp = O(n−α). There have been various generalizations of

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this result [29, 2,3, 36, 10,11,12, 38, 25,21,24, 7]. Especially, Chandra [3]

generalizes the result of Quade and prove that if f ∈Lip (α, p) for α ∈(0,1]

and 1< p <∞, thenkf −Nn(f)kp =O(n−α), when (pn) is a monotonic se- quence of positive real numbers such that (n+ 1)pn=O(Pn). Later, Leindler [28] generalizes the result of Chandra, imposing weaker assumptions on the sequence (pn). Let’s give some results that are as close as possible to our work in the case of 1< p <∞.

Theorem 1. [28] Let f ∈Lip (α, p) for α ∈(0,1], 1< p <and let (pn) be a sequence of positive real numbers. If one of the conditions

i) 0< α <1 and (pn)∈AM DS,

ii) 0< α <1, (pn)∈AM IS and(n+ 1)pn=O(Pn), iii) α= 1 and

n−1

P

k=1

k|∆pk|=O(Pn), iv) α= 1,

n−1

P

k=0

|∆pk|=O(Pn) and (n+ 1)pn=O(Pn), holds, then

kf−Nn(f)kp=O n−α.

Mittal and his collaborators [30] extend the results of Leindler by using matrix transforms of functions in Lip (α, p), when the matrixA= (an,k) such that (an,k)∈N IS or (an,k)∈N DS. In the weighted case Guven [11] proves this result using more general matrix transforms, namely for the matrices A= (an,k) such that (an,k)∈AM IS or (an,k)∈AM DS. As a result we can deduce the following theorem from Theorems 1 and 2 proved in [11] for the matrix transforms defined as:

Tn(f, A) :=

n

X

k=0

an,kSk(f) (x), n= 0,1,2, . . .

Theorem 2 ([11]). Let f ∈ Lip (α, p) for α ∈ (0,1], 1 < p <and A = (an,k) be a lower triangular matrix with non-negative entries such that

s(A)n −1=O(n−α). If one of the conditions

i) 0< α <1,(an,k)∈AM DS and (n+ 1)an,0=O(1),

ii) 0< α <1, (an,k)∈AM IS and (n+ 1)an,r =O(1) where r is integer part of n/2,

iii) α= 1 and

n−1

P

k=1

|∆an,k−1|=O n−1, holds, then

kf −Tn(f, A)kp =O n−α.

This theorem was generalized to weighted and nonweighted Lebesgue space with variable exponent in [18,37,13], respectively. At the same time, it can be observed [36] that theorem similar to Theorem 2 is also true under the assumption s(A)n = 1.

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Theorem 3 ([36]). Let f ∈ Lip (α, p) for α ∈ (0,1], 1 < p <and A= (an,k)be a lower triangular matrix with non-negative entries ands(A)n = 1.

If one of the conditions

i) 0< α <1,(an,k)∈AM DU M S,

ii) 0< α <1, (an,k)∈AM IU M S and (n+ 1)an,n=O(1), iii) α= 1 and

n−2

P

k=0

|∆kAn,k|=O n−1, holds, then

fTn(A)(f)

p =O n−α.

2. MAIN RESULTS

In this work, we estimate the error of trigonometric approximation by ma- trix transformsTn(A)(f) inf ∈Lip (α, p(·), θ),when p(·)∈ P0(T), α∈(0,1]

andθ >0. We obtain the generalization of the above mentioned results. The- orem proved by us is stronger than the previous ones, because we prove it by imposing weaker assumptions. Main results are following.

Theorem 4. Let f ∈ Lip (α, p(·), θ) for α ∈ (0,1], p(·) ∈ P0(T), θ > 0 and A = (an,k) be a lower triangular matrix with non-negative entries and s(A)n = 1. If one of the conditions

i) 0< α <1,(an,k)∈AM DU M S,

ii) 0< α <1, (an,k)∈AM IU M S and (n+ 1)an,n=O(1), iii) 0< α <1 and

n−1

P

k=−1

|∆an,k|=O n−1 where an,−1= 0, iv) α= 1 and

n−2

P

k=0

|∆kAn,k|=O n−1, holds, then

fTn(A)(f)

p(·),θ =O n−α.

Theorem 4 is more general than the corresponding theorems given in [37].

In the case of p(·) =const and θ= 0 it was proved in [36].

LetPn=Pnk=0pk, for a sequence (pn) of positive real numbers and let Pn,k = k+11

n

X

j=n−k

pj.

