823–833 DOI: 10.18514/MMN.2018.2233 APPROXIMATION BY N ¨ORLUND AND RIESZ TYPE DEFERRED CES `ARO MEANS IN THE SPACE HP

Vol. 19 (2018), No. 2, pp. 823–833 DOI: 10.18514/MMN.2018.2233

APPROXIMATION BY N ¨ORLUND AND RIESZ TYPE DEFERRED CES `ARO MEANS IN THE SPACE H_P^.!/

U ˇGUR DE ˇGER AND HILAL BAYINDIR Received 15 February, 2017

Abstract. The deferred Ces`aro transformations which have useful properties not possessed by the Ces`aro transformation was considered by R.P. Agnew in [1]. In [9], Deˇger and K¨uc¸¨ukaslan introduced a generalization of deferred Ces`aro transformations by taking account of some well known transformations such as Woronoi-N¨orlund and Riesz, and considered the degree of ap- proximation by the generalized deferred Ces`aro means in the spaceH.˛; p/,p1,0 < ˛1 by concerning with some sequence classes. In 2014, Nayaket al.studied the rate of convergence problem of Fourier series by deferred Ces`aro mean in the spaceHp^.!/introduced by Daset al.

in [7].

In this study we shall give the degree of approximation by the generalized deferred Ces`aro means in the spaceH_p^.!/. Therefore the results given in [17] are generalized according to the summability method.

2010Mathematics Subject Classification: 40G05; 41A25; 42A05; 42A10 Keywords: deferred Ces`aro means, N¨orlund means, generalized H¨older metric

1. INTRODUCTION

Letf be a2 periodic function andf 2LpWDLp.0; 2/forp1and let sn.fIx/D1

2a0C

n

X

kD1

.akcoskxCbksinkx/

n

X

kD0

Ak.fIx/

partial sum of the first.nC1/terms of the Fourier series off 2Lp.p1/at a point x.

The degree of approximation of the sums s_n.fIx/ in different spaces has been studied by many authors. In [20], Quade investigated approximation properties of the partial sums of Fourier series inLp norms. Chandra in [5] and Leindler in [13]

generalized the results of Quade using the Woronoi-N¨orlund and the Riesz means of Fourier series for some sequence classes. A similar problems were studied for more general means in [8] and [15].

c 2018 Miskolc University Press

The space

H_˛D ff 2C₂ W jf .x/ f .y/j Kjx yj^˛; 0 < ˛1g

whereC2 is space of all2-periodic and continuous functions defined on Rwith the supremum norm andK is a positive constant, not necessarily the same at each occurrence, is a Banach space (see Pr¨osdorff, [19]) with the normk k^˛defined by

kfk^˛D kfk^CCsup

x¤y

^˛f .x; y/ (1.1)

where

^˛f .x; y/D jf .x/ f .y/j

jx yj^˛ .x¤y/;

by convention⁰f .x; y/D0and

kfk^C D sup

x2Œ ;jf .x/j:

The metric generated by the norm (1.1) onH˛is called the H¨older metric. Pr¨osdorff has studied the degree of approximation by Ces`aro means in the H¨older metric and proved the following theorem.

Theorem 1([19]). Letf 2H˛.0 < ˛1/and0ˇ < ˛1:Then kn.f / fkˇ DO.1/

n^{ˇ ˛} ; 0 < ˛ < 1I n^{ˇ 1}lnn ; ˛D1 wheren.f /is the Ces`aro means of the Fourier series off.

The case ˇD0 in Theorem 1 is due to Alexits [2]. Leindler introduced more general classes than the H¨older classes of2-periodic continuous functions, gener- alized the results of Pr¨osdorff [11]. Chandra obtained a generalization of Theorem1 by considering the Woronoi-N¨orlund transform [4]. In [16] Mohapatra and Chandra considered the problem by a matrix means of the Fourier series off 2H˛.

Further generalizations of the H¨older metric was given in [6] and [7]. In [6], Das et al. studied the results regarding the degree of approximation by infinite matrix means involved in the deferred Ces`aro means in a generalized H¨older metric.

