Approximation by Nörlund and Riesz means in weighted Lebesgue space with
variable exponent
Article in Communications · May 2019 DOI: 10.31801/cfsuasmas.460449 CITATIONS 0 READS 88 1 author:
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Balikesir University
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C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 68, N umb er 2, Pages 2014–2025 (2019) D O I: 10.31801/cfsuasm as.460449
ISSN 1303–5991 E-ISSN 2618-6470
http://com munications.science.ankara.edu.tr/index.php?series= A 1
APPROXIMATION BY NÖRLUND AND RIESZ MEANS IN WEIGHTED LEBESGUE SPACE WITH VARIABLE EXPONENT
AHMET TESTICI
Abstract. We investigate the approximation properties of Nörlund and Riesz means of trigonometric Fourier series are investigated in the subset of weighted Lebesgue space with variable exponent.
1. Introduction and Main results
Let T := [0; 2 ] and let p ( ) : T ! [1; 1) be a Lebesgue measurable 2 peri-odic function. We suppose that the considered exponent functions p ( ) satisfy the condition
1 < p := ess inf
x2T p (x) ess supx2T
p (x) := p+< 1.
In addition to this requirement if there exist a positive constant c such that
jp (x) p (y)j ln (1= jx yj) c; x; y 2 T; 0 < jx yj 1=2
then we say that p ( ) 2 P0(T). The variable exponent Lebesgue space Lp( )(T) is de…ned as the set of all Lebesgue measurable 2 periodic functions f such that
p( )(f ) := R2
0 jf (x)j p(x)
dx < 1: Equipped with the norm kfkp( )= inf
n
> 0 : p( )(f = ) 1 o
Lp( )(T), p ( ) 2 P0(T) becomes a Banach space. The fundamental properties of Lebesgue spaces with variable exponent are explained in monographs [3, 4, 5].
For a given weight ! we de…ne the weighted variable Lebesgue space Lp( )! (T) as the set of all measurable 2 periodic functions f such that f ! 2 Lp( )(T). The norm of Lp( )! (T) can be de…ned as kfkp( );! := kf!kp( ).
If p ( )=constant, then Lp( )! (T) coincides with the weighted Lebesgue spaces Lp
!(T). In this case ApMuckenhoupt class becomes important point. In order
Received by the editors: September 17, 2018; Accepted: April 19, 2019.
2010 Mathematics Subject Classi…cation. Primary 42A10, 41A25; Secondary 41A30.
Key words and phrases. Weighted Lebesgue space with variable exponent, Lipschitz class, Nörlund mean, Riesz mean, Muckenhoupt weight, Fourier series.
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to prove boundedness of some operators and crucial theorems of harmonic analy-sis in weighted Lebesgue spaces it is needed that weight function ! belongs to ApMuckenhoupt class. Similar situations are valid in the weighted Lebesgue space with variable exponent Lp( )! (T) : In our investigations we will use the Muckenhoupt weights class Ap( )(T) de…ned as
De…nition 1. For a given exponent p ( ) we say that ! 2 Ap( )(T) if sup Bj jBjj 1 ! Bj p( ) ! 1 Bj p0( )< 1, 1=p ( ) + 1=p 0( ) = 1,
where supremum is taken over all open intervals Bj T with the characteristic
functions Bj.
Let f 2 L1(T) and let
f (x) a0 2 + 1 X k=1 (akcos kx + bksin kx) (1)
be Fourier series of f where ak := ak(f ) = 1 2 Z f (t) cos ktdt and bk:= bk(f ) = 1 2 Z f (t) sin ktdt are Fourier coe¢ cients of f: Let also
u0(f ) (x) := a0
2 , uk(f ) (x) := akcos kx + bksin kx ; k = 1; 2; :::; n
where ak and bk are Fourier coe¢ cients of f . We denote the n th partial sums of the series (1) by Sn(f ) (x) := n X k=0 uk(f ) (x) ; n = 0; 1; 2; ::: .