Ifan,k :=pk/Pn, then Theorem 4implies

Corollary 5. Let f ∈Lip (α, p(·), θ) for α∈(0,1], p(·)∈ P0(T), θ >0 and (pn) be a sequence of positive real numbers. If one of the conditions i) 0< α <1,(pk)∈AM DU M S,

ii) 0< α <1, (pk)∈AM IU M S and (n+ 1)pn=O(Pn), iii) 0< α <1 and

n−1

P

k=−1

|∆pk|=O(Pn/n) where p−1= 0,

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iv) α= 1 and

n−2

P

k=0

|∆kPn,k|=O(Pn/n), holds, then

kf −Nn(f)kp(·),θ =O n−α. Ifpn:=Aβ−1n for someβ >0, where

Aβ0 = 1, Aβk = (β+1)(β+2)...(β+k)

k! , k≥1

thenNn(f) coincides with the generalized Ces`aro means σβn(f) (x) := 1

Aβn

n

X

k=0

Aβ−1n−kSk(f) (x) , n= 0,1,2, . . . Hence,Corollary 5implies

Corollary 6. Let f ∈Lip (α, p(·), θ) for α∈(0,1], p(·)∈ P0(T), θ >0 and β >0.Then

fσβn(f)

p(·),θ =O n−α.

Corollary 6 in the case of p(·) =const and θ = 0 was proved in [3] (non- weighted case) and in [10] (weighted case).

3. AUXILIARY RESULTS

Let

En(f)p(·),θ := infnkf−Tnkp(·),θ :Tn∈Πno

be the best approximation number offLp(·),θ0 (T) in the class Πnof trigono- metric polynomials of degree not exceeding n.

Lemma 7 ([37]). If fLp(·),θ0 (T), p(·) ∈ P0(T) and θ > 0, then the inequality

En(f)p(·),θ = OΩ (f,1/n)p(·),θ, n= 1,2, . . . , holds.

Lemma8 ([37]). Letp(·)∈ P0(T), θ >0∈(0,1].If f ∈Lip (α, p(·), θ), then the inequality

kf −Sn(f)kp(·),θ =O n−α, n= 1,2,3. . . , holds.

Lemma 9 ([37]). Let p(·) ∈ P0(T) and θ > 0. If f ∈ Lip (1, p(·), θ), then the inequality

kSn(f)−σn(f)kp(·),θ =On−1, n= 1,2,3. . . , holds.

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Lemma 10. Let p(·)∈ P0(T) and θ >0.If f ∈Lip (α, p(·), θ), α∈(0,1), then the inequality

kf −σn(f)kp(·),θ =O n−α, n= 1,2,3. . . , holds.

Proof. Let f ∈ Lip (α, p(·), θ), α ∈ (0,1), p(·) ∈ P0(T) and θ > 0. The conjugate function of fL1(T) is defined as

fe(x) := π1

π

Z

−π f(t) 2 tant−x2 dt.

Since fLp(·),θ0 (T), p(·) ∈ P0(T) and θ > 0, Theorem 2.1 given in [5]

provides that there exists a constant c1(p, θ) > 0, depending on p(·) and θ such that

fe

p(·),θc1(p, θ)kfkp(·),θ.

Using this inequality and the standard technics developed in [39] we obtain the inequality

(1) kσn(f)kp(·),θc2(p, θ)kfkp(·),θ for some constantc2(p, θ)>0.

Let Tn0(f) be the best approximation trigonometric polynomial to f in Lp(·),θ0 (T),that isfTn0(f)p(·),θ =En(f)p(·),θ forn= 0,1,2. . .. Applying (1), the Minkowski inequality andLemma 7we have

kf−σn(f)kp(·),θfTn0(f)

p(·),θ+Tn0(f)−σn(f)

p(·),θ

= En(f)p(·),θ+σn

Tn0(f)−f

p(·),θ

= En(f)p(·),θ+O(Tn0(f)−f

p(·),θ)

= O(En(f)p(·),θ) =OΩ (f,1/n)p(·),θ

= O n−α.

Thus, lemma is proved.

Lemma11 ([36]). Let 0< α <1andA= (an,k)be infinite lower triangular matrix with non-negative entries and s(A)n = 1. If one of the conditions i) (an,k)∈AM DU M S,

ii) (an,k)∈AM IU M S and(n+ 1)an,n=O(1), holds then

n

X

k=0

(k+ 1)−αan,n−k =O(n+ 1)−α.

