2. Approximation of functions by matrix-Euler summability means of Fourier series in generalized Holder metric

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 3 (2012.), Pages 12-22

APPROXIMATION OF FUNCTIONS BY MATRIX-EULER SUMMABILITY MEANS OF FOURIER SERIES IN

GENERALIZED H ¨OLDER METRIC

(COMMUNICATED BY HU¨SEIN BOR)

SHYAM LAL AND SHIREEN

Abstract. In this paper, a new estimate for the degree of trignometric ap-proximation of a function f ∈ Hr(w), (r ≥ 1) class by Matrix-Euler means (∆.E1) of its Fourier Series has been determined.

1. Introduction

The degree of approximation of a functionf belonging toLipαclass by N¨orlund summability method (N, pn) has been determined by several investigators like

Khan[6], Qureshi[7, 8], Chandra[11], Leindler[9], Stepants[2] and Lal[16]. Working in quite different direction, Totik[17, 18], Mazhar[14], Totik and Mazhar[15] and Chandra[12] have studied the approximation of functions in H¨older space H(w). But till now no work seems to have done to obtain the degree of approximation of functions f ∈Hr(w), (r≥1), by Matrix-Euler (∆.E1) product summability means.

In an attempt to make an advance study in this direction, in this paper, a new estimate for degree of trignometric approximation of a functionf ∈Hr(w), (r≥1)

space has been determined. It is important to note thatHr(w), (r≥1), is a

gen-eralization of H(w)_{, H}

(α),r and H(α) spaces. Some important applications of main

theorem has been investigated.

2. Definition and Notations

Letf(x) be a 2πperiodic function, integrable in the Lebesgue sense over [o,2π] and belonging toHr(w) class. Let the Fourier series off(x) is given by

f(x) =1 2a0+

∞

X

n=1

(ancosnx+bnsinnx) (1)

2010Mathematics Subject Classification. 42B05, 42B08.

Key words and phrases. Trigonometric approximation, Fourier series, (E,1) means, matrix summability means and (∆.E1)- summability means, generalized Minkowski’s inequality, gener-alized H¨older metric.

c

2012 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted April 17, 2012. Published date June 10, 2012.

withnthpartial sumssn(f;x).

LetP∞

n=0un be an infinite series havingnthpartial sumsn=P n ν=0uν.

LetT = (an,k) be an infinite triangular matrix satisfying the condition of regularity

(Silverman-T¨oeplitz [10]) i.e.

(i). Pn

k=0an,k= 1 asn→ ∞,

(ii). an,k= 0 fork > n ,

(iii). Pn

k=0|an,k| ≤M , a finite constant.

The sequence-to-sequence transformation

t∆_n =

n X

k=0

an,ksk= n X

k=0

an,n−ksn−k

defines the sequencet∆

n of triangular matrix means of the sequence{sn}, generated

by the sequence of coefficients (an,k).

If t∆

n →s as n→ ∞ then the series P∞

n=0un is summable tos by triangular

matrix ∆- method (Zygmund[1], p.74).

Let E_n(1) = 1 2n

n X

k=0 _n

k

sk. If E (1)

n → s as n → ∞, then P∞_n=0un is said to be

summable tosby the Euler’s method,E1(Hardy[5]).

The triangular matrix ∆-transform of E1 transform defines the (∆.E1) transform t∆E

n of the partial sums sn of the seriesP

∞

n=0un by

t∆E_n =

n X

k=0

an,kE1k= n X

k=0 an,k

1 2k

k X

ν=0 _k

ν

sν.

Ift∆E

n →sas n→ ∞, P∞

n=0un is said to be summable (∆.E1) tos.

sn →s ⇒ E(1)n =

1 2n

n X

ν=0 _n

ν

sν→s as n→ ∞, E1 method is regular,

⇒ t∆_n(E_n(1)) =t∆E_n →s as n→ ∞, ∆ method is regular,

⇒ (∆.E1) method is regular.

Some important particular cases of triangular matrix-Euler means (∆.E1) are

(i). (H,_n+11 ).(E1) means, whenan,k =_{(n−k+1) log}1 _n.

(ii). (N, pn).E1 means, whenan,k = pn−k

Pn , wherePn = Pn

k=0pk6= 0.

(iii). (N, pn, qn).E1 means, whenan,k= pn−kqk

Rn ,

whereRn=P n

k=0pkqn−k6= 0.