Let (pn)1n=0 be sequence of positive real numbers. We de…ne the Nörlund and Riesz means of the series (1), respectively,
Nn(f ) (x) := 1 Pn n X m=0 pn mum(f ) (x) ; and Rn(f ) (x) := 1 Pn n X m=0 pmum(f ) (x) :
where Pn = Pnm=0pm and P 1 = p 1 := 0. In the case of pn = 1 for all n = 0; 1; 2; ::; the both of Nn(f ) and Rn(f ) means coincide with the Cesàro mean n(f ),
de…ned as n(f ) (x) := 1 n + 1 n X m=0 um(f ) (x) .
De…nition 2. Let f 2 Lp( )! (T), p ( ) 2 P0(T) and ! ( ) 2 Ap( )(T). We de…ne the modulus of smoothness as (f; )p( );!:= sup jhj 1 h Z h 0 [f (x + t) f (x)] dt p( );! ; > 0.
The correctness of this de…nition follows from the boundedness of the maximal operator M : f ! Mf (x) := sup B•x 1 jBj Z B jf (t)j dt
in the Lp( )! (T), where B is any open subinterval of T (see, [7]). So we have that if ! ( ) 2 Ap( )(T), then the maximal operator M is bounded in L
p( )
! (T), p ( ) 2 P0(T). In this case there exist a positive constant c1(p) such that the inequality
kMfkp( );! c1(p) kfkp( );! (2)
holds for every f 2 Lp( )! (T). By this fact if f 2 Lp( )! (T) ; p ( ) 2 P0(T) and ! ( ) 2 Ap( )(T), then there exists a positive constant c2(p) such that
(f; )p( );! c2(p) kfkp( );! : (3)
Moreover, it can be shown that if f; g 2 Lp( )! (T), then (f + g; )p( );! (f; )p( );!+ (g; )p( );! and also lim
!0 (f; )p( );!= 0: W!p( );r(T); r = 1; 2; :::; denotes the class of all Lebesgue measurable 2 periodic and r 1 times continuously di¤erentiable functions such that f(r)2 Lp( )
! (T).
We de…ne the variable exponent Lipschitz class Lipp( )r ( ; !), 0 < 1, as Lipp( )r ( ; !) := f 2 W!p( );r(T) : f(r);
p( );!= O ( ) ; > 0 .
In the classical case the approximation properties of n(f ) in classical Lipschitz
classes where 1 p < 1 and 0 < 1 were investigated by Quade in [8]. The
Quade’s results were generalized by Mohapatra and Russel [9], Chandra [10, 11] and Leindler [12]. In [11] under the some conditions related with the sequence (pn)1n=0 Chandra proved satisfactory results about approximation by the Nn(f ) and Rn(f ) means in in classical Lipschitz classes where 1 p < 1 and 0 < 1. Guven carried and extended the results obtained in [11] to weighted Lipschitz classes where 1 < p < 1 (see, [13, 14]). In the Lebesgue space with variable exponent space Guven and Isra…lov investigated the approximation properties of Nn(f ) and
Rn(f ) means for Lipschitz classes in [15]. After that Guven extended this results to triangular matrix transforms in [16]. In weighted Lebesgue space with variable exponent Isra…lov and Testici were investigated the approximation properties of matrix transform of Fourier series in [18]. In the Lebesgue space with variable exponent approximation Nörlund and Riesz submethods were studied in [20]. In [21] the results obtained in [13] generalize to weighted Lorentz space for the derivatives of functions.
In this work we investigate the approximation properties of the Nörlund and Riesz means of the Fourier series in Lipp( )r ( ; !), 0 < 1 where p ( ) 2 P0(T) ; ! 2 Ap( )(T), also it is important to emphasize that obtained results in this work can be considered that generalizations of the given results in [15].