In [9], the authors considered the degree of approximation to functions in this space with respect to the norm in the space given in [6] by the deferred Woronoi- N¨orlund means and the deferred Riesz means of the Fourier series of the functions.

The modulus of continuity off 2C2 is defined by

!.f; ı/WD sup

0<jhjı

jf .xCh/ f .x/j: According to this, the class of functionsH^!is defined by

H^!WD ff 2C2 W!.f; ı/DO.!.ı//g where!.ı/is a modulus of continuity.

There are numerous papers related to the degree of approximation inH^! space such as Leindler [11], Mazhar and Totik [14], Daset al.[7], Khatri and Mishra [10].

A further generalization ofH^!space has been given by Das, Nath and Ray in [7].

They defined the following notations: Iff 2Lp.0; 2/,p1, then denote H_p^.!/WD ff 2Lp.0; 2/; p1WA.f; !/ <1g

where!is a modulus of continuity and A.f; !/WDsup

t¤0

jjf . Ct / f ./jjp

!.jtj/

wherek:kpwill denoteLp-norm with respect toxand is defined by kfkpWD

1 2

Z 2

0 jf .xj^pdx

1 p

: The norm in the spaceHp^.!/is defined by

jjfjjp^.!/WD jjfjj^pCA.f; !/:

Given the spacesH_p^.!/andH_p^.v/, if ^{!.t /}_{v.t /} is nondecreasing , then H_p^.!/H_p^.v/Lp p1

since

jjfjjp^.v/max

1;!.2/

v.2/

jjfjjp^.!/:

Especially, the leading studies related to degree of approximation inH_p^.!/ space can be found in [7] and [12].

2. DEFERREDCESARO MEAN AND ITS GENERALIZATION`

The concept of deferred Ces`aro mean has been given by Agnew as following (see [1]). LetaD.an/andbD.bn/be sequences of nonnegative integers with conditions an< bn nD1; 2; 3; ::: (2.1) and

nlim!1bnD C1: (2.2)

The deferred Ces`aro mean,D_a^b, determined byaandb DnDD_a^bDSa_nC1CSa_nC2C CS_bn

bn an D 1

bn an bn

X

kDanC1

S_k; where.Sk/is a sequence of real or complex numbers.

Since each D_a^b with conditions (2.1) and (2.2) satisfies the Silverman-Toeplitz conditions, every D_a^b is regular. Note that D_a^b involves, except in caseanD0all n, means of deferred elements of .Sn/. It is also known that D_{n 1}ⁿ is the identity transformation andD₀ⁿis the.C; 1/transformation. The basic properties ofD^b_a can be found in [1]. By considering deferred Ces`aro means, Deˇger and K¨uc¸¨ukaslan gave the following notations with conditions (2.1) and (2.2) in [9].

Let.pn/be a positive sequence of real numbers. Then D^b_aNn.fIx/D 1

P₀^{bn an 1}

bn

X

mDanC1

p_{bn m}Sm.fIx/;

and

D_a^bRn.fIx/D 1 P_a^bn

nC1 bn

X

mDa_nC1

pmSm.fIx/

where

P₀^{bn an 1}D

bn an 1

X

kD0

p_k ¤0; P_a^bn

nC1D

bn

X

kDanC1

p_k¤0 and

Sn.fIx/D 1

Z

f .xCt / Dn.t / dt;

in which

Dn.t /Dsin.nC¹2/t 2sin ₂^t :

These two methods are calleddeferred Woronoi-N¨orlund means,.D_a^bN; p/, and deferred Riesz means, .D_a^bR; p/, with respect to Sm.fIx/, respectively. In case bnDnandanD0, the methodsD_a^bNn.fIx/andD_a^bRn.fIx/give us the classicaly known Woronoi-N¨orlund and Riesz means, respectively. Provided thatpnD1for all n.0/, both of them yield deferred Ces`aro means

D_a^b.fIx/D 1 bn an

b_n

X

mDanC1

Sm.fIx/

ofS_m.f; x/.

In addition to this, if bnDn, anD0 andp_k D1 for these two methods, then they coincide with Ces`aro method C1. In the event that anD0, .bn/ is a strictly increasing sequence of positive integers withb.0/D0andp<sub>k</sub>D1, then they give us Ces`aro submethod which is obtained by deleting a set of rows from Ces`aro matrix (see [3,8,18]).