Let C2π denote the Banach space of all 2π-periodic and continuous functions

defined on [0,2π] under the supremum norm. For 0< α≤1, let

H(α)={f ∈C2π:|f(x+t)−f(x)|=O(|t| α

The spaceH(α)is a Banach space (Pr¨ossdorfff [13]) under the norm

kfk_(α) = sup

0≤x≤2π

|f(x)|+ sup

x,t t6=0

|f(x+t)−f(x)|

|t|α , 0< α≤1

= kfk_∞+ sup

x,t t6=0

|f(x+t)−f(x)|

|t|α , 0< α≤1.

The metric induced by the normkk_(α)onH(α)is called the H¨older metric. Clearly

(H(α),kfk(α)) is a Banach space which decreases asαincreases, i.e., H(α)⊆H(β)⊆C2π, for 0≤β < α≤1,

and

kfk_(β)≤(2π)α−βkfk_(α). In general,

sup

0≤x≤2π

|f(x)| 6= sup

x,t t6=0

|f(x+t)−f(x)|

|t|α , 0< α≤1.

We define the normkk_rby

kfk_r=





 n

1 2π

R2π 0 |f(x)|

r dxo

1

r

for 1≤r <∞

ess sup

0<x<2π

|f(x)| forr=∞.

LetLr[0,2π] =

f : [0,2π]→R:

Z 2π

0

|f(x)|rdx <∞

, r≥1 , be the space of all 2π-periodic, integrable functions and for allt

H(α),r= (

f ∈Lr[0,2π] :

Z 2π

0

|f(x+t)−f(x)|rdx

1

r

=O(|t|α)

)

.

The spaceH(α),r, r≥1,0< α≤1 is a Banach space under the norm kk(α),r:

kfk_(α),r =kfk_r+ sup

t6=0

kf(.+t)−f(.)k_r

(|t|α) .

kfk_(0),r=kfk_r.

The metric induced by the normkk_(α),r onH(α),r is called H¨older continuous with

degreer.

Easily, it can be obtained by

kfk_(β),r≤(2π)α−βkfk_(α),r, 0≤β < α≤1, r≥1. Sincef ∈H(α),r if and only ifkfk(α),r<∞, we have

H(α),r⊆H(β),r⊆Lr[0,2π], 0≤β < α≤1, r≥1.

For f ∈ Lr[0,2π], r ≥ 1, the integral modulus of continuity is defined by

wr(f, δ) = sup 0<t≤δ

₁

2π

Z 2π

0

|f(x+t)−f(x)|rdx

r1

1≤r <∞and ifr=∞, then w(f, δ) =w∞(f, δ) = sup

o<t≤δ

max

x |f(x+t)−f((x)|forf ∈C2π.

It is known (Zygmund [1], p.45) thatwr(f, δ)→0 asδ→0.

Let w : [0,2π] → _R be an arbitrary function with w(t) >0 for 0 < t ≤2π and lim

t→0+w(t) =w(0) = 0.

The class of functionH(w)_{has been defind by Leindler [9] as}

H(w)={f ∈C2π:|f(x+t)−f(x)|=O(w(t))}

wherewis a modulus of continuity, that is,wis a postive non-decreasing continuous function with the property: w(0) = 0, w(t1+t2)≤w(t1) +w(t2).

We defineH_r(w)=

(

f ∈Lr[0,2π] : 1≤r <∞, sup

t6=0

kf(.+t)−f(.)k_r

w(t) <∞

)

andkfk(w)_r =kfk_r+sup

t6=0

kf(.+t)−f(.)k_r

w(t) , r≥1.Clearlykk

(w)

r is a norm onH (w) r .

The completeness of the spaceHr(w)can be discussed considering the completeness

ofLr _(r≥1).

kfk(v)_r =kfk_r+ sup

t6=0

kf(.+t)−f(.)k_r

v(|t|) , r≥1.If

w(t)

t tends to zero ast→o + _then

f0(x) exists and is zero everywhere and f is constant. Letw(t)_v(t)be positive non decreasing.

Thenkfk(v)_r ≤max1,w(2π)_v(2π)kfk(w)_r <∞.Thus,

H_r(w)⊆H_r(v)⊆Lr, r≥1

Remarks.

(i). If we takew(t) =tα_thenH(w)_{reduces toH}

(α) class.

(ii). By taking w(t) =tα_{in H}(w)

r , it reduces toH(α),r.

