Degree of Approximation of Functions in Lipschitz Class with Muckenhoupt Weights by Matrix Means

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Degree of Approximation of Functions in

Lipschitz Class with Muckenhoupt Weights by

Matrix Means

Uaday Singh and Shailesh Kumar Srivastava

Abstract—In this paper, we investigate the approximation properties of the matrix means of trigonometric Fourier series of f belonging to weighted Lipschitz class Lip(α, p, w) with Muckenhoupt weights generated byT ≡(an,k) under relaxed conditions. Our theorem extends some of the previous results pertaining to the degree of approximation of functions in weighted Lipschitz classLip(α, p, w)and the ordinary Lipschitz classLip(α, p).

Index Terms—Fourier series, Matrix means, Muckenhoupt class, ClassLip(α, p, w).

I. INTRODUCTION

A

MEASURABLE 2π-periodic function w : [0,2π] → [0,∞] is said to be a weight function if the set

w−1_({0_,_∞}) _{has the Lebesgue measure zero. We say that}

f ∈Lp_w[0,2π](=Lp_w), the weighted Lebesgue space of all measurable2π-periodic functions if

kfk_p,w=

Z 2π

0

|f(x)|pw(x)dx

1/p

<∞, 1≤p <∞.

Let 1 < p < ∞. A weight function w belongs to the Muckenhoupt class Ap if

sup I



 1 |I|

Z

I

w(x)dx



 

 1 |I|

Z

I

[w(x)]−1/(p−1)dx



 p−1

<∞,

where the supremum is taken over all intervalsI with length |I| ≤ 2π. The weight functions belonging to Ap class, introduced by Hunt et al. [1], play an important role in different fields of mathematical analysis.

Letw∈Ap andf ∈Lpw. The modulus of continuity of the functionf is defined by

Ω(f, δ)p,w = sup

|h|≤δ

k∆h(f)kp,w, δ >0,

where (∆hf)(x) =_h1 Rh

0 |f(x+t)−f(x)|dt. The existence of the modulus of continuity of f ∈ Lp

w follows from the boundedness of the Hardy-Littlewood maximal function in the space Lp

w [2]. The modulus of continuity Ω(f, ·)p,w defined by Ky [3] is non-decreasing, non-negative, continuous function such that

lim

δ→0Ω(f, δ)p,w= 0

Manuscript received July 05, 2013; revised October 08, 2013. This research was supported by the Council of Scientific and Industrial Research, (CSIR), New Delhi, India (Award No.-09/143(0706)/2009-EMR-I) in the form of fellowship to the second author.

U. Singh and S. K. Srivastava are with the Department of Mathematics, In-dian Institute of Technology Roorkee, Roorkee -247667 (India) (e-mails: [email protected], ph.: +919453551769, [email protected], ph.: +919760197682).

and Ω(f1+f2, ·)p,w≤ Ω(f1, ·)p,w+ Ω(f2, ·)p,w. The modulus of continuity Ω(f, ·)p,w is defined in this way, since the spaceLp

w is non-invariant, in general, under the usual shiftf(x)→f(x+h). We note that in the casew≡ 1, the modulus of continuity Ω(f, ·)p,w and the classical integral modulus of continuity wp(f, ·) are equivalent [3]. The weighted Lipschitz class Lip(α, p, w)for0< α≤1is defined by

Lip(α, p, w) ={f ∈Lpw: Ω(f, δ)p,w=O(δα), δ >0}. Forw(x) = 1∀x∈[0,2π] the weightedLip(α, p, w)class reduces to well known Lipschitz classLip(α, p).

Letf ∈Lp[0,2π] (p≥1) be a2π-periodic function. Then, forn∈N∪ {0} we write

sn(f;x) = a0/2 +

Xn

k=1(akcoskx+bksinkx) = Xn

k=0uk(f;x), u0(f;x) =s0(f;x) =a0/2, the (n+ 1)th_{partial sum of Fourier series of} _f _{at point} _x_, which is a trigonometric polynomial of order (or degree)n. The matrix means of the Fourier series off are defined by the sequence to sequence transformation

τn(f;x) =τn(x) =

Xn

k=0an,ksk(f;x), n∈N∪ {0}, (1) where T ≡ (an,k) is a lower triangular matrix with non-negative entries such that an,−1 = 0, An,k = Pn_r=kan,r andAn,0= 1 ∀n≥0. The Fourier series off is said to be