There exist some inclusions that established relation between deferred method and Ces`aro method. Before giving this inclusions without detail, we need to give some definition. If.an/and.bn/satisfy, in addition to (2.1) and (2.2), the condition

an

bn anDO.1/ (2.3)

for alln, then (D^b_a) is properly deferred ; such a transformation is called a proper (D_a^b) (see [1]). Therefore we know that

.C; 1/D.an; bn/ if and only if .D^b_a/ is properly deferred, and

D.an; n/.C; 1/:

On the other hand,

D.an; n/.C; 1/ if and only if .D_a^b/ is properly deferred.

In2014, Nayaket al. has studied the rate of convergence problem of Fourier series by deferred Ces`aro mean in the generalized H¨older metric (H_p^.w/) and gave the fol- lowing theorem.

Theorem 2 ([17]). Let v and w be moduli of continuity such that ^w_v is non- decreasing andf 2H_p^.w/; p1. Let

qnD.2jC1/pnC2j where j is a positive integer. Then

.i /kDn.Sn.fI:// f .:/kp^.v/DO 1

pnC1

C O.1/

.pnC1/² Z

.pnC1/

w.t / t³v.t /dt:

If in addition _{t v.t /}^{w.t /} is non-increasing then

.i i /kDn.Sn.fI:// f .:/kp^.v/D O.1/

.pnC1/² Z

.pnC1/

w.t / t³v.t /dt and a fortiori

.i i i /kDn.Sn.fI:// f .:/kp^.v/DO

w.=.pnC1//

v.=.pnC1//

:

Taking into account of this theorem and the methods given in [9], we shall improve the results in [17] on the degree of approximation by the generalized deferred Ces`aro means in the spaceH_p^!. Therefore the results given in [17] are generalized according to the summability method.

3. THEOREMS

In this section we will give two theorems on the degree of approximation by de- ferred Woronoi-N¨orlund means and deferred Riesz means in generalized H¨older met- ric.

Theorem 3. Letvandwbe moduli of continuity such that^w_v is nondecreasing and f 2H_p^.w/; p1. LetbnD.2jC1/anC2j where j is a positive integer. Moreover let.pn/be a positive sequence and the conditions

.bn an/p_bnDO.P_a^bn

nC1/ (3.1)

and

bn 1

X

mDanC1

jm.pm/j DO.jp_bn panC1j/ (3.2) are satisfied wherem.pm/Dpm pmC1. Then

kD_a^bRn.fI:/ f .:/kp^.v/

DO

1Cjp_bn pa_nC1j pb_n

( 1

anC1C 1 .anC1/²

Z

.anC1/

w.t / t³v.t /dt

)

Proof. By definition of the deferred Riesz means, we have D^b_aRn.fI:/ f .:/D 1

P_a^bn

nC1 b_n

X

mDanC1

pm.Sm.fI:/ f .://:

By elementary methods, we get IWDD^b_aRn.fI:/ f .:/D 1

P_a^b_nⁿ_C1

bn an 1

X

mD0

pmCanC1.SmCanC1.fI:/ f .://:

Using Abel’s transformation, we see that I D 1

P_a^b_nⁿ_C1

bn an 2

X

mD0

ŒpmCanC1 pmCanC2Œ

m

X

kD0

.S_kCanC1.fI:/ f .://

C 1 P_a^bn

nC1

bn an 1

X

kD0

Œ.S_kCanC1.fI:/ f .://p_bnDWI1CI2 (3.3) By considering the second term in right side of the above equality, we write

I2D p_bn P_a^bn

nC1

b_n a_n bn an

bn

X

kDanC1

.S_k.fI:/ f .://Dp_bn.bn an/ P_a^bn

nC1

.D_a^b.Sn.fI:// f .://:

