Tauberian Theorems for Certain Regular Processes Page: 3
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preserving Z-Z matrix (Definition 1.4) that transforms every
rearrangement of x into k.
In 1970 1. J. Maddox (11) obtained what might be con-
sidered as the ultimate improvement of Buck's theorem. He
considered a matrix A which summed every subsequence of a
diVergent sequence x and showed that A must be Schur (Defi-
nition 1.3, Theorem 1.6). Since the class of Schur matrices
is disjoint from the class of regular matrices, Buck's
theorem follows as a corollary. Recently, Dawson (6) has
obtained an analog to this result of Maddox involving
stretchings.. The second and third chapters of this paper
contain theorems which follow the pattern established by
Maddox and Dawson. In the second chapter an analog is proved
in which "subsequence" is replaced with "rearrangement"
(Theorem 2.3). The third chapter deals with absolute sum-
mability, and a theorem is obtained which has Fridy's char-
acterization of X as a corollary. This theorem shows that
if x is in c0 (the space of all null complex sequences)
but not in k and the matrix A transforms every rearrangement
of x into Z, then A is not sum-preserving t-Z (Theorem 3.2).
In addition, the following question proposed by J. A.
Fridy (8, p. 9) is answered in the affirmative. Is a.null
sequence x necessarily in t in case there is a sum-preserving
Z-k matrix A such that Ay is in k for every subsequence y
of x? (Theorem 3.1).
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Keagy, Thomas A. Tauberian Theorems for Certain Regular Processes, dissertation, August 1975; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc501087/m1/8/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .