Theory of Some Sequence Spaces and Matrix Transformations: A Review

INTRODUCTION

The study of convergence of series is an art. The work done prior to the time of Leonard Euler (1707-1783), was concerned essentially with orthodox examination of convergence and those series which did not converge were of little or no importance. Augustin-Louis Cauchy (1789-1875) and Neil H. Abel (1802-1829) were amorvg the pioneers in introducing a rigorous foundation for the algebra of infinite scries. In his famous treatise; "Analyse Algebriques", Cauchy (1821) gave a rigorous definitive formulation of the new classical concept of "convergence" for infinite series. Given an infinite series N. Cauchy defined the „sum‟ of as the number l. Since there cannot exist two such numbers J, this definition of the „sum‟ of an infinite series is unique. A series which has a sum in Cauchy‟s sense is said to be convergent. A series which is not convergent is called divergent. So greatly emphasized by the idea of this now too familiar concept of convergence that infinite series which diverged were usually dropped as being without any mathematical significance. Indeed, the so-called „divergent series‟ were a sort of taboo in the eyes of many mathematicians including even a genius like Abel, who is known to have remarked: "Divergent series are the inventions of the devil and it is shameful to base on them any demonstration what so ever". For generalizing the notion of convergence, one can, for example, replace the sequence (xn) of partial sums of any given series by the sequence of „Arithmetic Means‟ or (C, 1) - means of (xn), denoted by where, If we call l the „(C,l) - sum‟ of (or the (C,l) -limit of (xn)) and say that the sequence (xn), or the series , is (C,l) - convergent (or (C, 1) - summable) to I. We observe that (i) may be summable (C, 1) to l whenever or (xn) converges to l and (ii) may be summable (C,l) even in certain cases in which fails to converge, e.g; 1-1 + 1-1+..., although divergent, is summable (C,l) to 1/2. More generally, we can replace by the sequence (yn) defined by the sequence-to-sequence matrix transformation (or A - mean): the series on the right defining yn being assumed to converge to y for each n, where is the element of matrix in the nth row and kth column. The sequence (xk), or the series will be said to be summable A to the sum l, if which were obtained by Cesaro (1890), Borel (1921) and others. It was however the celebrated German mathematician Toeplitz (1881-1940) who, in 1911, brought the methods of linear space theory to bear on problems connected with matrix transformations on sequence spaces. Toeplitz characterized all those infinite matrices which map the space c into itself, leaving the limit of each convergent sequence invariant. To be explicit, he gave the necessary and sufficient conditions on A for (as ), whenever (as ).

In 1970, Shiue studied and discussed the Cesaro sequence spaces and which were defined in Problem 2 in New Archives for Mathematics, 3, XVII, No. 1 (1968) as: with the finite norms : and respectively. In 1977, Lim defined the space ces(p) for with Inf as : where denotes a sum over the ranges which is paranormed by where with In 1977, Johnson and Mohapatra (1979) generalized the space to defined by Where is positive sequence of real numbers and is such that If qn=l, for all n, then ces(p, q) reduces to ces(pn) studied by Lim. Also if pn=p and qn=l for all n, then ces (p,q) reduces to cesp defined and discused by Shiue (1970).

An Orlicz function is a function which is continuous, non-decreasing and convex with ,, for x > 0 and as If convexity of Orlicz function M is replaced by then this function is called modulus function, defined and discussed by Ruckle and Maddox . An Orlicz function M is said to satisfy condition for all values of u, if there exists a constant K>0, such that The - condition is equivalent to for all values of u and for l > 1. An Orlicz function M can always be represented in the following integral form: where q(t), known as kernel of M, is right differentiable for for q(t) is non-decreasing and , as Lindenstrauss and Tzafriri (1971) used the idea of Orlicz functions to construct an Orlicz sequence space defined by for some }, The space with the norm : is a Banach space. We use throughout a sequence of positive real numbers

H. Kizmaz (1981) has introduced and studied the difference sequence spaces Recently, Mikail Et has defined the spaces Qamaruddin (2006) has studied the following difference sequence spaces of Orlicz type with the aid of certain convex function M, » We define the new difference Orlicz space in the same manner as Mikail Et With the help of sequence of Orlicz functions we also define

We know the following results. Theorem A. if, and only if, the condition

(1)

Theorem B. if, and only if, the condition :

(2)

holds; if, and only if, the condition holds; if, and only if, the condition

(4)

Converges uniformly in m, and

(5)

converges for all k, hold. Theorem C. if, and only if, the condition

(6)

holds. Theorem D. if, and only if, the conditions (1),

(7)

converges, hold. with for all if, and only if, the conditions (1),

(8)

For all k and

(9)

hold. Theorem E. if, and only if, the conditions (5) and

(10)

Nanda characterized the matrices and which are the generalizations of Theorem F. if, and only if, the conditions:

(11)

For some integer there exists such that

(12)

and there exists such that

(13)

hold Theorem G. if, and only if, the conditions and for all integer,

(14)

Converges uniformly in n, hold. Theorem H. if, and only if, the conditions (12) and (15) for all integers,there exists an integer, such that

(15)

hold. Our aim here is to characterize the matrix classes where is one of the sequence spaces and . We also investigate some matrix transformations in the case

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Faculty, Department of Commerce, PT. JLN Govt. College, Faridabad