A sequence of positive real numbers (pn)1n=0is called almost monotone increasing if there exists a constant K, depending only on the sequence (pn)1n=0 such that for
all n m the inequality
pm Kpn
holds. Almost monotone increasing sequences are denoted by (pn)1n=02 AMIS. Along this work we will use the notations
gn:= gn gn+1, mg (n; m) := g (n; m) g (n; m + 1) ,
and f = O (g) means that there exists some positive constant c such that f cg. Moreover c( ); c1( ); c2( ); :::; denote the constants (in general di¤erent in the di¤erent relations) depending in general on the parameters given in the brackets and independent of n.
Our main results are following:
Theorem 3. Let p ( ) 2 P0(T) ; ! ( ) 2 Ap( )(T) ; 0 < < 1; and let (pn)1n=0 be a sequence of positive real numbers such that (pn)1n=02 AMIS and
(n + 1)r+1pn= O (Pn) . (4)
If f 2 Lipp( )r ( ; !), then the estimate
kf Nn(f )kp( );!= O n ( +r) ; n = 1; 2; :::; holds.
Theorem 4. Let p ( ) 2 P0(T) ; ! ( ) 2 Ap( )(T) and let (pn)1n=0 be a sequence of positive real numbers such that
n 1 X k=0 j pkj = O Pn nr+1 . (5)
If f 2 Lipp( )r (1; !) ; then the estimate kf Nn(f )kp( );!= O n
holds.
Theorem 5. Let p ( ) 2 P0(T) ; ! ( ) 2 Ap( )(T) ; and let (pn)1n=0 be a sequence of positive real numbers such that
n 1 X m=0 Pm m + 1 = O Pn (n + 1)r+1 ! : (6)
If f 2 Lipp( )r ( ; !), then the estimate kf Rn(f )kp( );! = O n
(1+r) ; n = 1; 2; :::;
holds.
2. Auxiliary results
In weighted Lebesgue space with variable exponent the approximation problems were studied using some di¤erent type modulus of smoothness in [17], [1], [2]. In these works the weight function ! satis…es the condition that ! p0 2 A
(p( )=p0)0 for some 1 < p0 < p . After that under the more intelligible condition, namely ! 2 Ap( ); the direct and inverse theorems of approximation theory in the weighted Lebesgue space with variable exponent were proved in [18] and [19], respectively. For the formulations of the results obtained in this work we need some auxiliary results proved in [18, 19].
Let n be the class of trigonometric polynomials of degree not exceeding n. The best approximation number of f 2 Lp( )! (T) is de…ned as
En(f )p( );!:= inf n
kf Tnkp( );!: Tn2 n
o
; n = 0; 1; 2; :: ,
and if En(f )p( );! = kf Tnkp( );!, then Tn 2 n is called the best approximation trigonometric polynomial to f in Lp( )! (T).
Lemma 6. [18] Let p ( ) 2 P0(T) and ! ( ) 2 Ap( )(T). Then there exists a positive constant c3(p) such that the inequality
kSn(f )kp( );! c3(p) kfkp( );! ; n = 1; 2; :::; holds for every f 2 Lp( )! (T) :
Lemma 7. [19] Let p ( ) 2 P0(T) and ! ( ) 2 Ap( )(T). Then there exists a positive constant c4(p) such that the inequality
k n(f )kp( );! c4(p) kfkp( );! ; n = 1; 2; :::; holds for every f 2 Lp( )! (T) :
Lemma 8. [18] If f 2 Lp( )! (T) ; p ( ) 2 P0(T), ! ( ) 2 Ap( )(T), then the estimate En(f )p( );!= O (f; 1=n)p( );! ; n = 1; 2; :::;
holds.
Lemma 9. [19] If f 2 W!p( );1(T); p ( ) 2 P0(T), ! ( ) 2 Ap( )(T), then there exist a positive constant c5(p) such that the inequality
kf Sn(f )kp( );! c5(p) n En(f 0) p( );! ; n = 1; 2; :::; holds.
Lemma 10. Let p ( ) 2 P0(T) ; ! ( ) 2 Ap( )(T) and 0 < 1. If f 2
Lipp( )r ( ; !), then the estimate
En(f )p( );! = O n
( +r) ; n = 1; 2; :::; holds.
