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

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Volume 6, Issue 2, Article 40, 2005

LOCALIZATION OF FACTORED FOURIER SERIES

HÜSEY˙IN BOR

DEPARTMENT OFMATHEMATICS

ERCIYESUNIVERSITY

38039 KAYSERI, TURKEY

bor@erciyes.edu.tr

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

Received 20 April, 2005; accepted 02 May, 2005 Communicated by L. Leindler

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.

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

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

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 p_n

k−1

|tn−tn−1|^k<∞.

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

ksummabil- ity is the same as|C,1|(resp.

N , p¯ _n

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

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126-05

summability N , p¯ n

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

∆λ_n=λ_n−λ_n+1.

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

(ancosnt+bnsinnt)≡

∞

X

n=1

An(t).

It is well known that the convergence of the Fourier series at t = x is 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.

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 replacing 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(λn)is a convex sequence such thatP

n⁻¹λn is convergent, then the summa- bility|R,logn,1|of the seriesP

An(t)λnlognat a point can be ensured by a local property.

3. THEMAINRESULT

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. Let k ≥ 1. If(λ_n) is a non-negative and non-increasing sequence such that Pp_nλ_nis convergent, then the summability

N , p¯ _n

k of the seriesP

A_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 thatP

p_nλ_nis con- vergent, where (p_n) is a sequence of positive numbers such that P_n → ∞as n → ∞, then P_nλ_n=O(1)asn → ∞andP

P_n∆λ_n <∞.

Proof. Since(λ_n)is non-increasing, we have that P_mλ_m =λ_m

m

X

n=0

p_n=O(1)

m

X

n=0

p_nλ_n =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. Sinceλ_n≥λ_n+1, we obtain

m

X

n=0

P_n∆λ_n ≤P_mλ_m+

m

X

n=0

p_nλ_n

=O(1) +O(1) =O(1) as m → ∞.

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

p_nλ_n is convergent, where(p_n)is a sequence of positive numbers such thatP_n →

∞asn→ ∞, then the seriesP

a_nλ_nP_nis summable N , p¯ _n

k. Proof. Let(T_n)be the sequence of( ¯N , p_n)means of the seriesP

a_nλ_nP_n. Then, by definition, we have

T_n= 1 Pn

n

X

v=0

p_v

v

X

r=0

a_rλ_rP_r = 1 Pn

n

X

v=0

(P_n−Pv−1)a_vλ_vP_v.

Then, forn ≥1, we have

T_n−Tn−1 = p_n P_nPn−1

n

X

v=1

Pv−1P_va_vλ_v.

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

P_nPn−1 n−1

X

v=1

P_vP_vs_v∆λ_v− p_n P_nPn−1

n−1

X

v=1

P_vs_vp_vλ_v

− 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 for k > 1, to complete the proof of Lemma 3.3, 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 indiceskandk⁰, where ¹_k + _k¹0 = 1andk >1, we get that

m+1

X

n=2

P_n p_n

k−1

|T_n,1|^k≤

m+1

X

n=2

p_n P_nPn−1

(_n−1 X

v=1

|s_v|^kP_vP_v∆λ_v

) ( 1 Pn−1

n−1

X

v=1

P_vP_v∆λ_v )k−1

. Since

n−1

X

v=1

P_vP_v∆λ_v ≤P_n−1

n−1

X

v=1

P_v∆λ_v,

it follows by Lemma 3.2 that 1

Pn−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

P_n p_n

k−1

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

m+1

X

n=2

p_n P_nPn−1

n−1

X

v=1

|s_v|^kP_vP_v∆λ_v

=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 Theorem 3.1 and Lemma 3.2. Again

m+1

X

n=2

P_n p_n

^k−1

|T_n,2|^k ≤

m+1

X

n=2

p_n P_nP_n−1

(_n−1 X

v=1

|s_v|^k(P_vλ_v)^kp_v

) ( 1 P_n−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

|s_v|^k(P_vλ_v)^kp_v

=O(1)

m

X

v=1

|s_v|^k(P_vλ_v)^kp_v

m+1

X

n=v+1

p_n PnPn−1

=O(1)

m

X

v=1

|s_v|^k(P_vλ_v)^kp_v P_v

=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 3.1 and Lemma 3.2. Using the fact that P_v < P_v+1, similarly we have that

m+1

X

n=2

P_n p_n

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

P_n pn

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 Lemma 3.2. Therefore, we get that

m

X

n=1

P_n pn

k−1

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

This completes the proof of Lemma 3.3.

In the particular case if we take p_n = 1for all values of n in Lemma 3.3, then we get the following corollary.

Corollary 3.4. Letk ≥ 1 and and s_n = O(1). If (λ_n)is a non-negative and non-increasing sequence such thatP

λ_nis convergent, then the seriesP

na_nλ_nis summable|C,1|_k.

Proof of Theorem 3.1. Since the behaviour of the Fourier series, as far as convergence is con- cerned, for a particular value of xdepends on the behaviour of the function in the immediate neighbourhood of this point only, hence the truth of Theorem 3.1 is a consequence of Lemma 3.3. If we takepn= 1for all values ofnin this theorem, then we get a new local property result

concerning the|C,1|_ksummability.

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. London Math. Soc., 25 (1950), 67–72.