(iii). If we taker→ ∞thenHr(w)class reduces toH(w).

The degree of approximation of a functionf :_R→_Rby a trignometric polyno-mialtn(x) = 1₂a0+P_ν=1∞ (aνcosνx+bνsinνx) of ordernis defined by (Zygmund

[1], p.114-115)

En(f) = minktn−fk_r.

We write,

φ(x, t) =f(x+t) +f(x−t)−2f(x),∆an,k=an,k−an,k+1, 0≤k≤n−1.

Kn∆E=

1 2π

n X

k=0 an,k

3. Theorem

In this paper, we prove the following theorem:

Theorem 3.1. Let A = (an,k) be a regular lower triangular infinite matrix such

that

n−1 X

k=0

|∆an,k|=O

₁

n+ 1

,(n+ 1)|an,n|=O(1). (2)

If f : [0,2π] →R be a 2π-periodic, Lebesgue integrable and belonging to the

gen-eralized classHr(w),r≥1;w,v be modulus of continuity and w(t)_v(t) be positive,

non-decreasing then the degree of approximation off by triangular matrix-Euler means

t∆En = Pn

k=0an,k₂1k Pk

ν=0 k ν

sν of its Fourier series (1) is given by

kt∆E_n −fk(v)r =O

1 (n+ 1)

Z π

1

n+1

w(t) t2_v(t)dt

!

. (3)

4. Lemmas

Following Lemmas are required to prove the theorems:

Lemma 4.1. For0< t≤(n+ 1)−1, K_n∆E(t) =O(n+ 1).

Proof. For 0< t≤(n+ 1)−1_{, sin}t 2 ≥

t

π, sinnt≤nt,|cost| ≤1. We have

K_n∆E(t) =

1 2π

n X

k=0 an,k

sin (n−k+ 1)(₂t)cosn−k₍t 2)

sin (₂t)

≤ 1

2π n X

k=0

|an,k|

(n−k+ 1)(t₂)cosn−k(₂t)

(_πt)

≤ 1

4(n+ 1)

n X

k=0

|an,k|

≤ M

4 (n+ 1) = O(n+ 1).

Lemma 4.2. For(n+ 1)−1< t < π, K_n∆E(t) =O_(n+1)t1 2

Proof. For (n+ 1)−1< t < π, sin₂t ≥ t

π, using Abel’s lemma, we get

K_n∆E(t) = 1 2π n X k=0 an,k

sin (n−k+ 1)(₂t)cosn−k(t₂) sin (t

2) ≤ 1 2t

n−1 X

k=0

(an,k−an,k+1) k X

ν=0

sin (n−ν+ 1)

t 2

cosn−ν

t 2

+an,n n X

k=0

sin (n−k+ 1)

_t

2

cosn−k

_t 2 ≤ 1 2t

"n−1 X

k=0

|∆an,k|

sin (2n−k+ 2)(₄t) sin (n+ 1)(₄t) sin (t

4)

+|an,n|

sin (n+ 2)(t₄) sin (n+ 1) t₄ sin ₄t

# ≤ π t2 "n−1

X

k=o

|∆an,k|+|an,n| #

max

0≤k≤n

sin (2n−k+ 2)

_t

2

sin(n+ 1)

_t 2 = π t2

"n−1 X

k=o

|∆an,k|+|an,n| # = π t2 O ₁

n+ 1

+O

₁

n+ 1

by(2)

= O

₁

(n+ 1)t2

.

5. Proof of the Theorem3.1

Following Titchmarsh [4],sk(f;x) of Fourier series (1) is given by

sk(f;x)−f(x) =

1 2π

Z π

0

φ(x, t)sin k+

1 2

t

sin t₂ dt, k= 0,1,2... .

Then 1 2n n X k=0 n k

(sk(f;x)−f(x)) =

1 2π

Z π

0

φ(x, t) 1 2n n X k=0 n k

_{sin k+}1 2

t

sin ₂t dt

or E_n1(x)−f(x) = 1 2π

Z π

0

φ(x, t) 1 2n_{sin (}t

2) ( Im n X k=0 _n k

ei(k+12)t ) dt = 1 2π Z π 0

φ(x, t) 1 2n_{sin (}t

2) ( Im n X k=0 _n k

eiktei2t ) dt = 1 2π Z π 0

φ(x, t) 1 2n_{sin (}t

2) n

Im 1 +eit n

.ei2t o dt = 1 2π Z π 0

φ(x, t)sin

(n+ 1) (₂t) cosn₍t 2)

Now

t∆E_n (x)−f(x) = 1 Pn

n X

k=0

an,kEn1−k(x)−f(x)

= 1 2π

Z π

0

φ(x, t)

n X

k=0 an,k

sin

(n−k+ 1)(₂t) cosn−k₍t 2)

sin (₂t) dt.