T-summable tos(x), if τn(f;x)→s(x) as n→ ∞. If for every convergent sequence {sn}, limn→∞ sn = s implies limn→∞τn = s, then matrix T is said to be regular. In particular, if

an,k =

pn−k/Pn, 0 ≤k≤n, 0, k > n,

and an,k=

pk/Pn, 0 ≤k≤n,

0, k > n, (2)

where Pn(= Pn

k=0pk 6= 0) → ∞as n → ∞ and

P−1= 0 =p−1, then the summability matrixT reduces to N¨orlund and Riesz matrices, respectively, and τn(f;x) in (1) defines corresponding meansNn(f;x)andRn(f;x). In case ofan,k = 1/(n+ 1)for 0≤k≤n andan,k = 0 for

k > n, theτn(f;x) reduces to Ces`aro means of order one, denoted byσn(f;x).

A positive sequence a = {an,k} is called almost monotonically decreasing with respect to k, if there exists a constant K = K(a), depending on the sequence a only,

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such that an,p ≤ Kan,m for all p ≥ m and we write that a ∈ AM DS. Similarly a = {an,k} is called almost monotonically increasing with respect tok, ifan,p≤Kan,m for all p ≤ m and we write that a ∈ AM IS [4, 5]. We note that every monotone sequence is an almost monotone sequence.

We also write ∆kan,k =an,k−an,k+1 and[x], the greatest integer contained in x.

The Fourier series and trigonometric polynomials play an important role in various scientific and engineering fields, e.g., Lo and Hui [6] use the Fourier series expansion in a very nice way. Based upon the Fourier series expansion, they propose a simple and easy-to-use approach for computing accurate estimates of Black-Scholes double barrier option prices with time-dependent parameters.

Chandra [7] has studied the approximation properties of the meansNn(f;x)andRn(f;x)inLip(α, p),1≤p <∞, 0< α≤1 with monotonicity conditions on the means gen-erating sequence {pn} and proved kNn(f;x)−f(x)kp =

O(n−α_{) =} _k_R

n(f;x)−f(x)kp, n = 1,2,3.... Mittal et al. [8] generalized the paper of Chandra [7] partially, and extended its Theorem 1 and Theorem 2 (ii) to matrix means with|Pn_k=0an,k−1|=O(n−α). On the other hand, Leindler [4] has relaxed the condition of monotonicity on {pn} and proved some of the results of Chandra [7] for almost monotone weights{pn}. Mittal et al. [5] extended the results of Leindler [4] to matrix means with almost monotone sequence {an,k} and row sums 1. Following [9], [10] and [11], recently Guven [12] has extended some of the results of Chandra [7] in another direction. He has extended Lipschitz class Lip(α, p) to Lip(α, p, w), and proved the weighted version of the Theorem 1 and Theorem 2 of Chandra [7] for 1 < p < ∞, 0 < α ≤1 i.e., for f ∈Lip(α, p, w)and monotone{pn} he proved

kNn(f;x)−f(x)k_p,w

= O(n−α) =kRn(f;x)−f(x)kp,w, n= 1,2,3, ... . Following Mittal et al. [5], very recently, Singh and Sonker [13] have studied the degree of approximation of periodic functions in generalized H¨older metric space through matrix means of Fourier series, where matrix

T ≡ (an,k) has almost monotone rows. On the other hand, Guven [14, Theorem 1 and Theorem 2] has given the weighted version of [4] and some of the results of [8] by assuming T ≡ (an,k) almost monotone with |Pn_k=0an,k−1| = O(n−α). We note that condition |Pn_k=0an,k−1| =O(n−α) was not used by Leindler [4] as Pn_k=0an,k = 1 for N¨orlund matrix. It also appears that the author in [14] have followed many conditions and calculations as given in [5] and [13] without citing them.

Apart from this Nigam and Sharma [15] applied the concept of (C,1)(E, q) summability method and establish a new theorem on degree of approximation of a function

f ∈ Lip(ξ(t), r) class. Very recently, the authors in [16] obtained the degree of approximation of functions conju-gate to the function belonging to weighted Lipschitz class

W(Lp, ξ(t))and find the error of approximation free fromp

and sharper that the earlier obtained results in this direction.