Hence by the condition (3.1) and Theorem2-(i), we obtain kI2kp^.v/ jp_bn.bn an/

P_a^bn

nC1

jkD^b_a.Sn.fI:// f .:/kp^.v/

DO.1/kD^b_a.Sn.fI:// f .:/kp^.v/

DO 1

anC1

C O.1/

.anC1/² Z

.anC1/

w.t /

t³v.t /dt: (3.4) Now let us consider the first term in (3.3). Then we have

I1D 1 P_a^bn

nC1

bn an 2

X

mD0

Œ

m

X

kD0

.S_kCanC1.fI:/ f .://ŒpmCanC1 pmCanC2

D 1 P_a^b_nⁿ_C1

bn 1

X

mDanC1

m.pm/

m an 1

X

kD0

.S_kCanC1.fI:/ f .://

D 1 P_a^bn

nC1 bn 1

X

mDa_nC1

m.pm/m an

m an m

X

kDanC1

.S_k.fI:/ f .://

D 1 P_a^bn

nC1 b_n 1

X

mDanC1

.m an/m.pm/ŒD_a^m.Sn.fI:// f .:/ (3.5) By considering (3.5) and Theorem2-(i), we write

kI1kp^.v/ 1 P_a^bn

nC1

.bn an/

bn 1

X

mDanC1

jm.pm/jkD_a^m.Sn.fI:// f .:/kp^.v/

D.bn an/ P_a^bn

nC1

( O

1 anC1

C O.1/

.anC1/² Z

.anC1/

w.t / t³v.t /dt

) _{bn 1} X

mDanC1

jm.pm/j: Owing to condition (3.1) and condition (3.2) in the last term, we obtain

kI1kp^.v/Djpb_n panC1j pbn

( O

1 anC1

C O.1/

.anC1/² Z

.anC1/

w.t / t³v.t /dt

)

: (3.6) Combining (3.3), (3.4) and (3.6), we get the desired result . Therefore the proof is

completed.

Corollary 1. Under conditions of Theorem3, iff 2Hp^.w/forp1and _{t v.t /}^{w.t /} is nonincreasing then

(i)kD_a^bRn.fI:/ f .:/kp^.v/DO.1/

1C^jpbn_p^p_bn^anC1j

1 .anC1/²

R

.anC1/

w.t / t³v.t /dt

and furthermore

(ii)kD_a^bRn.fI:/ f .:/kp^.v/DO.1/

1C^jpbn_p^p_bn^anC1j_w.=.a

nC1//

v.=.anC1//.

Theorem3and its corollaries are very important due to the relations between Riesz type deferred method and Riesz method. Let us recall some results establishing these relations given in [9]. Suppose that the sequences.an/and.bn/satisfy the conditions (2.1) and (2.2). If the condition

p1Cp2C Cpa_n

P_a^bn

nC1

DO.1/ (3.7)

satisfy for .pn/, then we shall say that .D_a^bR; p/ is properly deferred and such a transformation is called a proper.D_a^bR; p/. We see that if.pn/D1for allnwith the conditions (2.1) and (2.2), then .D_a^bR; p/ and the condition (3.7) are reduced.D/

and the condition (2.3), respectively. In [9], we know that ”.R; p/.D_a^bR; p/if and only if.D^b_aR; p/ is proper”. On the other hand, we have ”.Dⁿ_aR; p/.R; p/” and

”.D_aⁿR; p/.R; p/if and only if.Dⁿ_aR; p/is proper”.

Theorem 4. Letv andw be moduli of continuity such that ^w_v is nondecreasing andf 2Hp^.w/,p1. Let

bnD.2j C1/anC2j

where j is a positive integer. Moreover let.p_n/be a positive sequence and the condi- tions

.bn an/p_{bn an 1}DO.P₀^{bn an 1}/ (3.8) and

bn 1

X

mDanC1

j_m.p_{bn m}/j DO.jp_{bn an 1} p₀j/ (3.9) are satisfied. Then

kD_a^bNn.fI / f ./kp^v

DO

p0C jpb_n a_n 1 p0j p_{bn an 1}

( 1

anC1C 1 .anC1/²

Z

.anC1/

w.t / t³v.t /dt

)

