ALMOST SUMMABLE SEQUENCES

King[3] introduced and examined the concepts of almost A-summable sequence, 
almost conservative matrix and almost regular matrix By following King, in this paper we introduce 
and examine the concepts of θ-almost A-summable sequence, θ-almost conservative matrix and θ-almost regular matrix


I. INTRODUCTION
Let A=(ak) be an infinite matrix of complex numbers and let x={x.be a sequence of complex numbers.The sequence {A,,(x)} defined by s called the A-transform of x whenever the series converges for n=1,2,3, The sequence x is stud to be A-summable to L if A. (x)} converges to L Let m denote the linear space of bounded sequences A sequence x m s saad to be almost converget to L if lim L uniformly m .m .ooA sequence x e m is said to be almost A-summable to L ff the A-transform of x is almost converget to L. The matrix A is said to be almost conservative if x e mphes that the A-transform of x is almost convergent A s said to be almost regular if the A-transform of x almost convergent to the limit of x for each x , where is the linear space of convergent sequences.
In the sequel the following notation is used C denotes the complex numbers, Z denotes the integers and N denotes positive integers.If x={x is an elemet of c, then [Ixl[ is defined by })x[[ suP{Ix k N} The linear space of all continuous linear funcaonals oriels denoted by c*.
By a lacunary sequence we shall mean an increasing integer sequence 0 (k,) such that k 0 and h, k -k_as r---> oc Throughout ths paper the intervals determined by O (kr) will be denoted by I, (k,_, ,k, ]. Freedman, Sember and Raphael[2] defined the space N o m the following manner: For any lacunary sequence 0 (k,) { hm lx_Lt= 0 forsomeL} Qu,t,-+my, the pac Mo w ,r,t,odud by O ad M,h,a[q a bow { hm llx } Mo: x (x,): r ,-.-,, -+ oc z-, .Lt 0, for some L, uniformly n m F. NURAY King[3], Introduced and examined the concepts of almost A-summable sequence, almost conservat,ve matrix and almost regular matrix In the present note, we introduce and examine the concepts of 0-almost A-summable sequence, 0-almost conservative matrix and 0-almost regular matrix.
2. DEFINITIONS AND TI:IEOREMS Definition A sequence x ts said to be 0-almost convergent to L f, for any lacunary sequence /9, hm x --,.m=L tmJformlynm r -h k,-_l, Definition 2. A sequence x is stud to be 0-almost A-summable to L f, for any lacunary sequence 0, the A-transform of x is 0-almost convergent to L.
Definition3.The matrix A m stud to be 0-almost conservanve if x mphes that the Atransform of x is 0-almost convergent.A Is stud to be 0-almost regular f the A-transform of x s 0- almost convergent to the hmit of X for each x .
Theorem 1.Let A=(a,) be an infinite matrix and let 0 be a lacunary sequence.Then the matrix A s 0-almost conservative if and only f 0) anj, rN -<c n=0,1,2,.
(,) there exasts an a C such that hm X"'-"a --> cz., z.., ,/J.k a uruformly n n, and r ,., .-:,k- (m) there exmts an s C, k=O, 1,2 such that lim , , a,j. a umformly n n. r-->och .,-,Proof.Suppose that A is 0-almost conservative.Fix n N. Let Hence t c* Snce A is 0-almost conservat,ve lm ,(x) t(x) uniformly in n.It follows that {t,(x)} ,s bounded for x e and fixed n r --> oo {II',oli} bou dod by,h pnnciple For each p N, define the sequence v=v(n,r) by v,.{sgn _ , a ,, O < _ k < _ p p .< k Then v s c, jlvjj where b=hmk Xk But (e)=a and m()= ,k=0,1,2 by(n) d (m), respsecnvely.Hence hm (x)(x) exs for each x c n=0,1,2,.with t(xb a a. + a, xe Snce t c* for each and n, t h the fo ,.,(x) bt(e) tr(e )] + x.t,e, ( It s ey to see from (1) d ( 2) that the convergence of {/(x)} to t(x) is umfo m n, mnce t.(e) a d t.(e a. unifoly tn n Therefore A ts 0-almost conseave and the theorem s proved Theorem 2 Let A=(k) be an tnfimte matrix and let be a lacuna sequence Then the a,, 0 umformly m n k 0,1,2, r -cc h, j,:l, Proof.Suppose that A is 0-almost regularThen A s 0-almost conservative so that 0v) must hold by Theorem (v) and (vi) must hold since the A-transform of the sequences et and e must be 0- almost convergent to 0 and 1, respecnvely.
Now suppose that (iv),(v) and (v0 hold Then A s 0-almost concervanve by Theorem Therefore limr t(x)--t(x) uniformly n n for each x e c The representanon (1) gves t(x)=hmk Xk.Hence A is 0-almost regular.This proves the theorem In the applicanons of summability theory to function theory it s mportant to know the reedon in which the sequence of partial sums of the geometric series m A-summable to l/(l-z) for a gven matrix A The following theorem s helpful m determining the region m which the sequence of pamal sums of the geometric series is O-almost A-summable to l/(l-z) Theorem 3. Let A be an infimte matrix such that (v) holds The sequence of parnal sums of the geometric series is 0-almost A-summable to 1/(l-z) if and only if z f2 where { lim o :=0 tmifrmlyinn 1 '-2= Z: Z a,,_j Hence, Therefore Proof.Let {s (:)} denote the sequence of parnal sums of the geometric series Then I-l-z +-] t(x) Trr .... ,,0-: hm t., that wj,.c*, j--n= 1,2

1-- 2 limr
t= umfomly m n f and only f z 1-z A ,s a fundamental set m c t follows from an elementary result of functional analyms that hm (x)=t.(x)es and t. c*.Therefore h the fo Since