Necessary conditions for double summability factor theorem

Vol. 11 (2010), No. 2, pp. 183–189

NECESSARY CONDITIONS FOR DOUBLE SUMMABILITY FACTOR THEOREM

EKREM SAVAS¸

Received 20 December, 2009

Abstract. We obtain necessary conditions for the seriesP P cmn, which is absolutely sum- mable of order k by a doubly triangular matrix method A, to be such thatP P cmnmn is absolutely summable of order k by a doubly triangular matrix B.

2000 Mathematics Subject Classification: 40F05, 40D25

Keywords: absolute summability factors, doubly triangular summability

1. INTRODUCTION

A doubly infinite matrix AD .a^mnij/ is said to be doubly triangular if amnij D 0 for i > m and j > n. The mn-th terms of the A-transform of a double sequencefs^mng is defined by

TmnD

m

X

i D0 n

X

j D0

amnijsij:

A series P P cmn, with partial sums smn is said to be absolutely A-summable, of order k 1, if

1

X

mD1 1

X

nD1

.mn/^{k 1}j
¹¹Tm 1;n 1j^k<1; (1.1) where for any double sequencefumng, and for any four-fold sequence famnijg, we define

11umnD u^mn u_mC1;n u_m;nC1C umC1;nC1;

11amnij D a^mnij a_mC1;n;i;j a_m;nC1;i;jC amC1;nC1;i;j;

ijamnij D amnij a_{m;n;i C1;j} a_{m;n;i;j C1}C am;n;i C1;j C1;

i 0amnij D a^mnij a_{m;n;i C1;j}; and

0jamnij D a^mnij a_{m;n;i;j C1}:

The one-dimensional version of (1.1) appears in [1].

2010 Miskolc University Pressc

Associated with A there are two matrices A and OA defined by Na^mnij D

m

X

Di n

X

Dj

amn; 0 i  m; 0  j  n; m; n D 0; 1; : : : ; and

Oamnij D
11Nam 1;n 1;i;j; 0 i  m; 0  j  n; m; n D 1; 2; : : : : It is easily verified that Oa0000D Na0000D a0000. In [3] it is shown that

Oa^mnij D

i 1

X

D0 j 1

X

D0

11am 1;n 1;;: Thus Oamni 0D Oamn0j D 0.

Let xmndenote the mn-th term of the A-transform of the sequence of partial sums fs^mng of the seriesP P cmn.

Then

xmnD

m

X

i D0 n

X

j D0

amnijsij D

m

X

i D0 n

X

j D0

amnij i

X

D0 j

X

D0

c

D

m

X

D0 n

X

D0 m

X

i D

n

X

j D

amnuvc

D

m

X

D0 n

X

D0

Na^mnc;

and a direct calculation verifies that XmnWD
¹¹xm 1;n 1D

m

X

D1 n

X

D1

Oa^mnc; since

Nam 1;n 1;m; D am 1;n 1;;nD Oa^{m;n 1;;n}D Oa^{m 1;n;m;n}D 0:

In a recent paper Savas and Rhoades[2] obtained sufficient conditions for the series P P cmn , which is absolutely summable of order k by a doubly triangular matrix method A, to be such that P P cmnmn is absolutely summable of order k by a doubly triangular matrix B.

In this paper we obtain necessary conditions for the series P P cmn, which is absolutely summable of order k by a doubly triangular matrix method A, to be such thatP P cmnmnis absolutely summable of order k by a doubly triangular matrix B.

2. MAINTHEOREM

Theorem 1. LetA and B be doubly triangular matrices with A satisfying

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
^uvOa^mnuvj^kD O.M^k.Oa^uvuv//; (2.1) where

M.Oa^uvuv/WD maxfj Oa^uvuvj; j
^u0OauC1v;u;vj; j
^0vOauvC1;u;vjg:

Then the necessary conditions of the fact that the jAjk summability of P P cmn

implies thejBjk summability ofP P cmnmnare the following items:

(i) j Ob^uvuvuvj D O.M. Oa^uvuv//,

(ii) j
^u0Ob_uC1;v;u;vuvj D O.M. Oa^uvuv//, (iii) j
^0vOb_u;vC1;u;vuvj D O.M. Oa^uvuv//, (iv)

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
^uvObmnuvuvj^kD O..u/^{k 1}M^k.Oa^uvuv//,