Let

ln(x) =t∆En (x)−f(x) = Z π

0

φ(x, t)K_n∆E(t)dt.

Then

ln(x+y)−ln(x) = Z π

0

(φ(x+y, t)−φ(x, t))Kn∆E(t)dt.

By generalized Minkowski’s inequality (Chui[3], p.37), we get

kln(.+y)−ln(.)kr≤ Z π

0

kφ(.+y, t)−φ(., t)k_r

K_n∆E(t) dt

=

Z _n1₊₁

0

kφ(.+y, t)−φ(., t)k_r

K_n∆E(t)

dt+

Z π

1

n+1

kφ(.+y, t)−φ(., t)k_r

K_n∆E(t)

dt

= I1+I2. (4)

Clearly

|φ(x+y, t)−φ(x, t)| ≤ |f(x+y+t)−f(x+y)|+|f(x+y−t)−f(x+y)|

+|f(x+t)−f(x)|+|f(x−t)−f(x)|.

Applying Minkowski’s inequality, we have

kφ(.+y, t)−φ(., t)k_r ≤ kf(.+y+t)−f(.+y)k_r+kf(.+y−t)−f(.+y)k_r

+kf(.+t)−f(.)k_r+kf(.−t)−f(.)k_r

= O(w(t)). (5)

Also

kφ(.+y, t)−φ(., t)k_r ≤ kf(.+y+t)−f(.+t)k_r+kf(.+y−t)−f(.−t)k_r

+2kf(.+y)−f(.)k_r

= O(w(|y|)). (6)

Forv is positive, non decreasing,t≤ |y|, we obtained

kφ(.+y, t)−φ(., t)k_r = O(w(t))

= O

v(t)

_w(t)

v(t)

= O

v(|y|)

_w(t)

v(t)

Since w(t)_v(t) is positive, non-decreasing, ift≥ |y|, then w(t)_v(t) ≥ w(_v(||_yy_||₎), so that

kφ(.+y, t)−φ(., t)k_r = O(w(|y|))

= O

v(|y|)

_w(t)

v(t)

. (7)

Using lemma (4.1) and (7) we obtain

I1 =

Z n+11

0

kφ(.+y, t)−φ(., t)k_r

K_n∆E(t) dt

= O

Z _n+11

0

v(|y|)w(t)

v(t)(n+ 1)dt

!

= O (n+ 1)v(|y|)

Z _n1₊₁

0

w(t) v(t)dt

!

= O



(n+ 1)v(|y|)

w 1 n+1

v_n+11

Z _n+11

0 dt





= O



v(|y|)

w 1 n+1

v_n+11



. (8)

Also,using Lemma (4.2) and (7) we get

I2 = Z π

1

n+1

kφ(.+y, t)−φ(., t)k_r

K_n∆E(t) dt

= O

Z π

1

n+1

v(|y|)w(t) v(t)

1 (n+ 1)t2dt

!

= O 1 n+ 1

Z π

1

n+1

v(|y|) w(t) t2_v(t)dt

!

. (9)

By (4), (8) and (9), we have

kln(.+y)−ln(.)kr = O 

v(|y|)

w 1 n+1

v 1 n+1





+O 1 n+ 1

Z π

1

n+1

v(|y|) w(t) t2_v(t)dt

!

.

Thus,

sup

y6=0

kln(.+y)−ln(.)k_r

v(|y|) = O





w_n+11

v_n+11





+O 1 n+ 1

Z π

1

n+1

w(t) t2_v(t)dt

!

Clearly

|φ(x, t)| = |f(x+t) +f(x−t)−2f(x)| ≤ |f(x+t)−f(x)|+|f(x−t)−f(x)|

Applying Minkowski’s inequality, we have

kφ(., t)k_r ≤ kf(.+t)−f(.)k_r+kf(.−t)−f(.)k_r

= O(w(t)). (11) Using(11), Lemma (4.1),Lemma (4.2) we obtain

kln(.)k_r =

t∆E_n −f _r≤

Z _n1₊₁

0

+

Z π

1

n+1 !

kφ(., t)k_r

K_n∆E(t) dt

=

Z n+11

0

kφ(., t)k_r

(K_n∆E(t) dt+

Z π

1

n+1

kφ(., t)k_r K_n∆E(t)

dt

= O (n+ 1)

Z _n+11

0

w(t)dt

!

