On the local properties of factored Fourier series ∗

On the local properties of factored Fourier series ^∗

H¨ useyin Bor

Abstract

In the present paper, a theorem on local property of|N , p¯ n, θn|ksumma- bility of factored Fourier series which generalizes a result of Bor [3] has been proved.

1 Introduction

LetPan be a given infinite series with partial sums (sn). We denote bytn the n-th (C,1) mean of the sequence (nan). A seriesPan is said to be summable

|C,1|_k,k≥1 , if (see [6],[8])

∞

X

n=1

1

n |tn|^k<∞. (1.1)

Let (p_n) be a sequence of positive numbers such that Pn=

n

X

v=0

pv→ ∞ as n→ ∞, (P_−i=p_−i= 0, i≥1). (1.2) The sequence-to-sequence transformation

σn = 1 P_n

n

X

v=0

pvsv (1.3)

defines the sequence (σn) of the Riesz mean or simply the ( ¯N , pn) mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [7]). The series Pan is said to be summable|N , p¯ n|_k, k≥1,if (see [2])

∞

X

n=1

(P_n/p_n)^k−1|∆σ_n−1|^k<∞, (1.4) where

∆σ_n−1=− pn

P_nP_n−1

n

X

v=1

P_v−1a_v, n≥1. (1.5)

∗Mathematics Subject Classifications: 40G99, 42A24, 42B24.

Key words: Absolute summability, infinite series, local property, Fourier series, c

2009 Universiteti i Prishtines, Prishtine, Kosov¨e.

Submitted November 2008, Published January 2009.

15

In the special case pn = 1 for all values of n, |N , pn |_k summability is the same as |C,1|_k summability. Also,if we takek = 1 andpn = 1/(n+ 1), then summability|N , p¯ n|k is equivalent to the summability|R,logn,1|. Let (θn) be any sequence of positive constants. The seriesP

an is said to be summable

|N , p¯ n, θn|k, k≥1,if (see [12])

∞

X

n=1

θ^k−1_n |∆σ_n−1|^k<∞. (1.6) If we take θ_n = ^P_pⁿ

n, then | N , p¯ _n, θ_n |k summability reduces to | N , p¯ _n |k

summability. Also, if we take θ_n =n and p_n = 1 for all values of n, then we get | C,1 |_k summability. Furthermore, if we takeθ_n =n, then | N , p¯ _n, θ_n |_k summability reduces to|R, p_n|_k (see [4]) summability. A sequence (λ_n) is said to be convex if ∆²λn ≥0 for every positive integern, where ∆²λn = ∆(∆λn) and ∆λn =λn−λn+1.

Letf(t) be a periodic function with period 2π,and integrable (L) over (−π, π).

Without any loss of generality we may assume that the constant term in the Fourier series off(t) is zero, so that

Z π

−π

f(t)dt= 0 (1.7)

and

f(t)∼

∞

X

n=1

(a_ncosnt+b_nsinnt) =

∞

X

n=1

A_n(t). (1.8)

2 Known result

Mohanty [11] has demonstrated that the summability|R,logn,1|of

XAn(t)/log(n+ 1), (2.1)

at t = x, is a local property of the generating function of P

An(t). Later on Matsumoto [9] improved this result by replacing the series (2.1) by

XAn(t)/log log(n+ 1)¹⁺, >0. (2.2) Generalizing the above result Bhatt [1] proved the following theorem.

Theorem A. If (λn) is a convex sequence such thatP

n⁻¹λn is convergent, then the summability|R,logn,1|of the seriesPA_n(t)λ_nlognat a point can be ensured by a local property.

Also, Mishra [10] has proved the following most general theorem on this matter.

Theorem B.If (p_n) is a sequence such that

Pn=O(npn) (2.3)

Pn∆pn=O(pnpn+1), (2.4) then the summability |N , p¯ n|of the series

∞

X

n=1

A_n(t)λ_nP_n/np_n (2.5)

at a point can be ensured by local property, where (λ_n) is as in Theorem A.

On the other hand Bor [3] has generalized Theorem B for|N , p¯ n |ksummability in the following form.

Theorem C. Letk≥1 and (pn) be a sequence such that the conditions (2.3) and (2.4) of Theorem B are satisfied. Then the summability | N , p¯ n |k of the series (2.5) at a point can be ensured by local property , where (λn) is as in Theorem A.

3 Main result

The aim of this paper is to generalize Theorem C for|N , p¯ n, θn|ksummability.

We shall prove the following theorem.

Theorem. Let k ≥1 and (pn) be a sequence such that the conditions (2.3)- (2.4) of Theorem B are satisfied. If (θn) is any sequence of positive constants such that

m

X

v=1

θ_vp_v Pv

k−1

1

v(λ_v)^k =O(1) (3.1)

m

X

v=1

θvpv

Pv

k−1

∆λv=O(1) (3.2)

m

X

v=1

θvpv

P_v k−1

1

v(λ_v+1)^k=O(1) (3.3) and

m+1

X

n=v+1

θnpn

Pn

k−1

pn

PnP_n−1 =O (

θvpv

Pv

k−1

1 Pv

)

, (3.4)

then the summability|N , p¯ n, θn|k of the series (2.5) at a point can be ensured by local property, where (λn) is as in Theorem A.

It should be noted that if we takeθn =^P_pⁿ

n,then we get Theorem C. In this case the conditions (3.1)-(3.3) are obvious and the condition (3.4) reduces to

m+1

X

n=v+1

pn

P_nP_n−1 =O 1

P_v

,

which always holds.

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

Lemma 1 ([10]). If the sequence (pn) is such that the conditions (2.3) and (2.4) of Theorem B are satisfied, then

∆(Pn/npn) =O(1/n). (3.5) Lemma 2 ([5]). If (λn) is a convex sequence such thatPn⁻¹λnis convergent, then (λn) is non-negative and decreasing, andn∆λn →0 asn→ ∞.

Lemma 3. Letk≥1 .If (sn) is bounded and all conditions of the Theorem are satisfied, then the series

∞

X

n=1

AnλnPn/npn (3.6)

is summable|N , p¯ n, θn|k,where (λn) is as in Theorem A.

Remark. Since (λ_n) is a convex sequence, therefore (λ_n)^k is also convex se- quence andP(1/n)(λ_n)^k <∞.

Proof of Lemma 3. Let (T_n) denotes the ( ¯N , p_n) mean of the series (3.6).

Then, by definition, we have Tn= 1

P_n

n

X

v=0

pv v

X

r=0

arλrPr/rpr= 1 P_n

n

X

v=0

(Pn−P_v−1)avλvPv/vpv. Then

Tn−T_n−1= pn

PnP_n−1

n

X

v=1

P_v−1Pv

avλv

vpv

, n≥1, (P₋₁= 0).

By Abel’s transformation, we have Tn−Tn−1 = − pn

PnP_n−1

n−1

X

v=1

pvPvsvλv

1 vpv

+ p_n

PnP_n−1

n−1

X

v=1

PvsvPv∆λv

1 vpv

+ p_n

PnPn−1 n−1

X

v=1

P_vλ_v+1∆(P_v/vp_v)s_v+s_nλ_n1 n

= Tn,1+Tn,2+Tn,3+Tn,4, say.

To prove the lemma, by Minkowski’s inequality, it is sufficient to show that

∞

X

n=1

θ^k−1_n |T_n,r|^k<∞, f or r= 1,2,3,4. (3.7)