(v)

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j Obm;n;uC1;vC1_uC1;vC1j^kD O

P₁

mDuC1

P₁

nDvC1.mn/^{k 1}j Oam;n;uC1;vC1j^k . Proof. For k 1 define

1

X

mD1 1

X

nD1

.mn/^{k 1}jY^mnj^k<1; (2.2) whenever

1

X

mD1 1

X

nD1

.mn/^{k 1}jX^mnj^k<1; (2.3) where

YmnD
¹¹ym 1;n 1;

ymnD

m

X

i D0 n

X

j D0

Nbmnijcijij:

The space of sequences satisfying (2.3) is a Banach space if it is normed by kXk D

jX⁰⁰j^kC jX⁰¹j^kC jX¹⁰j^kC

1

X

mD1 1

X

nD1

.mn/^{k 1}jX^mnj^k1=k

: (2.4)

We shall also consider the space of sequencesfY^mng that satisfy (2.2). This space is also a BK-space with respect to the norm

kY k D

jY⁰⁰j^kC jY⁰¹j^kC jY¹⁰j^kC

1

X

mD1 1

X

nD1

.mn/^{k 1}jY^mnj^k1=k

: (2.5)

The transformation Pm i D0

Pn

j D0Nbmnijcijij maps sequences satisfying (2.3) into sequences satisfying (2.2). By the Banach-Steinhaus Theorem there exists a constant K > 0 such that

kY k  KkXk: (2.6)

For fixed u; v; the sequencefc^ijg is defined by c^uvD cuC1;vC1D 1; cuC1;vD cu;vC1D 1; cij D 0, otherwise,

XmnD 8 ˆˆ ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ ˆˆ :

0; m u; n < v;

0; m < u; n v;

Oa^mnuv; mD u; n D v;

u0Oamnuv; mD u C 1; n D v;

0vOa^mnuv; mD u; n D v C 1;

uvOa^mnuv; m > u; n > v;

and

YmnD 8 ˆˆ ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ ˆˆ :

0; m u; n < v;

0; m < u; n v;

Obmnuv_uv; mD u; n D v;

u0Obmnuvuv; mD u C 1; n D v;

0vObmnuvuv; mD u; n D v C 1;

uvObmnuvuv; m > u; n > v:

From (2.4) and (2.5) it follows that kXk Dn

.uv/^{k 1}j Oauvuvj^kC ..u C 1/v/^{k 1}j
u0a_uC1;v;u;vj^k (2.7) C .u.v C 1//^{k 1}j
0va_u;vC1u;vj^k

C

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
uvOamnuvj^ko1=k

; and

kY k Dn

.uv/^{k 1}Obuvuvuvj^kC ..u C 1/v/^{k 1}j
^u0Ob_uC1;v;u;vuvj^k (2.8) C .u.v C 1//^{k 1}j
0vOb_u;vC1u;vuvj^k

C

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
^uvOa^mnuvuvj^ko1=k

: Substituting (2.7) and (2.8) into (2.6), along with (2.1), gives

.uv/^{k 1}j Obuvuvuvj^kC ..u C 1/v/^{k 1}j
u0Ob_uC1v;u;vuvj^k C.u.v C 1//^{k 1}j
^0vOb_u;vC1u;vuvj^k

C

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
^uvObmnuvuvj^k K^kn

.uv/^{k 1}j Oa^uvuvj^k

..uC 1/v/^{k 1}j
^u0OauC1v;u;vj^kC .u.v C 1//^{k 1}
u0Oau;vC1u;vj^k C

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
^uvOa^mnuvj^ko D K^kfO.1/.uv/^{k 1}M^k.Oa^uvuv/g:

The above inequality implies that each term of the left hand side is O.f.uv/^{k 1}M^k.Oa^uvuv/g/.

Using the first term one obtains

.uv/^{k 1}j Ob^uvuvuvj^kD O.f.uv/^{k 1}M^k.Oa^uvuv/g/:

Thus

j Obuvuvuvj D O.M. Oauvuv//;

which is condition (i).

In a similar manner one obtains conditions (ii) - (iv).

