A note on |N, p, q|

A note on |N, p, q|

, (1 ≤ k ≤ 2) summability of orthogonal series

Xhevat Z. Krasniqi

Department of Mathematics and Computer Sciences

Avenue ”Mother Theresa ” 5, Prishtin¨e, 10000, Republic of Kosova xheki00@hotmail.com

Received: 3.6.2010; accepted: 20.9.2010.

Abstract. In this paper we present some results on |N, p, q|k, (1 ≤ k ≤ 2) summability of orthogonal series.

Keywords: Orthogonal series, N¨orlund summability.

MSC 2000 classification:Primary 42C15, 40F05, Secondary 40G05.

1 Introduction

LetP_∞

n=0an be a given infinite series with its partial sums {sn}. Then, let p denotes the sequence {pn}. For two given sequences p and q, the convolution (p ∗ q)n is defined by

(p ∗ q)n= Xn m=0

pmq_n−m= Xn m=0

p_n−mqm.

When (p ∗ q)n 6= 0 for all n, the generalized N¨orlund transform of the sequence {sn} is the sequence {t^p,qn } obtained by putting

t^p,q_n = 1 (p ∗ q)n

Xn m=0

p_n−mqmsm.

The infinite seriesP∞

n=0anis absolutely summable (N, p, q)kof order k, if for k ≥ 1 the series X∞

n=0

n^k−1|t^p,qn − t^p,qn−1|^k

converges, and we write in brief

X∞ n=0

an∈ |N, p, q|k.

We note that for k = 1, |N, p, q|ksummability is the same as |N, p, q| summability introduced by Tanaka [3].

Let {ϕn(x)} be an orthonormal system defined in the interval (a, b). We assume that f(x) belongs to L²(a, b) and

f (x) ∼ X∞ n=0

anϕn(x), (1)

http://siba-ese.unisalento.it/ c 2010 Universit`a del Salento

where an=Rb

af (x)ϕn(x)dx, (n = 0, 1, 2, . . . ).

We write (see [4])

Rn:= (p ∗ q)ⁿ, R^jn:=

Xn m=j

pn−mqm

and

Rⁿ⁺¹_n = 0, R⁰_n= Rn. Also we put

Pn:= (p ∗ 1)n= Xn m=0

pm and Qn:= (1 ∗ q)n= Xn m=0

qm.

Our main purpose of the present paper is to study the |N, p, q|ksummability of the orthogonal series (1), for 1 ≤ k ≤ 2, and to deduce as corollaries all results of Y. Okuyama [4].

Throughout this paper K denotes a positive constant that it may depends only on k, and be different in different relations.

2 Main Results

We prove the following theorem.

Theorem 2.1 If for 1 ≤ k ≤ 2 the series X∞

n=0

( n^2−k2

Xn j=1

R^j_n

Rn −R^j_n−1 R_n−1

!2

|aj|² )^k₂

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, p, q|k almost everywhere.

Proof. For the generalized N¨orlund transform t^p,q_n (x) of the partial sums of the orthogonal seriesP_∞

n=0anϕn(x) we have that t^p,q_n (x) = 1

Rn

Xn m=0

p_n−mqm

Xm j=0

ajϕj(x)

= 1 Rn

Xn j=0

ajϕj(x) Xn m=j

p_n−mqm

= 1 Rn

Xn j=0

R^j_najϕj(x)

wherePm

j=0ajϕj(x) are partial sums of order k of the series (1).

As in [4] page 163 one can find that

△t^p,qn (x) := t^p,q_n (x) − t^p,q_n−1(x) = Xn j=1

R^j_n

Rn −R^j_n−1 R_n−1

!

ajϕj(x).

Using the H¨older’s inequality and orthogonality to the latter equality, we have that Z b

a

|△t^p,qn (x)|^kdx ≤ (b − a)^1−k2

Z b a

|t^p,qn (x) − t^p,q_n−1(x)|²dx

^k2

= (b − a)^1−k2

" _n X

j=1

R^j_n

Rn −R^j_n−1 R_n−1

!2

|a^j|²

#^k₂ .

Hence, the series X∞ n=1

n^k−1 Z b

a

|△t^p,qn (x)|^kdx ≤ (b − a)^1−k2 X∞ n=1

n^k−1

" _n X

j=1

R^jn

Rn−R^j_n−1 Rn−1

!2

|aj|²

#^k₂ (2)

converges by the assumption. According to the Lemma of Beppo-Levi the proof of the theorem

ends. ^QED

For k = 1 in theorem 2.1 we have the following result.

Corollary 2.2 [4] If the series X∞ n=0

( _n X

j=1

R^jn

Rn−R^j_n−1 R_n−1

!2

|aj|² )¹₂

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, p, q| almost everywhere.

Let us prove now another two corollaries of the Theorem 2.1.

Corollary 2.3 If for 1 ≤ k ≤ 2 the series X∞

n=0

n^1−1kpn

PnP_n−1

!k( _n X

j=1

p²_n−j

Pn

pn −Pn−j

p_n−j

2

|aj|² )^k₂

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, p|k almost everywhere.

Proof. After some elementary calculations one can show that R^j_n

Rn −R^j_n−1

R_n−1 = pn

PnP_n−1

Pn

pn −P_n−j p_n−j

 p_n−j

for all qn= 1, and the proof follows immediately from Theorem 2.1. ^QED Remark 2.4 We note that:

1. If pn= 1 for all values of n then |N, p|ksummability reduces to |C, 1|ksummability 2. If k = 1 and pn= 1/(n + 1) then |N, p|k is equivalent to |R, log n, 1| summability.

Corollary 2.5 If for 1 ≤ k ≤ 2 the series X∞

n=0

n^1−1kqn

QnQ_n−1

!k( _n X

j=1

Q²_j−1|aj|² )^k₂

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, q|kalmost everywhere.

Proof. From the fact that

R^jn

Rn −R^j_n−1

R_n−1 = −qnQj−1

QnQ_n−1

for all pn= 1, the proof follows immediately from Theorem 2.1. ^QED Also, putting k = 1 in Corollaries 2.3 and 2.5 we obtain

Corollary 2.6 [1] If the series X∞

n=0

pn

PnP_n−1 ( _n

X

j=1

p²_n−j

Pn

pn −P_n−j p_n−j

2

|a^j|² )¹₂

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, p| almost everywhere.

Corollary 2.7 [2] If the series X∞ n=0

qn

QnQ_n−1 ( _n

X

j=1

Q²_j−1|aj|² )¹₂

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, q| almost everywhere.

If we put

w^(k)(j) := 1 j^k2−1

X∞ n=j

n^2k R^jn

Rn−R^j_n−1 R_n−1

!2

(3) then the following theorem holds true.

Theorem 2.8 Let 1 ≤ k ≤ 2 and {Ω(n)} be a positive sequence such that {Ω(n)/n}

is a non-increasing sequence and the series P_∞

n=1 1

nΩ(n) converges. Let {pn} and {qn} be non-negative. If the series P∞

n=1|an|²Ω^k2−1(n)w^(k)(n) converges, then the orthogonal series P∞

n=0anϕn(x) ∈ |N, p, q|k almost everywhere, where w^(k)(n) is defined by (3).

Proof. Applying H¨older’s inequality to the inequality (2) we get that X∞

n=1

n^k−1 Z b

a

|△t^p,qn (x)|^kdx ≤

≤ K X∞ n=1

n^k−1

" _n X

j=1

R^j_n

Rn −R^j_n−1 R_n−1

!2

|aj|²

#^k₂

= K X∞ n=1

1 (nΩ(n))^2−k2

"

nΩ^k2−1(n) Xn j=1

R^jn

Rn−R^j_n−1 R_n−1

!2

|aj|²

#^k₂

≤ K X∞ n=1

1 (nΩ(n))

!^2−k₂ " _∞ X

n=1

nΩ^2k−1(n) Xn j=1

R^jn

Rn −R^j_n−1 R_n−1

!2

|aj|²

#^k₂

≤ K ( _∞

X

j=1

|a^j|² X∞ n=j

nΩ^2k−1(n) R^j_n

Rn−R^j_n−1 R_n−1

!2)^k₂

≤ K ( _∞

X

j=1

|aj|²

Ω(j) j

²_{k−1 ∞}X

n=j

n^2k R^j_n

Rn−R^j_n−1 R_n−1

!2)^k₂

= K ( _∞

X

j=1

|aj|²Ω^k2−1(j)w^(k)(j) )^k₂

,

which is finite by assumption, and this completes the proof. ^QED Finally, as a direct consequence of the theorem 2.8 is the following (k = 1).

Corollary 2.9 [4] Let {Ω(n)} be a positive sequence such that {Ω(n)/n} is a non-increasing sequence and the seriesP∞

n=1 1

nΩ(n) converges. Let {pⁿ} and {qⁿ} be non-negative. If the series P∞

n=1|an|²Ω(n)w⁽¹⁾(n) converges, then the orthogonal seriesP∞

n=0anϕn(x) ∈ |N, p, q| almost everywhere, where w⁽¹⁾(n) is defined by

w⁽¹⁾(j) :=1 j

X∞ n=j

n² R^j_n

Rn−R_n−1^j R_n−1

!2

.

References

[1] Y. Okuyama, On the absolute N¨orlund summability of orthogonal series, Proc. Japan Acad. 54, (1978), 113-118.

[2] Y. Okuyama and T. Tsuchikura, On the absolute Riesz summability of orthogonal series, Analysis Math. 7, (1981), 199-208.

[3] M. Tanaka, On generalized N¨orlund methods of summability, Bull. Austral. Math. Soc.

19, (1978), 381-402.

[4] Y. Okuyama, On the absolute generalized N¨orlund summability of orthogonal series, Tamkang J. Math. Vol. 33, No. 2, (2002), 161-165.

[5] Y. Okuyama, Absolute summability of Fourier series and orthogonal series, Lecture Notes in Math. No. 1067, Springer-Verlag New York Tokyo 1984.