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Pointwise and norm convergence of a class of biorthogonal expansions


Author: Harold E. Benzinger
Journal: Trans. Amer. Math. Soc. 231 (1977), 259-271
MSC: Primary 42A60
DOI: https://doi.org/10.1090/S0002-9947-1977-0442588-2
MathSciNet review: 0442588
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Abstract: Let $ \{ {u_k}(x)\} ,\{ {v_k}(x)\} ,k = 0, \pm 1, \ldots ,0 \leqslant x \leqslant 1$, be sequences of functions in $ {L^\infty }(0,1)$, such that $ ({u_k},{v_j}) = {\delta _{kj}}$. Let $ {\phi _k}(x) = \exp \;2k\pi ix$. It is shown that if for a given p, $ 1 < p < \infty $, the sequence $ \{ {u_k}\} $ is complete in $ {L^p}(0,1)$, and $ \{ {v_k}\} $ is complete in $ {L^q}(0,1),pq = p + q$, and if the $ {u_k}$'s, $ {v_j}$'s are asymptotically related to the $ {\phi _k}$'s, in a sense to be made precise, then $ \{ {u_k}\} $ is a basis for $ {L^p}(0,1)$, equivalent to the basis $ \{ {\phi _k}\} $, and for every f in $ {L^p}(0,1)$ a.e. This result is then applied to the eigenfunction expansions of a large class of ordinary differential operators.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0442588-2
Keywords: Biorthogonal expansions, maximal function mapping, eigenfunction expansions
Article copyright: © Copyright 1977 American Mathematical Society

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