We acknowledge valuable discussions with Dr. J. Lindenstrauss of Yale University. An infinite matrix A is called conservative if for each convergent sequence x the transformed sequence Ax is convergent. It is a classical theorem that a necessary and sufficient condition that a matrix A be conservative is that: (1) \\A\\ =supi Ey" i \aa\ < °°> (2) lim,-,«, EyTM iaa exists, (3) lim,-,M an exists ...

The concepts of convergence and divergence, while not defined explicitly until early 19th century, have been studied since the third century BC. Throughout this time, mathematicians have largely focused upon convergent sequences and series, having seemingly little analytical reason to study series that diverged. However, as the area of mathematical analysis developed over the past few centuries...

We investigate the effect of four-dimensional matrix transformation on new classes of double sequences. Stretchings of a double sequence is defined, and this definition is used to present a four-dimensional analogue of D. Dawson’s copy theorem for stretching of a double sequence. In addition, themultidimensional analogue of D. Dawson’s copy theorem is used to characterize convergent double sequ...

Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is....

Karamata's Tauberian theorem relates the asymptotics of a nondecreasing right-continuous function to that of its Laplace-Stieltjes transform, using regular variation. This paper establishes the analogous Tauberian theorem for matrix-valued functions. Some applications to time series analysis are indicated.