: (3.10) Proof. Since

JWDD_a^bNn.fI:/ f .:/D 1 P₀^{bn an 1}

b_n a_n 1

X

mD0

p_{bn m an 1}.SmCanC1.fI:/ f .://;

we get that

JD 1

P₀^{bn an 1}

bn an 2

X

mD0

Œ

m

X

kD0

.S_kCanC1 f .://Œp_{bn m an 1} p_{bn m an 2}

C 1

P₀^{bn an 1}

bn an 1

X

kD0

Œ.S_kCanC1.fI:/ f .:/p0DWJ1CJ2 (3.11) by Abel’s transformation. Let us considerJ2. Since

J2Dp0.bn an/ P₀^{bn an 1}

.D_a^b.Sn.fI:// f .://;

we have

kJ2kp^.v/O.1/ p0

p_{bn an 1} ( 1

anC1C 1 .anC1/²

Z

.anC1/

w.t / t³v.t /dt

)

(3.12) by considering (3.8) and Theorem2-(i). Let us estimateJ1. By elementary methods, we know that

J1D 1 P₀^{bn an 1}

bn 1

X

mDanC1

.p_{bn m} p_{bn m 1}/

m an 1

X

kD0

.S_kCanC1.fI:/ f .://

D 1

P₀^{bn an 1}

bn 1

X

mDanC1

m.p_{bn m}/.m an/ŒD_a^m.Sn.fI:// f .:/:

Taking into account of (3.8), (3.9) and Theorem2-(i), we write kJ1kp^.v/O

jp_{bn an 1} p0j p_{bn an 1}

( 1

a_nC1C 1 .a_nC1/²

Z

.anC1/

w.t / t³v.t /dt

)

: (3.13) Therefore, we get (3.10) by collecting of (3.11)-(3.13).

Corollary 2. Under conditions of Theorem 4, if f 2Hp^.w/, p1 and _{t v.t /}^{w.t /} is nonincreasing then

(i)kD_a^bN_n.fI:/ f .:/kp^.v/DO.1/_.a ¹

nC1/²

_p

0Cjp_{bn an 1} p₀j p_{bn an 1}

R

.anC1/

w.t / t³v.t /dt and furthermore

(ii)kD_a^bNn.fI:/ f .:/k^p.v/DO.1/.^w.=.a_v.=.a^nC1//

nC1///_p

0Cjp_{bn an 1} p0j p_{bn an 1}

.

Remark1. If we takep_kD1in Theorem3, Theorem4, Corollary1and Corollary 2, then all of results coincide with Theorem2.

4. CONCLUSION

Several studies have been carried out on the degree of approach in the Hp^.w/

spaces. The means used in these studies include the results obtained by approach- ing to functions in this space using the classically known Ces`aro, Woronoi-N¨orlund and Riesz means. These results depend either directly related to the degree of polyno- mials, or to the upper and lower limits of the method such as deferred Ces`aro means.

In this study we have shown that how the speed of the approach is being affected, based on the sequences that determines the Woronoi-N¨orlund and Riesz means by considering the deferred type of the Woronoi-N¨orlund and Riesz mean introduced in [9]. It is evident that the results generalized previous work on these spaces.

REFERENCES

[1] R. P. Agnew, “On deferred Ces´aro means,”Ann. Math. (2), vol. 33, no. 3, pp. 413–421, 1932, doi:

10.2307/1968524.

[2] G. Alexits, “Convergence problems of orthogonal series,” (International Series of Monographs on Pure and Applied Mathematics. Vol. 20.) New York-Oxford-London-Paris: Pergamon Press. IX, 350 p. (1961), 1961.

[3] D. H. Armitage and I. J. Maddox, “A new type of Ces`aro mean,”Analysis, vol. 9, no. 1-2, pp.

195–204, 1989.

[4] P. Chandra, “On the generalized Fejer means in the metric of H¨older space,”Math. Nachr, vol.

109, pp. 39–45, 1982, doi:10.1002/mana.19821090105.

[5] P. Chandra, “Trigonometric approximation of functions inLp-norm,”J. Math. Anal. Appl, vol.

275, no. 1, pp. 13–26, 2002, doi:10.1016/S0022-247X(02)00211-1.

[6] G. Das, T. Ghosh, and B. Ray, “Degree of approximation of functions by their Fourier series in the generalized H¨older metric,”Proc. Indian Acad. Sci., Math. Sci, vol. 106, no. 2, pp. 139–153, 1996, doi:10.1007/BF02837167.

[7] G. Das, A. Nath, and B. K. Ray, “An estimate of the rate of convergence of Fourier series in the generalized H¨older metric,” inAnalysis and applications. Outcome of the conference dedicated to Professor Tribikram Pati on the occasion of his 70th birthday, Ujjain, India, 1999. New Delhi:

Narosa Publishing House; Boca Raton, FL: Chapman and Hall/CRC, 2002, pp. 43–60.

[8] U. Deˇger and M. Kaya, “On the approximation by Ces´aro submethod,”Palest. J. Math, vol. 4, no. 1, pp. 44–56, 2015.

[9] U. Deˇger and M. K¨uc¸¨ukaslan, “A generalization of deferred Ces`aro means and some of their applications,”J. Inequal. Appl, vol. 2015, p. 16, 2015, doi:10.1186/s13660-014-0532-0.

[10] K. Khatri and V. N. Mishra, “Degree of approximation by the.TE¹/means of Conjugate Series of Fourier Series in the H¨older metric,”Iran J Sci Technol Trans Sci, vol. 2017, p. 9, 2017, doi:

https://doi.org/10.1007/s40995-017-0272-3.

[11] L. Leindler, “Generalizations of Proessdorf’s theorems,”Stud. Sci. Math. Hung, vol. 14, pp. 431–

439, 1979.

[12] L. Leindler, “A relaxed estimate of the degree of approximation by Fourier series in generalized H¨older metric,”Anal. Math, vol. 35, no. 1, pp. 51–60, 2009, doi:10.1007/s10476-009-0106-6.

[13] L. Leindler, “Trigonometric approximation inLp-norm,”J. Math. Anal. Appl, vol. 302, no. 1, pp.

129–136, 2005, doi:10.1016/j.jmaa.2004.07.049.

[14] S. Mazhar and V. Totik, “Approximation of continuous functions byT-means of Fourier series,”

J. Approx. Theory, vol. 60, no. 2, pp. 174–182, 1990, doi:10.1016/0021-9045(90)90082-2.

[15] V. N. Mishra and L. N. Mishra, “Trigonometric approximation of signals (functions) inLp-norm,”

Int. J. Contemp. Math. Sci, vol. 7, no. 17-20, pp. 909–918, 2012.

[16] R. Mohapatra and P. Chandra, “Degree of approximation of functions in the H¨older metric,”Acta Math. Hung, vol. 41, pp. 67–76, 1983, doi:10.1007/BF01994063.

[17] L. Nayak, G. Das, and B. Ray, “An estimate of the rate of convergence of Fourier series in the generalized H¨older metric by deferred Ces`aro mean,”J. Math. Anal. Appl, vol. 420, no. 1, pp.

563–575, 2014, doi:10.1016/j.jmaa.2014.06.001.

[18] J. A. Osikiewicz, “Equivalence results for Ces`aro submethods,”Analysis, M¨unchen, vol. 20, no. 1, pp. 35–43, 2000, doi:10.1524/anly.2000.20.1.35.

[19] S. Pr¨ossdorf, “Zur konvergenz der fourierreihen h¨olderstetiger funktionen,” Mathematische Nachrichten, vol. 69, no. 1, pp. 7–14, 1975, doi: 10.1002/mana.19750690102. [Online].

Available:http://dx.doi.org/10.1002/mana.19750690102

[20] E. Quade, “Trigonometric approximation in the mean,”Duke Math. J, vol. 3, pp. 529–543, 1937, doi:10.1215/S0012-7094-37-00342-9.

Authors’ addresses

Uˇgur Deˇger

Mersin University, Faculty of Science and Literature, Department of Mathematics, 33343, Mersin, Turkey

E-mail address:degpar@hotmail.com(udeger@mersin.edu.tr)

Hilal Bayindir

Mersin University, Institute of Science and Literature, Department of Mathematics, 33343, Mersin, Turkey

E-mail address:hilalbayindir2@gmail.com