JJ II

(1)

volume 6, issue 2, article 40, 2005.

Received 20 April, 2005;

accepted 02 May, 2005.

Communicated by:L. Leindler

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

LOCALIZATION OF FACTORED FOURIER SERIES

HÜSEY˙IN BOR

Department of Mathematics Erciyes University

38039 Kayseri, Turkey EMail:bor@erciyes.edu.tr

URL:http://fef.erciyes.edu.tr/math/hbor.htm

c

2000Victoria University ISSN (electronic): 1443-5756 126-05

(2)

Localization of Factored Fourier Series

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of12

J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005

http://jipam.vu.edu.au

Abstract

In this paper we deal with a main theorem on the local property of|N, p¯ _n|_k summability of factored Fourier series, which generalizes some known results.

2000 Mathematics Subject Classification:40D15, 40G99, 42A24, 42B15.

Key words: Absolute summability, Fourier series, Local property.

1. Introduction

LetP

a_nbe a given infinite series with partial sums(s_n). Let(p_n)be a sequence of positive numbers such that

(1.1) P_n =

n

X

v=0

p_v → ∞ as n→ ∞, (P−i =p−i = 0, i≥1).

The sequence-to-sequence transformation

(1.2) t_n= 1

P_n

n

X

v=0

p_vs_v

defines the sequence (t_n)of the( ¯N , p_n)means of the sequence(s_n)generated by the sequence of coefficients(p_n).

The seriesP

a_nis said to be summable N , p¯ _n

k, k ≥1,if (see [2]) (1.3)

∞

X

n=1

P_n pn

k−1

|t_n−t_n−1|^k <∞.

In the special case whenp_n = 1/(n+ 1)for all values ofn (resp. k = 1),

N , p¯ _n

k summability is the same as|C,1|(resp.

N , p¯ _n

) summability. Also if we takek = 1andpn = 1/(n+ 1), summability

N , p¯ n

k is equivalent to the summability|R,logn,1|. A sequence(λ_n)is said to be convex if∆²λ_n ≥0for every positive integern, where∆²λ_n = ∆λ_n−∆λ_n+1 and∆λ_n=λ_n−λ_n+1.

(4)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of12

Letf(t)be a periodic function with period2πand 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

(1.4)

Z π

−π

f(t)dt= 0

and

(1.5) f(t)∼

∞

X

n=1

(a_ncosnt+b_nsinnt)≡

∞

X

n=1

A_n(t).

It is well known that the convergence of the Fourier series at t = xis a local property of the generating function f (i.e. it depends only on the behaviour of f in an arbitrarily small neighbourhood ofx), and hence the summability of the Fourier series att =xby any regular linear summability method is also a local property of the generating functionf.

(5)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of12

2. Known Results

Mohanty [4] has demonstrated that the|R,logn,1|summability of the factored Fourier series

(2.1) X A_n(t)

log(n+ 1)

att =x, is a local property of the generating function off, whereas the|C,1|

summability of this series is not. Matsumoto [3] improved this result by replac- ing the series (2.1) by

(2.2) X An(t)

{log log(n+ 1)}^δ, δ >1.

Generalizing the above result Bhatt [1] proved the following theorem.

Theorem A. If (λ_n) is a convex sequence such that P

n⁻¹λ_n is convergent, then the summability |R,logn,1|of the seriesP

A_n(t)λ_nlogn at a point can be ensured by a local property.

(6)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of12

3. The Main Result

The aim of the present paper is to prove a more general theorem which includes of the above results as special cases. Also it should be noted that the conditions on the sequence (λ_n) in our theorem, are somewhat more general than in the above theorem.

Now we shall prove the following theorem.

Theorem 3.1. Letk ≥1. If(λ_n)is a non-negative and non-increasing sequence such that P

p_nλ_n is convergent, then the summability N , p¯ _n

k of the series PA_n(t)λ_nP_nat a point is a local property of the generating functionf.

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

Lemma 3.2. If (λ_n)is a non-negative and non-increasing sequence such that Ppnλnis convergent, where (pn)is a sequence of positive numbers such that P_n→ ∞asn→ ∞, thenP_nλ_n=O(1)asn→ ∞andP

P_n∆λ_n<∞.

Proof. Since(λ_n)is non-increasing, we have that Pmλm =λm

m

X

n=0

pn =O(1)

m

X

n=0

pnλn =O(1) as m → ∞.

Applying the Abel transform to the sumPm

n=0p_nλ_n, we get that

m

X

n=0

P_n∆λ_n=

m

X

n=0

p_nλ_n−P_mλ_m+1.

(7)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of12

Sinceλ_n≥λ_n+1, we obtain

m

X

n=0

Pn∆λn≤Pmλm+

m

X

n=0

pnλn

=O(1) +O(1) =O(1) as m → ∞.

Lemma 3.3. Let k ≥ 1 and s_n = O(1). If (λ_n) is a non-negative and non- increasing sequence such thatP

p_nλ_nis convergent, where (p_n)is a sequence of positive numbers such that P_n → ∞asn → ∞, then the seriesP

a_nλ_nP_n is summable

N , p¯ _n k.

Proof. Let (T_n) be the sequence of ( ¯N , p_n) means of the series P

a_nλ_nP_n. Then, by definition, we have

T_n= 1 P_n

n

X

v=0

p_v

v

X

r=0

a_rλ_rP_r= 1 P_n

n

X

v=0

(P_n−P_v−1)a_vλ_vP_v.

Then, forn≥1, we have

T_n−Tn−1 = pn

P_nPn−1 n

X

v=1

Pv−1P_va_vλ_v.

(8)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of12

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

P_nPn−1 n−1

X

v=1

PvPvsv∆λv− p_n P_nPn−1

n−1

X

v=1

Pvsvpvλv

− p_n P_nPn−1

n−1

X

v=1

P_vp_v+1s_vλ_v+1+s_np_nλ_n

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

By Minkowski’s inequality fork >1, to complete the proof of Lemma3.3, it is sufficient to show that

(3.1)

∞

X

n=1

(P_n/p_n)^k−1|T_n,r|^k <∞, for r = 1,2,3,4.

Now, applying Hölder’s inequality with indicesk andk⁰, where ¹_k+_k¹0 = 1and k > 1, we get that

m+1

X

n=2

Pn

p_n k−1

|Tn,1|^k

≤

m+1

X

n=2

p_n P_nPn−1

(_n−1 X

v=1

|sv|^kPvPv∆λv

) ( 1 Pn−1

n−1

X

v=1

PvPv∆λv

)^k−1 . Since

n−1

X

v=1

PvPv∆λv ≤Pn−1 n−1

X

v=1

Pv∆λv,

(9)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of12

it follows by Lemma3.2that 1

P_n−1

n−1

X

v=1

P_vP_v∆λ_v ≤

n−1

X

v=1

P_v∆λ_v =O(1) as n → ∞.

Therefore

m+1

X

n=2

Pn

p_n k−1

|Tn,1|^k =O(1)

m+1

X

n=2

pn

P_nPn−1 n−1

X

v=1

|sv|^kPvPv∆λv

=O(1)

m

X

v=1

|s_v|^kP_vP_v∆λ_v

m+1

X

n=v+1

p_n P_nPn−1

=O(1)

m

X

v=1

P_v∆λ_v =O(1) as m → ∞, by virtue of the hypotheses of Theorem3.1and Lemma3.2. Again

m+1

X

n=2

Pn

p_n k−1

|T_n,2|^k ≤

m+1

X

n=2

pn

P_nPn−1

(_n−1 X

v=1

|s_v|^k(P_vλ_v)^kp_v

) ( 1 Pn−1

n−1

X

v=1

p_v )^k−1

=O(1)

m+1

X

v=2

p_n P_nP_n−1

n−1

X

v=1

=O(1)

m

X

v=1

m+1

X

n=v+1

p_n PnPn−1

(10)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of12

=O(1)

m

X

v=1

|s_v|^k(P_vλ_v)^kp_v Pv

=O(1)

m

X

v=1

|s_v|^k(P_vλ_v)^k−1p_vλ_v

=O(1)

m

X

v=1

p_vλ_v =O(1) as m → ∞,

in view of the hypotheses of Theorem 3.1and Lemma3.2. Using the fact that P_v < P_v+1, similarly we have that

m+1

X

n=2

P_n pn

^k−1

|T_n,3|^k =O(1)

m

X

v=1

p_v+1λ_v+1 =O(1) as m → ∞.

Finally, we have that

m

X

n=1

Pn

p_n k−1

|T_n,4|^k=

m

X

n=1

|s_n|^k(P_nλ_n)^k−1p_nλ_n

=O(1)

m

X

n=1

p_nλ_n=O(1) as m→ ∞,

by virtue of the hypotheses of the theorem and Lemma3.2. Therefore, we get that

m

X

n=1

Pn

p_n k−1

|T_n,r|^k =O(1) as m → ∞, for r = 1,2,3,4.

This completes the proof of Lemma3.3.

(11)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of12

In the particular case if we takep_n = 1 for all values ofn in Lemma 3.3, then we get the following corollary.

Corollary 3.4. Let k ≥ 1 and and s_n = O(1). If (λ_n) is a non-negative and non-increasing sequence such that P

λ_n is convergent, then the series Pnanλnis summable|C,1|_k.

Proof of Theorem3.1. Since the behaviour of the Fourier series, as far as con- vergence is concerned, for a particular value of xdepends on the behaviour of the function in the immediate neighbourhood of this point only, hence the truth of Theorem3.1is a consequence of Lemma3.3. If we takepn = 1for all values ofnin this theorem, then we get a new local property result concerning the

|C,1|_ksummability.

(12)

Hüseyin Bor

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of12

References

[1] S.N. BHATT, An aspect of local property of|R,logn,1|summability of the factored Fourier series, Proc. Nat. Inst. Sci. India, 26 (1960), 69–73.

[2] H. BOR, On two summability methods, Math. Proc. Cambridge Philos Soc., 97 (1985), 147–149.

[3] K. MATSUMOTO, Local property of the summability|R,logn,1|, Tôhoku Math. J. (2), 8 (1956), 114–124.

[4] R. MOHANTY, On the summability|R,logw,1|of Fourier series, J. Lon- don Math. Soc., 25 (1950), 67–72.