II. MAINRESULT

In the present paper, we continue the work of [5] and [13] and prove weighted version of the theorem of [5] for

p >1 which extend the result of Leindler [4] to weighted version as well as matrix version forp >1. Our theorem also extends theorems of Guven [12] to matrix means τn(f;x) under the relaxed conditions of monotonicity and replaces the two theorems of Guven [14] by a single theorem for Pn

k=0an,k = 1. More precisely, we prove:

Theorem 1. Let f ∈Lip(α, p, w), p > 1, w∈Ap and let

T ≡ (an,k) be an infinite lower triangular regular matrix and satisfies one of the following conditions:

(i) 0< α <1,{an,k} ∈AM IS ink,

(ii) 0< α <1,{an,k} ∈AM DS ink and(n+ 1)an,0=

O(1),

(iii) α= 1andPn_k=0−1(n−k)|∆kan,k|=O(1),

(iv) α= 1,Pn_k=0|∆kan,k|=O(an,0)with(n+ 1)an,0=

O(1),

(v) 0< α≤1, Pn_k=0−1

∆k

_A

n,0−An,k+1 k+1

=O

1 n+1

.

Then

kf(x)−τn(f;x)kp,w=O((n+ 1)

−α

), n= 0,1,2... (3) III. LEMMAS

To prove our theorem, we need the following lemmas.

Lemma 1 ([12]). Let1< p <∞,w∈Ap and 0< α≤1. Then the estimate

kf(x)−sn(f;x)kp,w =O((n+ 1)

−α

), n= 0,1,2, ..., (4) holds for everyf ∈Lip(α, p, w).

Lemma 2 ([12]). Let 1 < p <∞ and w ∈ Ap. Then, for

f ∈Lip(1, p, w)the estimate

ksn(f;x)−σn(f;x)kp,w=O((n+ 1)

−1₎_{, n}_{= 0}_,₁_,₂_{, ... ,} (5) holds.

Lemma 3 ([5]). Let either {an,k} ∈ AM IS or {an,k} ∈

AM DS with(n+ 1)an,0=O(1). Then, for0< α <1,

Xn

k=0(k+ 1)

−α

an,k=O((n+ 1)− α

).

Proof.Letr= [n/2]and{an,k} ∈AM IS, then Pn

k=0(k+ 1)

−α_a n,k

≤Kan,r

Xr

k=0(k+ 1)

−α_{+ (}_r_{+ 1)}−αXn

k=r+1an,k ≤ Kan,r(r+ 1)1−α+ (r+ 1)−α

Xn

k=0an,k ≤ K(r+ 1)−α(r+ 1)an,r+ (r+ 1)−αAn,0 = O(r+ 1)−α=O((n+ 1)−α),

in view of(r+1)an,r≤(n−r+1)an,r≤K(an,r+an,r+1+

...+an,n)≤An,0 and(r+ 1)−α=O((n+ 1)−α). If{an,k} ∈AM DS and(n+ 1)an,0=O(1), then

Xn

k=0(k+ 1)

−α_a

n,k ≤ Kan,0

Xn

k=0(k+ 1)

−α

= O((n+ 1)−α).

This completes the proof of Lemma 3.

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IV. PROOF OFTHEOREM1

We prove the cases (i) and (ii) together by using Lemma 1 and Lemma 3. Since

τn(f;x)−f(x) =

Xn

k=0an,k{sk(f;x)−f(x)}, kτn(f;x)−f(x)kp,w ≤

Xn

k=0an,kksk(f;x)−f(x)kp,w = Xn

k=0an,kO(k+ 1)

−α

= O((n+ 1)−α).

Next, we consider the case (iv). By Abel’s transformation andan,n+1= 0,

τn(f;x) =

Xn

k=0an,ksk(f;x) = Xn

k=0an,k

Xk

i=0ui(f;x)

= Xn

k=0An,kuk(f;x), and thus

sn(f;x)−τn(f;x) =

Xn

k=0(1−An,k)uk(f;x) =Xn

k=1k

−1₍_A

n,0−An,k)kuk(f;x). Hence, again by Abel’s transformation and An,n+1= 0, we get

sn(f;x)−τn(f;x) =

Xn

k=1(∆kk

−1₍_A

n,0−An,k))×

Xk

i=1iui(f;x) + (n+ 1)

−1Xn

k=1kuk(f;x). Therefore,

ksn(f;x)−τn(f;x)kp,w ≤Xn

k=1

∆kk−1(An,0−An,k)

Xk

i=1iui(f;x) _p,w

+ (n+ 1)−1

Xn

k=1kuk(f;x)

_p,w. (6)

Also

sn(f;x)−σn(f;x) = (n+ 1)−1Xn

k=0((n+ 1)uk(f;x)−sk(f;x)) = (n+ 1)−1Xn

k=1kuk(f;x), which implies

Xn

k=1kuk(f;x)

_p,w= (n+ 1)kσn(f;x)−sn(f;x)kp,w = O(1), (7) in view of Lemma 2.