Proof. Let f 2 Lipp( )r ( ; !). Since f 2 W!p( );r(T) and r = 1; 2; :::; Lemma 9 implies that En(f )p( );! c5(p) n En(f 0) p( );! ; n = 1; 2; :::; Thus, consecutively r times, using this inequality, we have
En(f )p( );!
c6(p)
nr En f
(r)
p( );! : (7)
By Lemma 8 and (7) we obtain En(f )p( );! c6(p) nr En f (r) p( );! c7(p) nr f (r); 1=n p( );! = O n ( +r) :
Lemma 11. Let p ( ) 2 P0(T) ; ! ( ) 2 Ap( )(T) and 0 < 1: If f 2 Lipp( )r ( ; !), then the estimate
kf Sn(f )kp( );!= O n
( +r) ; n = 1; 2; 3:::;
holds.
Proof. Let Lipp( )r ( ; !) ; r = 1; 2; :::; and let Tn (n = 0; 1; 2; :::) be the best ap-proximation trigonometric polynomial to f in Lp( )! (T). Applying Lemma 10 we have
kf Tnkp( );!= En(f )p( );! = O n ( +r) :
By Lemma 6 for n = 1; 2; 3:::; we obtain
kf Sn(f )kp( );! kf Tnkp( );!+ kTn Sn(f )kp( );! = kf Tnkp( );!+ kSn(Tn) Sn(f )kp( );!
= O kf Tnkp( );!
= O n ( +r) .
Lemma 12. Let p ( ) 2 P0(T) ; ! ( ) 2 Ap( )(T) : If f 2 Lipp( )r (1; !), then the estimate
kSn(f ) n(f )kp( );! = O n (1+r) ; n = 1; 2; 3:::; holds.
Proof. Let f 2 Lipp( )r (1; !) ; r = 1; 2; :::; and let Tn (n = 0; 1; 2; :::) be the best ap-proximation trigonometric polynomial to f in Lp( )! (T). Since f 2 Lp( )! (T) applying Lemma 7 and Lemma 10 we have
kf n(f )kp( );! kf Tnkp( );!+ kTn n(f )kp( );!
= kf Tnkp( );!+ k n(Tn f )kp( );!
= O kf Tnkp( );!
= O n (1+r) . (8)
By Lemma 11 for = 1 and (8) we obtain
kSn(f ) n(f )kp( );! kSn(f ) f kp( );!+ kf n(f )kp( );!
= O n (1+r) :
Lemma 13. Let (pn)1n=0 be a sequence of positive numbers. If (pn)1n=02 AMIS and (n + 1)r+1pn= O (Pn) ; then n X m=1 m ( +r)pn m= O n ( +r)Pn for r = 0; 1; 2; :::; and 0 < < 1.
Proof. Let r = 0; 1; 2; :::; and 0 < < 1. In the case of r = 0; Lemma 13 was proved in [12]. Similar way we can prove the other part of Lemma. Let k be integer part of n=2. If (pn)1n=02 AMIS and (n + 1)
r+1 pn= O (Pn) ; then n X m=1 m ( +r)pn m Kpn k X m=1 m + (k + 1) ( +r) n X m=k+1 pn m = O Pn=nr+1 k X m=1 m + O n ( +r) Pn
= O n ( +r) Pn.
3. Proofs of the main results Proof of the Theorem 3. Let 0 < < 1. Since
f (x) = 1 Pn n X m=0 pn mf (x) we have f (x) Nn(f ) (x) = 1 Pn n X m=0 pn mff (x) Sm(f ) (x)g : By Lemma 11, Lemma 13 and (4) we obtain
kf Nn(f )kp( );! 1 Pn n X m=0 pn mkf Sm(f )kp( );! = 1 Pn n X m=1 pn mO m ( +r) + pn Pn kf S0(f )kp( );! = 1 PnO n ( +r)P n + O (n + 1) (r+1) = O n ( +r) :
Proof of the Theorem 4. Let f 2 Lipp( )! (1; !) and n 1P k=1j p kj = O Pn=nr+1 . It is clear that Nn(f ) (x) = 1 Pn n X m=1 Pn mum(f ) (x) : By Abel transform Sn(f ) (x) Nn(f ) (x) = 1 Pn n X m=1 (Pn Pn m) um(f ) (x) = 1 Pn n X m=1 m Pn Pn m m m X k=1 kuk(f ) (x) ! + 1 n + 1 n X k=1 kuk(f ) (x) ; hence kSn(f ) Nn(f )kp( );! 1 Pn n X m=1 m Pn Pn m m m X k=1 kuk(f ) (x) p( );!