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4. PROOF OF MAIN RESULTS

Proof of Theorem 4. Letf ∈Lip (α, p(·), θ), α ∈(0,1], p(·) ∈ P0(T), θ > 0 and A = (an,k) be a lower triangular matrix with non-negative entries such that s(A)n = 1. We suppose that the conditions either i) or ii) holds. Since 0< α <1 and

f(x)−Tn(A)(f) (x) =

n

X

k=0

an,n−k(f(x)−Sk(f) (x)),

applying the Minkowski inequality, Lemma 8and Lemma 11, respectively we have that

fTn(A)(f)

p(·),θ

n

X

k=0

an,n−kkf −Sk(f)kp(·),θ

c3 n

X

k=0

an,n−k(k+ 1)−α=O n−α. Therefore, we provedTheorem 4 in the case of i) and ii).

Let 0 < α < 1 and

n−1

P

k=−1

|∆an,k| = O n−1, where an,−1 = 0. By Abel transformation ([1, p. 1]) and definition of σn(f) we have

f(x)−Tn(A)(f) (x) =

n

X

k=0

an,n−k(f(x)−Sk(f) (x))

=

n−1

X

k=0

(an,n−kan,n−k−1)

k

X

j=0

(f(x)−Sj(f) (x)) +an,0

n

X

k=0

(f(x)−Sk(f) (x))

=

n−1

X

k=0

(an,n−kan,n−k−1) (k+ 1) (f(x)−σk(f) (x)) +an,0(n+ 1) (f(x)−σn(f) (x)).

Using Minkowski’s inequality, Lemma 10we obtain that

fTn(A)(f)

p(·),θ

n−1

X

k=0

|an,n−kan,n−k−1|(k+ 1)kf−σk(f)kp(·),θ +an,0(n+ 1)kf−σn(f)kp(·),θ

c4 (n−1

X

k=0

|an,n−k−an,n−k−1|(k+ 1)1−α+an,0(n+1)1−α )

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c5(n+ 1)1−α (n−1

X

k=0

|an,n−kan,n−k−1|+an,0 )

= c6(n+ 1)1−α

n−1

X

k=−1

|∆an,k|=O n−α.

Hence, the iii) part of Theorem 4 is also proved. Finally, we prove the last part of theorem. Let α = 1 and

n−2

P

k=0

|∆kAn,k|=O n−1. Using twice Abel’s transformation we have

f(x)−Tn(A)(f) (x) =

n

X

k=0

an,n−k(f(x)−Sk(f) (x))

=

n−1

X

k=0

(Sk+1(f) (x)−Sk(f) (x))

k

X

j=0

an,n−j+ (f(x)−Sn(f) (x))

n

X

k=0

an,n−k

=

n−1

X

k=0

(Sk+1(f) (x)−Sk(f) (x))

n

X

j=n−k

an,j + (f(x)−Sn(f) (x))s(A)n

= f(x)−Sn(f) (x) +

n−1

X

k=0

uk+1(f) (x) (k+ 1)An,k

= f(x)−Sn(f) (x) +

n−2

X

k=0

(An,kAn,k+1)

k

X

j=0

(j+ 1)uj+1(f) (x)

+An,n−1 n−1

X

k=0

(k+ 1)uk+1(f) (x)

= f(x)−Sn(f) (x) +

n−2

X

k=0

(An,kAn,k+1)

k

X

j=0

(j+ 1)uj+1(f) (x)

+n1

n

X

j=1

an,j n−1

X

k=0

(k+ 1)uk+1(f) (x).

Hence, applying the Minkowski inequality we get

f−Tn(A)(f)

p(·),θ ≤ kf−Sn(f)kp(·),θ+

n−2

X

k=0

|An,k−An,k+1|

k+1

X

j=1

juj(f) p(·),θ

+1ns(A)n

n

X

k=1

kuk(f) p(·),θ

=kf−Sn(f)kp(·),θ+

n−2

X

k=0

|∆kAn,k|

k+1

X

j=1

juj(f) p(·),θ

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(2) + 1n

n

X

k=1

kuk(f) p(·),θ

.

Since

Sn(f) (x)−σn(f) (x) =

n

X

k=0

"

uk(f) (x)−n+11

k

X

ν=0

uν(f) (x)

#

=

n

X

k=0

1−n+1−kn+1 uk(f) (x)

= n+11

n

X

k=0

kuk(f) (x), using Lemma 9we have

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n

X

k=1

kuk(f) p(·),θ

= (n+ 1)kSn(f)−σn(f)kp(·),θ =O(1). Finally, combining (2), (3) and Lemma 8forα= 1 we obtain that

fTn(A)(f)

p(·),θ ≤ O n−1+

n−2

X

k=0

|∆kAn,k| O(1) + 1nO(1) =O n−1.

Thereby, all parts ofTheorem 4 are proved.

Acknowledgements. The authors would like to thank the anonymous referee.

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Received by the editors: March 11, 2021; accepted: June 4, 2021; published online: No- vember 8, 2021.

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