Using (7) in (6), we get ksn(f;x)−τn(f;x)kp,w≤

Xn

k=1

+(n+ 1)−1. (8) Now,

∆kk−1 (An,0−An,k) = An,0−An,k

k −

An,0−An,k+1

k+ 1

= k−1(k+ 1)−1(An,0−An,k−kan,k) = k−1(k+ 1)−1(Xk−1

i=0 an,i−kan,k). (9)

Next, we shall verify by induction that,

Xk−1

i=0 an,i−kan,k

≤Xk

i=1i|an,i−1−an,i|. (10) Fork= 1, we have

Xk−1

i=0 an,i−kan,k

=|an,0−an,1|= 1.|an,0−an,1|,

i.e., (10) is true for k= 1. Let us assume that (10) is true fork=m, then fork=m+ 1,

| Xm

i=0an,i−(m+ 1)an,m+1| =

Xm−1

i=0 an,i+an,m+man,m −man,m−(m+ 1)an,m+1| ≤

Xm−1

i=0 an,i−man,m

+ (m+ 1)|an,m−an,m+1| = Xm

i=1i|an,i−1−an,i|+ (m+ 1)|an,m−an,m+1| = Xm+1

i=1 i|an,i−1−an,i|.

Thus (10) is true for k = m+ 1, hence (10) is true for 1≤k≤n.

Using (9) and (10), we get Pn

k=1

≤ Xn k=1k

−1₍_k_{+ 1)}−1Xk

i=1i|an,i−1−an,i| ≤ Xn

i=1i|an,i−1−an,i|

X∞

k=ik

−1₍_k_{+ 1)}−1 = Xn

i=1|an,i−1−an,i|= Xn−1

k=0|∆kan,n−k| = O(an,0) =O((n+ 1)−1). (11) Combining (8) and (11), we get

ksn(f;x)−τn(f;x)k_p,w =O((n+ 1)−1). (12) Using Lemma 1 and (12), we have forα= 1

kf(x)−τn(f;x)kp,w ≤ kf(x)−sn(f;x)kp,w +ksn(f;x)−τn(f;x)kp,w =O((n+ 1)

−1₎_.

Herewith the case (iv) is proved.

For the proof of case (iii), we first verify that the condition Pn−1

k=0(n−k)|∆kan,k|=O(1), implies that

Xn

k=1

=O((n+ 1)−1). (13) From (9), we can write

Xn

k=1 | ∆kk

−1₍_A

n,0−An,k)| ≤ Xn

k=1k

−1₍_k_{+ 1)}−1Xk

i=1i|an,i−1−an,i| = Xn

k=1∆(k

−1 )Xk

i=1i|an,i−1−an,i|.

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By Abel’s transformation, we have

Xn

k=1|∆kk

−1₍_A

n,0−An,k)| ≤ Xn+1

k=1k

−1_.k_|_a

n,k−1−an,k| − 1

n+ 1

Xn+1

k=1k|an,k−1−an,k| = Xn+1

k=1

₁

k−

1

n+ 1

k|an,k−1−an,k| = Xn+1

k=1

_n₋_k_{+ 1}

n+ 1

|an,k−1−an,k| = Xn

k=0

_n

−k n+ 1

|an,k+1−an,k|

≤ 1

n+ 1 Xn−1

k=0(n−k)|∆kan,k|=O((n+ 1)

−1₎_, which verifies (13). Combining (8), (13) and Lemma 2, we get (3) forα= 1.

Finally, we prove the case (v). Using Lemma 1 and Abel’s transform

kτn(f;x)−f(x)kp,w≤

Xn

k=0an,kksk(f;x)−f(x)kp,w = O{Xn

k=0(k+ 1)

−α_a n,k}

= O{Xn−1

k=0∆k(k+ 1)

−α₍Xk i=0an,i) +(n+ 1)−αXn

i=0an,i} = O{Xn−1

k=0(An,0−An,k+1){(k+ 1)

−α₋₍_k_{+ 2)}−α_} +(n+ 1)−αAn,0}

= O{Xn

k=0(k+ 1)

−α₍_A

n,0−An,k+1)/(k+ 1)} +O(n+ 1)−α, (14) where by Abel’s transformation

Pn

k=0 (k+ 1)

−αAn,0−An,k+1

k+ 1 = Xn−1

k=0∆k

_A

n,0−An,k+1

k+ 1

Xk

i=0(i+ 1)

−α

+An,0−An,n+1

n+ 1

Xn

i=0(i+ 1)

−α

≤ Xn−1 k=0∆k

An,0−An,k+1

k+ 1

(k+ 1)1−α +(n+ 1)

1−α

n+ 1 ≤ (n+ 1)1−αXn−1

k=0∆k

_A

n,0−An,k+1

k+ 1

+(n+ 1)−α=O((n+ 1)−α), (15) in view of An,n+1= 0 and condition (v) of Theorem 1. Collecting (14) and (15), we get (3). Thus proof of Theorem 1 is complete.