+ 1 n + 1 n X k=1 kuk(f ) (x) p( );! : Since Sn(f ) (x) n(f ) (x) = 1 n + 1 n X k=1 kuk(f ) (x) ; by Lemma 12 we get 1 n + 1 n X k=1 kuk(f ) (x) p( );! = O n (1+r) : Hence kSn(f ) Nn(f )kp( );! 1 Pn n X m=1 m Pn Pn m m O m r + O n (1+r) : (9) By a simple computations we have
m Pn Pn m m = 1 m (m + 1) n X k=n m pk (m + 1) pn m !
and by induction one can easily obtain n X k=n m pk (m + 1) pn m m X k=1 k jpn k+1 pn kj : Thus, n X m=1 m Pn Pn m m m r n X m=1 m Pn Pn m m n X m=1 1 m (m + 1) m X k=1 k jpn k+1 pn kj ! = n X k=1 k jpn k+1 pn kj n X m=k 1 m (m + 1) ! n X k=1 k jpn k+1 pn kj 1 X m=k 1 m (m + 1) ! = n X k=1 jpn k+1 pn kj = n 1X k=1 j pkj = O Pn nr+1 : (10)
By (9) and (10) we have
kSn(f ) Nn(f )kp( );!= O n (1+r) : (11)
Combining Lemma 11 for = 1 and (11) we obtain
kf Nn(f )kp( );! kf Sn(f )kp( );!+ kSn(f ) Nn(f )kp( );!
= O n (1+r) :
Proof of the Theorem 6. By Abel transform, Rn(f ) (x) = 1 Pn n 1X m=0 fPm(Sm(f ) (x) Sm+1(f ) (x)) + PnSn(f ) (x)g = 1 Pn n 1X m=0 Pm( um+1(f ) (x)) + Sn(f ) (x) and hence we have
Rn(f ) (x) Sn(f ) (x) = 1 Pn n 1X m=0 Pmum+1(f ) (x) : (12)
Using Abel transform n 1X m=0 Pmum+1(f ) (x) = n 1 X m=0 Pm m + 1(m + 1) um+1(f ) (x) = n 1 X m=0 Pm m + 1 m X k=0 (k + 1) uk+1(f ) (x) ! + Pn n + 1 n 1X k=0 (k + 1) uk+1(f ) (x) : By Lemma 12 and (6) we get
n 1X m=0 Pmum+1(f ) (x) p( );! n 1 X m=0 Pm m + 1 m X k=0 (k + 1) uk+1(f ) (x) p( );! + Pn n + 1 n 1 X k=0 (k + 1) uk+1(f ) (x) p( );! = n 1X m=0 Pm m + 1 (m + 2) kSm+1(f ) m+1(f )kp( );! +PnkSn(f ) n(f )kp( );!
= n 1X m=0 Pm m + 1 O (m + 1) r + O Pn nr+1 = O Pn nr+1 : By (12) we have kRn(f ) Sn(f )kp( );! = 1 Pn n 1X m=0 Pmum+1(f ) (x) p( );! = 1 PnO Pn nr+1 = O n (r+1) :
Acknowledgements. This work was supported by TUBITAK Grant 114F422:
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Current address : Ahmet Testici: Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145, Balikesir, Turkey
E-mail address : testiciahmet@hotmail.com
ORCID Address: https://orcid.org/0000-0002-1163-7037
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