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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A biorthogonal expansion related to the zeros of an entire function


Author: Harold E. Benzinger
Journal: Proc. Amer. Math. Soc. 49 (1975), 135-142
MSC: Primary 47E05; Secondary 30A66, 34B25
MathSciNet review: 0365231
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Abstract: Let $ D(\lambda ) = {a_0} + \Sigma _{k = 1}^\infty {a_k}{e^{\lambda /k}}$, where the $ {a_k}$'s are given complex constants, with $ {a_0}{a_1} \ne 0,\Sigma _{k = 1}^\infty \vert{a_k}\vert < \infty $. Let $ \{ {\lambda _k}\} ,k = 1,2, \cdots $, denote the sequence of distinct zeros of $ D(\lambda )$, labeled in order of increasing modulus, and with multiplicities $ {m_k} \geq 1$. Let $ \{ {\phi _l}(x)\} $ denote the sequence of functions $ \{ {x^j}\exp ({\lambda _k}x):j = 0, \cdots ,{m_k} - 1;k = 1,2, \cdots ;0 < x \leq 1\} $. We show that for each $ p.1 \leq p < \infty $, there is a sequence $ \{ {\psi _l}(x)\} $ in $ {L^q}(0,1)\;(pq = p + q)$ such that $ ({\phi _l},{\psi _m}) = {\delta _{lm}}$. Then we show that $ \{ {\phi _l}\} $ is complete in $ {L^p}(0,1),1 \leq p < \infty $, and for $ 1 < p < \infty $, we find a subspace of $ {L^p}(0,1)$ such that the biorthogonal expansion $ f = \Sigma _{k = 1}^\infty (f,{\psi _k}){\phi _k}$ is valid in the norm of $ {L^p}(0,1)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0365231-3
PII: S 0002-9939(1975)0365231-3
Article copyright: © Copyright 1975 American Mathematical Society