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DOI: 10.24193/subbmath.2018.3.08

A note on the degree of approximation of functions belonging to certain Lipschitz class by almost Riesz means

Uaday Singh and Arti Rathore

Abstract. The problem of obtaining degree of approximation for the 2π−periodic functions in the weighted Lipschitz classW(Lp, ξ(t)) (p≥1) by Riesz means of the Fourier series have been studied by various investigators under Lp−norm.

Recently, Deepmala and Piscoran [Approximation of signals(functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier series, J. Inequal.

Appl., (2016), 2016:163. DOI 10.1186/s13660-016-1101-5] obtained a result on degree of approximation for weighted Lipschitz class by Riesz means. In this note, we extend this study to the weighted Lp−norm which in turn improves some of the previous results. We also derive some corollaries from our result.

Mathematics Subject Classification (2010):42A10, 42A24, 41A25.

Keywords:Fourier series, degree of approximation, weightedLp−norm, general- ized Minkowski inequality, almost Riesz means.

1. Introduction

Letf be a 2π−periodic function belonging to the spaceLp: =Lp[0,2π](p≥1).

Then the trigonometric Fourier series off is defined as f(x)∼a0

2 +

X

k=1

(akcoskx+bksinkx) (1.1)

and thenthpartial sum of the Fourier series off, given by sn(f;x) =a0

2 +

n

X

k=1

(akcoskx+bksinkx), n∈N withs0(f;x) =a0/2,

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is called the trigonometric polynomial of degree or ordern(see [11]). Throughout this paper||.||p will denote theLp norm defined by

||f||p=



 1

R

0 |f(x)|pdx1p

, p≥1;

ess sup

0≤x≤2π

|f(x)|, p=∞. (1.2)

The following subclasses ofLp[0,2π]−space are well known in the literature.

A functionf ∈Lipα,if

f(x+t)−f(x) =O(tα), for 0< α≤1, t >0.

A functionf ∈Lip(α, p),if

||f(x+t)−f(x)||p=O(tα), for 0< α≤1, p≥1, t >0.

Given a positive increasing functionξ(t),a functionf ∈Lip(ξ(t), p),if

||f(x+t)−f(x)||p=O(ξ(t)), forp≥1, t >0, andf ∈W(Lp, ξ(t)),if

f(x+t)−f(x)

sinβx 2

p

=O(ξ(t)), forβ ≥0, p≥1, t >0. (1.3) We have the following inclusions:

Lipα⊆Lip(α, p)⊆Lip(ξ(t), p)⊆W(Lp, ξ(t))

for all 0< α≤1 andp≥1. Khan [2] was the first to use the weight function of the form sinβp(x/2).

We obtain the degree of approximation of a function f ∈ Lp−space by a trigono- metric polynomial τn(f;x) of degree n in Lp−norm by measuring the deviation

||τn(f;x)−f(x)||p.This method of approximation is called the trigonometric Fourier approximation. Theτn(f;x) is called Fourier approximant off.

A bounded sequence{sn}is said to be almost convergent to a limits,if

n→∞lim Sn,m= lim

n→∞

1 n+ 1

m+n

X

k=m

sk

=s (1.4)

uniformly with respect tom(see [4]).

It can be easily verified that a convergent sequence is almost convergent and both the limits are same.

An infinite seriesP

un with the sequence of partial sums {sn} is said to be almost Riesz summable tos, if

τn,m= 1 Pn

n

X

k=0

pkSk,m→s as n→ ∞ (1.5)

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uniformly with respect tom(see [9]).

We also use the following notations

ψ(x, t) =ψ(t) =f(x+t)−2f(x) +f(x−t), andMn,m(t) = 1

n

X

k=0

pk (k+ 1)Pn

sin((k+ 2m+ 1)2t) sin((k+ 1)2t)

sin2(t/2) . (1.6)

2. Known results

Lorentz [4] was the first who introduced the concept of almost convergence of sequences. King [3] investigated the regularity conditions for the almost summability matrices. Mazhar and Siddiqui [5] applied the concept of almost convergence of se- quences to almost convergence of trigonometric sequences. In [7], Nanda introduced the spaces of strongly almost summable sequence spaces which happened to be com- plete paranormed spaces under certain conditions. The concept of almost convergence led to the formulation of various almost summability methods. After the definition of almost summability methods, Sharma and Qureshi [9] and Qureshi [8] determined the degree of approximation of certain functions by almost Riesz and almost N¨orlund means of their Fourier series. Working in the same direction, Mishra et. al. [6] deter- mined the degree of approximation of functions belonging toLip(α, p) class by almost Riesz means.

Recently, Deepmala and Piscoran [1] proved a theorem on the degree of approxima- tion for functions belnging to W(Lp, ξ(t))(p ≥ 1)−class using almost Riesz means of its Fourier series with non-negative, non-decreasing weightspn. They proved the following theorem:

Theorem 2.1. [1] Assume f is a 2π−periodic signal (f unction) and integrable in the sense of Lebesgue over [0,2π]. Then the degree of approximation of f ∈ W(Lp, ξ(t)) (p ≥ 1)−class with 0 ≤ β ≤ 1−1/p by an almost Riesz means of its Fourier series is given by

||τn,m(f;x)−f(x)||p=O

Pnβ+1/pξ(Pn−1)

, ∀n >0, (2.1) provided that the positive increasing function ξ(t)has the following features:

{ξ(t)/t}is non-increasing in t, (2.2) Z π/Pn

0

x(t)|

ξ(t) p

sinβp(t/2)dt

!1/p

=O(1) (2.3)

and

Z π π/Pn

t−δx(t)|

ξ(t) p

dt

!1/p

=O(Pnδ), (2.4)

whereδis an arbitrary number such that(β−δ)q−1>0, p−1+q−1= 1, 1≤p≤ ∞, and conditions(2.3) and(2.4) holds uniformly inx.

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Remark 2.2. We note that in the statement of Theorem 2.1, the authors have taken p≥1,but in the proof they have used the H¨older’s inequality for p >1. Therefore, the proof is not valid forp= 1.

Remark 2.3. Forp=∞,conditions (2.3) and (2.4) will not hold in the present form.

Remark 2.4. In view of the remarks of Zhang [[10],p.1140], we note that the assump- tion conditions 0≤β ≤1−1/pwith 1/p+ 1/q= 1 and (β−δ)q−1>0 of Theorem 2.1 imply thatδ <0. In this case, from condition (2.4), Theorem 2.1 is true for the functionf which is a constant almost everywhere and thus the result is trivial.

3. Reformulation of the problem and main result

Being motivated by the above remarks, we reconsider the problem of Theorem 2.1 and note that the authors defined the function classW(Lp, ξ(t))(p≥1) with the weight function sinβp(x/2) whereas the deviation ||τn,m(f;x)−f(x)||p is measured in ordinary Lp−norm. Actually, the function class W(Lp, ξ(t)) defined in (1.3) is a subclass of the weightedLp[0,2π]−space with the weight functionsinβp(x/2),so it is pertinent to measure the deviation in the weighted norm defined as

||f||p,β = 1

2π Z

0

|f(x)|psinβp(x/2)dx 1/p

, p≥1. (3.1)

In this paper, we reformulate the problem of Theorem 2.1 for the almost Riesz means and measure the deviation in the weighted norm defined in (3.1). More precisely, we prove

Theorem 3.1. Let f be a2π−periodic function in W(Lp, ξ(t))(p≥1)−class and let {pn} be a non-negative, monotonic sequence such that

(n+ 1) max{p0, pn}=O(Pn). (3.2) Then the degree of approximation off by almost Riesz means of its Fourier series is given by

||τn,m(f;x)−f(x)||p,β =O

ξ π

n+ 1

+ (n+ 1)−σ

, (3.3)

where the positive increasing functionξ(t)satisfies the condition

t−σξ(t)is non-decreasing for some0< σ <1. (3.4) Note that the conditions (2.3) and (2.4) of Theorem 2.1 have been relaxed in Theorem 3.1. Also, we prove the theorem for both non-decreasing and non-increasing sequence{pn} with condition (3.2).

4. Lemmas

We need the following lemmas for the proof of our theorem:

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Lemma 4.1. Let Mn,m(t)be given by (1.6). Then

Mn,m(t) =O(n+ 1), for 0< t≤ π (n+ 1).

Proof. For 0< t≤n+1π ,using sin(t/2)≥t/π and sinnt≤nsint,we have

|Mn,m(t)|=

1 2π

n

X

k=0

pk

(k+ 1)Pn

sin((k+ 2m+ 1)2t)sin((k+ 1)t2) sin2(2t)

≤ 1 2πPn

n

X

k=0

pk

(k+ 1)

(k+ 1)(k+ 2m+ 1) sin2(t2) sin2(2t)

= 1

2πPn

n

X

k=0

pk(k+ 2m+ 1)

=O(n+ 1).

Lemma 4.2. Let Mn,m(t)be given by (1.6). Then

Mn,m(t) =O 1

(n+ 1)t2

, for π

(n+ 1) < t≤π.

Proof. Forπ/(n+ 1)< t≤π,using sin(t/2)≥t/π and sinnt≤nsint,we have

|Mn,m(t)|=

1 2π

n

X

k=0

pk

(k+ 1)Pn

sin((k+ 2m+ 1)2t) sin((k+ 1)2t) sin2(2t)

≤ 1 2πPn

n

X

k=0

pk (k+ 1)

(k+ 1) sin(2t) sin((k+ 2m+ 1)2t) sin(2t)

π t

= 1

2πPnt

n

X

k=0

pksin

(k+ 2m+ 1)t 2

.

Then, using condition (3.2),monotonicity of{pn} and Abel’s lemma, we have Mn,m(t)

=O 1

(n+ 1)t2

, in view of

n

X

k=0

sin (k+ 2m+ 1)t 2

=O(1/t).

Lemma 4.3. Let g(x, t)∈Lp([a, b]×[c, d]), p≥1. Then,

Z b a

Z d c

g(x, t)dt

p

dx 1/p

≤ Z d

c

Z b a

g(x, t)

pdx

!1/p dt.

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This inequality is also known as the generalized form of Minkowski’s inequality [[11], p.19].

5. Proof of Theorem 3.1

Proof.Using the integral representation ofSk,m(f;x) and definition ofτn,m(f;x) given in (1.5),we have

τn,m(f;x)−f(x) = 1 Pn

n

X

k=0

pk{Sk,m(f;x)−f(x)}

= 1

2πPn

Z π 0

ψ(t)

n

X

k=0

pk (k+ 1)

[cosmt−cos(k+m+ 1)t]

2 sin2(2t) dt

= 1

2πPn

Z π 0

ψ(t)

n

X

k=0

pk

(k+ 1)

sin((k+ 2m+ 1)t/2) sin((k+ 1)t/2) sin2(2t) dt

= Z π

0

ψ(t)Mn,m(t)dt, which on applying Lemma 4.3 gives

||τn,m(f;x)−f(x)||p,β = 1

2π Z

0

Z π 0

ψ(t)Mn,m(t)dt

p

sinβp(x/2)dx 1/p

≤ Z π

0

1 2π

Z 0

|ψ(t)|psinβp(x/2)dx 1/p

|Mn,m(t)|dt.

Using the fact thatψ(t)∈W(Lp, ξ(t)) due tof ∈W(Lp, ξ(t)), we have

||τn,m(f;x)−f(x)||p,β= Z π

0

O(ξ(t))|Mn,m(t)|dt

=O(1)

Z π/(n+1) 0

ξ(t)|Mn,m(t)|dt+ Z π

π/(n+1)

ξ(t)|Mn,m(t)|dt

=I1+I2 (say). (5.1)

Using Lemma 4.1,increasing nature ofξ(t) and the mean value theorem for integrals, we have

I1=O(1)

Z π/(n+1) 0

ξ(t)|Mn,m(t)|dt=O(1)

Z π/(n+1) 0

(n+ 1)ξ(t)dt

=O

ξ π

n+ 1

. (5.2)

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Using Lemma 4.2, condition (3.4) and the mean value theorem for integrals, we have I2=O(1)

Z π π/(n+1)

ξ(t)|Mn,m(t)|dt

=O(1) 1 (n+ 1)

Z π π/(n+1)

tσ t2

ξ(t)

tσ dt=O(1)ξ(π)π−σ (n+ 1)

π n+ 1

σ−1

=O (n+ 1)−σ

. (5.3)

Collecting (5.1)−(5.3),we have

||τn,m(f;x)−f(x)||p,β =O

ξ π

n+ 1

+ (n+ 1)−σ

.

6. Corollaries

Forβ = 0,the weighted classW(Lp, ξ(t)) reduces toLip(ξ(t), p).Thus, we have the following corollary:

Corollary 6.1. Let f be a 2π−periodic function in Lip(ξ(t), p)(p≥ 1)−class and let {pn} be a non-negative, monotonic sequence such that

(n+ 1) max{p0, pn}=O(Pn).

Then the degree of approximation off by almost Riesz means of its Fourier series is given by

||τn,m(f;x)−f(x)||p=O

ξ π

n+ 1

+ (n+ 1)−σ

,

where the positive increasing functionξ(t)satisfies the condition t−σξ(t)is non-decreasing for some0< σ <1.

If β = 0 and ξ(t) = tα, then the weighted class W(Lp, ξ(t))(p≥ 1) reduces to the class Lip(α, p)(p ≥ 1). In this case, the function t−σξ(t) = tα−σ is increasing for 0< σ < α≤1. Thus, we have the following corollary:

Corollary 6.2. Letf be a2π−periodic function in Lip(α, p)(p≥1)−class and let{pn} be a non-negative, monotonic sequence such that

(n+ 1) max{p0, pn}=O(Pn).

Then the degree of approximation off by almost Riesz means of its Fourier series is given by

||τn,m(f;x)−f(x)||p=O (n+ 1)−σ

, 0< σ < α.

However, we can obtain the degree of approximation of a function f ∈ Lip(α, p) independently as under:

Puttingξ(t) =tαin (5.1),we have

I1=O(1)

Z π/(n+1) 0

(n+ 1)tαdt=O (n+ 1)−α

, 0< α≤1, (6.1)

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and

I2=O 1/(n+ 1) Z π

π/(n+1)

tα t2dt=

O((n+ 1)−α), 0< α <1;

O log(n+1)n+1

, α= 1. (6.2)

Combining (6.1) and (6.2), we have

||τn,m(f;x)−f(t)||p=

O((n+ 1)−α), 0< α <1;

O log(n+1)n+1

, α= 1.

If β = 0, ξ(t) = tα and p→ ∞, then the weighted classW(Lp, ξ(t))(p≥1) reduces to the class Lip(α).Thus, we have the following corollary:

Corollary 6.3. Let f be a2π−periodic function in Lip(α)class and let{pn}be a non- negative, monotonic sequence such that

(n+ 1) max{p0, pn}=O(Pn).

Then the degree of approximation off by almost Riesz means of its Fourier series is given by

||τn,m(f;x)−f(x)||=O (n+ 1)−σ

, 0< σ < α.

Independently, we can obtain

||τn,m(f;x)−f(x)||=

O((n+ 1)−α), 0< α <1;

O log(n+1)n+1

, α= 1.

References

[1] Deepmala, Piscoran, L.I.,Approximation of signals (functions) belonging to certain Lip- schitz classes by almost Riesz means of its Fourier series, J. Inequal. Appl., (2016), 2016:163.

[2] Khan, H.H., A note on a theorem of Izumi, Commun. Fac. Sci. Math, Ankara, Turkey, 31(1982), 123-127.

[3] King, J.P.,Almost summable sequences, Proc. Amer. Math. Soc.,17(1966), 1219-1225.

[4] Lorentz, G.G., A contribution to the theory of divergent series, Acta Math., 80(1948), 167-190.

[5] Mazhar, S.M., Siddiqui, A.H.,On almost summability of a trigonometric sequence, Acta Math. Hungar.,20(1969), no. 1-2, 21-24.

[6] Mishra, V.N., Khan, H.H., Khan, I.A., Mishra, L.N.,On the degree of approximation of signalsLip(α, p),(p≥1) class by almost Riesz means of its Fourier series, J. Class.

Anal.,4(2014), no. 1, 79-87.

[7] Nanda, S., Some sequence space and almost convergence, J. Austral. Math. Soc., A 22(1976), 446-455.

[8] Qureshi, K.,On the degree of approximation of a periodic functionf by almost N¨orlund means, Tamkang J. Math.,12(1981), no. 1, 35-38.

[9] Sharma, P.L., Qureshi, K., On the degree of approximation of a periodic function by almost Riesz means, Ranchi Univ. Math.,11(1980), 29-43.

[10] Zhang, R.J., On the trigonometric approximation of the generalized weighted Lipschitz class, Appl. Math. Comput.,247(2014), 1139-1140.

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[11] Zygmund, A.,Trigonometric Series, Cambridge University Press, Cambridge, 2002.

Uaday Singh

Department of Mathematics

Indian Institute of Technology Roorkee Roorkee-247667, India

e-mail:[email protected] Arti Rathore

Department of Mathematics

Indian Institute of Technology Roorkee Roorkee-247667, India

e-mail:[email protected]

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