Now, applying H¨older’s inequality, we have that

m+1

X

n=2

θ^k−1_n |Tn,1|^k ≤

m+1

X

n=2

θ^k−1_n pn

Pn

^k 1 P_n−1

n−1

X

v=1

|sv|^k pv

λvPv

vpv

^k

× ( 1

P_n−1

n−1

X

v=1

pv

)^k−1

= O(1)

m

X

v=1

pv

Pv

pv

k

(λv)^k 1 v^k

m+1

X

n=v+1

θnpn

Pn

k−1

pn

PnP_n−1

= O(1)

m

X

v=1

Pv

p_v k−1

(λv)^k 1 v^k

θvpv

P_v k−1

= O(1)

m

X

v=1

v^k−1(λ_v)^k 1 v^k

θ_vp_v Pv

k−1

= O(1)

m

X

v=1

θ_vp_v Pv

k−1

1

v(λ_v)^k =O(1) as m→ ∞,

by virtue of the hypotheses of the Theorem. Since

n−1

X

v=1

Pv∆λv ≤P_n−1

n−1

X

v=1

∆λv ⇒ 1 P_n−1

n−1

X

v=1

Pv∆λv≤

n−1

X

v=1

∆λv=O(1),

by Lemma 2, we have that

m+1

X

n=2

θ_n^k−1|Tn,2|^k ≤

m+1

X

n=2

θ^k−1_n pn

Pn

k

1 P_n−1

n−1

X

v=1

Pv

vpv

k

Pv∆λv|sv|^k

× ( 1

P_n−1

n−1

X

v=1

pv

)^k−1

= O(1)

m

X

v=1

Pv

vpv

^k 1 v^kPv∆λv

m+1

X

n=v+1

θnpn

Pn

^k−1 pn

PnP_n−1

= O(1)

m

X

v=1

P_v vpv

^k 1 v^k∆λ_v

θ_vp_v Pv

k−1

= O(1)

m

X

v=1

θ_vp_v Pv

k−1

∆λ_v

= O(1) as m→ ∞,

in view of the hypotheses of the Theorem and Lemma 2.

Using the fact that ∆(Pv/vpv) =O(1/v) by Lemma 1, we have that

m+1

X

n=2

θ_n^k−1|Tn,3|^k =

m

X

n=1

θ_n^k−1 pn

P_n ^k 1

P_n−1^k |

n−1

X

v=1

Pvλv+1∆(Pv/vpv)sv|^k

= O(1)

m+1

X

n=2

θ^k−1_n p_n

Pn

^k 1 P_n−1^k

(_n−1 X

v=1

P_v pv

p_vλ_v+11 v

)^k

= O(1)

m+1

X

n=2

θ^k−1_n pn

P_n ^k 1

P_n−1

n−1

X

v=1

Pv

p_v ^k

p_v(λ_v+1)^k 1 v^k

× ( 1

P_n−1

n−1

X

v=1

pv

)^k−1

= O(1)

m

X

v=1

P_v pv

k

p_v(λ_v+1)^k 1 v^k

m+1

X

n=v+1

θ_np_n Pn

k−1

p_n PnPn−1

= O(1)

m

X

v=1

Pv

pv

^k−1

(λv+1)^k 1 v^k

θvpv

Pv

^k−1

= O(1)

m

X

v=1

v^k−1(λv+1)^k 1 v^k

θvpv

P_v k−1

= O(1)

m

X

v=1

θvpv

P_v k−1

1

v(λ_v+1)^k =O(1) as m→ ∞, by virtue of the hypotheses of the Theorem. Finally, we have that

m

X

n=1

θ^k−1_n |T_n,4|^k =

m

X

n=1

θ^k−1_n (λ_n)^k 1 n^k

= O(1)

m

X

n=1

θ^k−1_n (λn)^k|sn |^k 1 n^k−1

1 n

= O(1)

m

X

n=1

θnpn

P_n

^k−1 1 n(λn)^k

= O(1) as m→ ∞,

in view of the hypotheses of the Theorem. Therefore we get that

m

X

n=1

θ^k−1_n |Tn,r|^k=O(1) as m→ ∞, f or r= 1,2,3,4.

which completes the proof of the Lemma 3.

Remark. If we takek= 1,then we get a result due to Mishra [10].

4 Proof of the Theorem

Since the behaviour of the Fourier series, as far as convergence is concerned, for a particular value ofxdepends on the behaviour of the function in the immediate neighborhood of this point only, hence the truth of the Theorem is necessary consequence of Lemma 3.

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H ¨USEY˙IN BOR

Department of Mathematics, Erciyes University,

38039 Kayseri, Turkey e-mail: [email protected]