### A note on |N, p, q|

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]

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−m}qm.

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

t^{p,q}_{n} = 1
(p ∗ q)n

Xn m=0

p_{n−m}qmsm.

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,q}n − t^{p,q}n−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 L^{2}(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^{j}n:=

Xn m=j

pn−mqm

and

R^{n+1}_{n} = 0, R^{0}_{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−}^{k}^{2}

Xn j=1

R^{j}_{n}

Rn −R^{j}_{n−1}
R_{n−1}

!2

|aj|^{2}
)^{k}_{2}

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−m}qm

Xm j=0

ajϕj(x)

= 1 Rn

Xn j=0

ajϕj(x) Xn m=j

p_{n−m}qm

= 1 Rn

Xn j=0

R^{j}_{n}ajϕj(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,q}n (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,q}n (x)|^{k}dx ≤ (b − a)^{1−}^{k}^{2}

Z b a

|t^{p,q}n (x) − t^{p,q}_{n−1}(x)|^{2}dx

^{k}2

= (b − a)^{1−}^{k}^{2}

" _{n}
X

j=1

R^{j}_{n}

Rn −R^{j}_{n−1}
R_{n−1}

!2

|a^{j}|^{2}

#^{k}_{2}
.

Hence, the series X∞ n=1

n^{k−1}
Z b

a

|△t^{p,q}n (x)|^{k}dx ≤ (b − a)^{1−}^{k}^{2}
X∞
n=1

n^{k−1}

" _{n}
X

j=1

R^{j}n

Rn−R^{j}_{n−1}
Rn−1

!2

|aj|^{2}

#^{k}_{2}
(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^{j}n

Rn−R^{j}_{n−1}
R_{n−1}

!2

|aj|^{2}
)^{1}_{2}

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−}^{1}^{k}pn

PnP_{n−1}

!k( _{n}
X

j=1

p^{2}_{n−j}

Pn

pn −Pn−j

p_{n−j}

2

|aj|^{2}
)^{k}_{2}

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−}^{1}^{k}qn

QnQ_{n−1}

!k( _{n}
X

j=1

Q^{2}_{j−1}|aj|^{2}
)^{k}_{2}

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, q|kalmost everywhere.

Proof. From the fact that

R^{j}n

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

p^{2}_{n−j}

Pn

pn −P_{n−j}
p_{n−j}

2

|a^{j}|^{2}
)^{1}_{2}

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

Q^{2}_{j−1}|aj|^{2}
)^{1}_{2}

converges, then the orthogonal series X∞ n=0

anϕn(x)

is summable |N, q| almost everywhere.

If we put

w^{(k)}(j) := 1
j^{k}^{2}^{−1}

X∞ n=j

n^{2}^{k} R^{j}n

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|^{2}Ω^{k}^{2}^{−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,q}n (x)|^{k}dx ≤

≤ 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}

#^{k}_{2}

= K X∞ n=1

1
(nΩ(n))^{2−k}^{2}

"

nΩ^{k}^{2}^{−1}(n)
Xn
j=1

R^{j}n

Rn−R^{j}_{n−1}
R_{n−1}

!2

|aj|^{2}

#^{k}_{2}

≤ K X∞ n=1

1 (nΩ(n))

!^{2−k}_{2} " _{∞}
X

n=1

nΩ^{2}^{k}^{−1}(n)
Xn
j=1

R^{j}n

Rn −R^{j}_{n−1}
R_{n−1}

!2

|aj|^{2}

#^{k}_{2}

≤ K
( _{∞}

X

j=1

|a^{j}|^{2}
X∞
n=j

nΩ^{2}^{k}^{−1}(n) R^{j}_{n}

Rn−R^{j}_{n−1}
R_{n−1}

!2)^{k}_{2}

≤ K
( _{∞}

X

j=1

|aj|^{2}

Ω(j) j

^{2}_{k}_{−1 ∞}X

n=j

n^{2}^{k} R^{j}_{n}

Rn−R^{j}_{n−1}
R_{n−1}

!2)^{k}_{2}

= K
( _{∞}

X

j=1

|aj|^{2}Ω^{k}^{2}^{−1}(j)w^{(k)}(j)
)^{k}_{2}

,

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 {p^{n}} and {q^{n}} be non-negative. If the series
P∞

n=1|an|^{2}Ω(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

n^{2} 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.