Using the sequence, defined by c_uC1;vC1D 1; and c^ij D 0 otherwise, yields

XmnD 8 ˆ<

ˆ:

0; m u C 1; n  v;

0; m u; n  v C 1;

Oam;n;uC1;vC1; m u C 1; n  v C 1 and

Y_mnD 8 ˆ<

ˆ:

0: m u C 1; n  v;

0; m u; n  v C 1;

Obm;n;uC1;vC1_uC1;vC1; m u C 1; n  v C 1:

The corresponding norms are kXk Dn X¹

mDuC1 1

X

nDvC1

.mn/^{k 1}j Oam;n;uC1;vC1j^ko1=k

and

kY k Dn X¹

mDuC1 1

X

nDvC1

.mn/^{k 1}j Obm;n;uC1;vC1_uC1;vC1j^ko1=k

: Applying (2.6), we have

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j Obm;n;uC1;vC1_uC1;vC1j^k

 K^kn X¹

mDuC1 1

X

nDvC1

.mn/^{k 1}j Oam;n;uC1;vC1j^ko

;

which implies (v). □

Every summability factor theorem becomes an inclusion theorem by setting each

mnD 1.

Corollary 1. Let A and B two doubly triangular matrices, A satisfying (2.1).

Then necessary conditions of the fact that thejAjksummability ofP P cmnimplies thejBjksummability ofP P cmnare the following items:

(i) j Ob^uvuvj D O.M. Oa^uvuv//,

(ii) j
u0Ob_uC1;v;u;vj D O.M. Oauvuv//, (iii) j
0vOb_u;vC1;u;vj D O.M. Oauvuv//, (iv)

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
uvObmnuvj^kD O..u/^{k 1}M^k.Oauvuv//, and

(v)

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j Obm;n;uC1;vC1j^k

D On X¹

mDuC1 1

X

nDvC1

.mn/^{k 1}j Oam;n;uC1;vC1j^ko1=k .

Proof. To prove Corollary1simply put mnD 1 in Theorem1. □ We shall call a doubly infinite matrix a product matrix if it can be written as the termwise product of two singly infinite matrices F and G; i.e., amnij D f^mignj for each i; j; m; n.

A doubly infinite weighted mean matrix P has nonzero entries pij=Pmn, where p00is positive and all of the other pij are nonnegative, and PmnWDPm

i D0

Pn j D0pij. If P is a product matrix then the nonzero entries are piqj=PmQn, where p0> 0; pi>

0 for i > 0; q0> 0; qi 0 for j > 0 and PmWDPm

i D0pi; QnWDPn

j D0qj. Now we have the following corollary

Corollary 2. LetB be a doubly triangular matrix, P a product weighted mean matrix satisfying

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}ˇ ˇ ˇ
uv

 pmqnPu 1Qv 1

PmPm 1QnQn 1

ˇ ˇ ˇ

k

D Opuqv

PuQv

 :

Then necessary conditions forP P cmnsummablejP jkto imply thatP P cmnmn

is summablejBjk are

(i) j Obuvuv_uvj D Opuqv

PuQv

 , (ii) j
^u0Ob_uC1;v;u;vuvj D Opuqv

PuQv

 , (iii) j
0vOb_u;vC1;u;vuvj D Opuqv

PuQv

 , (iv)

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j
^uvObuvuvuvj^k D O

.uv/^{k 1} puqv

P uQv

k , and

(v)

1

X

mDuC1 1

X

nDvC1

.mn/^{k 1}j Obm;n;uC1;vC1_uC1;vC1j^kD O.1/.

3. ACKNOWLEDGEMENTS

I wish to thank the referee for his careful reading of the manuscript and for his helpful suggestions.

REFERENCES

[1] B. E. Rhoades and E. Savas¸, “General summability factor theorems and applications,” Sarajevo J.

Math., vol. 1(13), no. 1, pp. 59–73, 2005.

[2] E. Savas¸ and B. E. Rhoades, “Double summability factor theorems and applications,” Math. In- equal. Appl., vol. 10, no. 1, pp. 125–149, 2007.

[3] E. Savas¸ and B. E. Rhoades, “Double absolute summability factor theorems and applications,”

Nonlinear Anal., vol. 69, no. 1, pp. 189–200, 2008.

Author’s address Ekrem Savas¸

Current address: Istanbul Ticaret University Department of Mathematics, Selman-i Pak Cad. 34672, Usk¨udar-Istanbul/Turkey¨

E-mail address: ekremsavas@yahoo.com