+O 1 (n+ 1)

Z π

1

n+1

w(t) t2 dt

!

= O

w

₁

(n+ 1)

+O 1 (n+ 1)

Z π

1

n+1

w(t) t2 dt

!

. (12)

Now, By (10) and (12)

kln(.)k(v)_r = kln(.)k_r+ sup y6=0

kln(.+y)−ln(.)kr v(|y|)

= O

w

₁

n+ 1

+O 1 (n+ 1)

Z π

1

n+1

w(t) t2 dt

!

+O





w 1 n+1

v 1 n+1



+O

1 (n+ 1)

Z π

1

n+1

w(t) v(t)t2dt

!

.

Using the fact thatw(t) =w(t)_v(t).v(t) ≤ v(π)w(t)_v(t), 0< t≤π, we get

kln(.)k (v)

r = O





w_n+11

v_n+11



+O

1 (n+ 1)

Z π

1

n+1

w(t) v(t)t2dt

!

. (13)

Sincewandv are modulus of continuity such that w(t)_v(t) is positive, non decreasing, therefore

1 n+ 1

Z π

1

n+1

w(t) v(t)t2dt≥

w_n+11

v_n+11

₁

n+ 1

Z π

1

n+1

dt t2 ≥

w_n+11

2v_n+11 .

Then

w_n+11

v_n+11

= O 1 (n+ 1)

Z π

1

n+1

w(t) t2_v(t)dt

!

By (13) and (14), we have

t

∆E

n −f

(v)

r = O

1 (n+ 1)

Z π

1

n+1

w(t) t2_v(t)dt

!

.

This completes the proof of theorem 3.1.

6. Applications

We obtain the following corollaries from the main Theorems.

Corollary 6.1. Let f ∈H(α),r ,r≥1,0< α≤1 then

t∆E_n −f

_(β),r = 





O_(n+1)1α−β

, 0≤β < α <1, Olog(n+1)π_n+1 , β = 0, α= 1.

Proof. If we takew(t) =tα_{, v(t) =t}β _{in theorem 3.1.}

Corollary 6.2. If we take an,k = _{(n−k+1) log}1 _n, in theorem 3.1, then degree of

approximation of a function f ∈Hr(w) by(H,_n+11 ).E1 means

tHEn =

1 logn

n X

k=0

1 n−k+ 1

1 2k

k X

ν=0 _k

ν

sν

of the fourier series (1) is given by

tHE_n −f

(v)

r = O

1 (n+ 1)

Z π

1

n+1

w(t) t2_v(t)dt

!

.

Corollary 6.3. If we take an,k = pn−k

Pn , wherePn= Pn

k=0pk 6= 0in theorem 3.1,

then degree of approximation of a functionf ∈Hr(w) by (N, pn).E1 means

tN E_n = 1 Pn

n X

k=0 pn−k

1 2k

k X

ν=0 _k

ν

sν

of the fourier series (1) is given by

tN E_n −f

(v)

r = O

1 (n+ 1)

Z π

1

n+1

w(t) t2_v(t)dt

!

.

Corollary 6.4. If we takean,k= pn−kqk

Rn , whereRn= Pn

k=0pkqn−k 6= 0in theorem

3.1, then degree of approximation of a function f ∈Hr(w) by (N, p, q).E1 means

tN E_n = 1 Rn

n X

k=0 pn−kqk

1 2k

k X

ν=0 _k

ν

sν

of the fourier series (1) is given by

tN E_n −f

(v)

r = O

1 (n+ 1)

Z π

1

n+1

w(t) t2_v(t)dt

!

Acknowledgements. Authors are grateful to the referee for his valuable sugges-tions and comments for improvement of this paper. Authors mention their sense of gratitude to Professor Huseyin Bor, Professor Faton Z. Merovci and Editorial Board of BMAA for supporting this work.

Shyam Lal, one of the author, is thankful to DST-CIMS, Banaras Hindu Uni-versity for encouragement to this work.

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Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi-221005, India