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The purpose of this paper is (1) to highlight some recent and heretofore
unpublished results in the theory of multiplier sequences and (2) to survey
some open problems in this area of research. For the sake of clarity of exposition,
we have grouped the problems in three subsections, although several of the problems
are interrelated. For the reader’s convenience, we have included the pertinent
definitions, cited references and related results, and in several instances, elucidated
the problems by...
The following problem, suggested by Laguerre’s Theorem (1884),
remains open: Characterize all real sequences {μk} k=0...∞
which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then
k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x).
In this paper this problem is solved under the additional assumption of a weak
growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞.
More precisely, it is established that...
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