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# A note on |N, p, q|

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k

### , (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 [email protected]

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

pmqn−m= Xn m=0

pn−mqm.

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

tp,qn = 1 (p ∗ q)n

Xn m=0

pn−mqmsm.

The infinite seriesP

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

n=0

nk−1|tp,qn − tp,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

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where an=Rb

af (x)ϕn(x)dx, (n = 0, 1, 2, . . . ).

We write (see [4])

Rn:= (p ∗ q)n, Rjn:=

Xn m=j

pn−mqm

and

Rn+1n = 0, R0n= 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

( n2−k2

Xn j=1

Rjn

Rn Rjn−1 Rn−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 tp,qn (x) of the partial sums of the orthogonal seriesP

n=0anϕn(x) we have that tp,qn (x) = 1

Rn

Xn m=0

pn−mqm

Xm j=0

ajϕj(x)

= 1 Rn

Xn j=0

ajϕj(x) Xn m=j

pn−mqm

= 1 Rn

Xn j=0

Rjnajϕj(x)

wherePm

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

As in [4] page 163 one can find that

△tp,qn (x) := tp,qn (x) − tp,qn−1(x) = Xn j=1

Rjn

Rn Rjn−1 Rn−1

!

ajϕj(x).

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

a

|△tp,qn (x)|kdx ≤ (b − a)1−k2

Z b a

|tp,qn (x) − tp,qn−1(x)|2dx

k2

= (b − a)1−k2

" n X

j=1

Rjn

Rn Rjn−1 Rn−1

!2

|aj|2

#k2 .

Hence, the series X n=1

nk−1 Z b

a

|△tp,qn (x)|kdx ≤ (b − a)1−k2 X n=1

nk−1

" n X

j=1

Rjn

RnRjn−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

Rjn

RnRjn−1 Rn−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

n1−1kpn

PnPn−1

!k( n X

j=1

p2n−j

Pn

pn Pn−j

pn−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 Rjn

Rn Rjn−1

Rn−1 = pn

PnPn−1

Pn

pn Pn−j pn−j

 pn−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.

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

n=0

n1−1kqn

QnQn−1

!k( n X

j=1

Q2j−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

Rjn

Rn Rjn−1

Rn−1 = −qnQj−1

QnQn−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

PnPn−1 ( n

X

j=1

p2n−j

Pn

pn Pn−j pn−j

2

|aj|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

QnQn−1 ( n

X

j=1

Q2j−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 jk2−1

X n=j

n2k Rjn

RnRjn−1 Rn−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|2k2−1(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).

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

n=1

nk−1 Z b

a

|△tp,qn (x)|kdx ≤

≤ K X n=1

nk−1

" n X

j=1

Rjn

Rn Rjn−1 Rn−1

!2

|aj|2

#k2

= K X n=1

1 (nΩ(n))2−k2

"

nΩk2−1(n) Xn j=1

Rjn

RnRjn−1 Rn−1

!2

|aj|2

#k2

≤ K X n=1

1 (nΩ(n))

!2−k2 " X

n=1

nΩ2k−1(n) Xn j=1

Rjn

Rn Rjn−1 Rn−1

!2

|aj|2

#k2

≤ K (

X

j=1

|aj|2 X n=j

nΩ2k−1(n) Rjn

RnRjn−1 Rn−1

!2)k2

≤ K (

X

j=1

|aj|2

Ω(j) j

2k−1 ∞X

n=j

n2k Rjn

RnRjn−1 Rn−1

!2)k2

= K (

X

j=1

|aj|2k2−1(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 Rjn

RnRn−1j Rn−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.

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