V. COROLLARIES

In order to justify the significance of our result, we prove that the following results are the particular cases of Theorem 1 for p >1. We also drive an analogous result of Theorem 1 for monotone {an,k}.

1) If we takean,k =pn−k/Pn fork≤nandan,k = 0for

k > n, then conditions (i) to (iv) of Theorem 1 reduce

to conditions (i) to (iv) of Theorem 1 of Leindler [4, p. 131], respectively, and τn(f;x) means reduces to Nn(f;x) means. Further, we note that Lip(α,p,1) ≡

Lip(α,p), p > 1. Thus our theorem generalizes Theo-rem 1 of [4], except for the casep= 1in two directions. 2) SinceLip(α,p,1)≡Lip(α,p), p >1and conditions of Theorem 1 of Mittal et al. [5, p. 4485] forp > 1 are included in the conditions (i) to (iv) of Theorem 1, our theorem includes weighted version of Theorem 1 of [5] forp >1.

3) Since every monotone sequence is almost monotone, the conditions (i) and (ii) of Theorem 1 are satisfied in case of monotonic {an,k}. Further, every sequence {an,k}non-decreasing with respect tokalways satisfies condition (iii) of Theorem 1, e. g.,

Pn−1

k=0 (n−k)|∆kan,k| = Xn−1

k=0(n−k)(an,k+1−an,k) = An,0−(n+ 1)an,0=O(1).

If{an,k}is non-increasing with respect to k, then (iv) of Theorem 1 is also true, e. g.,

Xn−1

k=0|∆kan,k| =

Xn−1

k=0(an,k−an,k+1) = an,0−an,n ≤an,0. Thus, we have the following analogous result of Theo-rem 1 for monotone{an,k}:

Theorem 2. Letf ∈Lip(α, p, w),p >1,w∈Ap and letT ≡(an,k)be an infinite regular triangular matrix and satisfies one of the following conditions:

(i) {an,k}is non-decreasing in k,

(ii) {an,k} is non-increasing ink and(n+ 1)an,0 =

O(1). Then (3) holds.

4) If we take an,k = pn−k/Pn for k ≤ n andan,k = 0 for k > n, then Theorem 2 reduces to Theorem 1 of Guven [12, p. 101].

5) Finally, if we takean,k =pk/Pnfork≤nandan,k = 0 for k > n, thenτn(f;x)means reduces to Rn(f;x) means; and

An,0−An,k+1 =

Xn

i=0an,i−

Xn

i=k+1an,i = Xn

i=0pi−

Xn

i=k+1pi

/Pn = Xk

i=0pi/Pn =Pk/Pn, so that

∆k

_A

n,0−An,k+1

k+ 1

= An,0−An,k+2

k+ 2 −An,0−An,k+1

k+ 1 = 1

Pn

_P

k+1

k+ 2 −

Pk

k+ 1

,

i. e., condition (v) of Theorem 1 reduces to condition (3) of Guven [12, Theorem 2]. Thus Theorem 1 under condition (v) extends Theorem 2 of Guven [12] to matrix means.

6) If we takean,k =Aβn−−1k /A β

n for k≤n andan,k = 0 fork > n(β >0), where

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Aβ₀ = 1, Aβ_k =β(β+ 1)...(β+k)

k! , k≥1, then matrix meansτn(f;x)reduces to Ces`aro means of orderβ >0 denoted byσβ

n(f;x) and defined as

σ_nβ(f;x) = 1

Aβn

Xn

k=0A β−1

n−ksk(f;x).

Hence, Corollary 3 of Guven [12, p. 102] can also be derived from Theorem 1.

7) In the light of remark of Guven [12, p. 102], we note that Theorems 1 and 2 also hold in reflexive weighted Orlicz spacesLM

w, which are discussed in [10] in detail. VI. CONCLUSION

Theorems of this paper are an attempt to formulate the problem of approximation of f ∈ Lip(α, p, w), p > 1 through trigonometric polynomials generated by the summa-bility means of the Fourier series off in a simpler manner. The case for p= 1 is still an open problem which can be addressed by making certain modifications in the definition of Muckenhoupt classAp.

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