Nonharmonic Fourier series and spectral theory

Author:
Harold E. Benzinger

Journal:
Trans. Amer. Math. Soc. **299** (1987), 245-259

MSC:
Primary 42A65; Secondary 34B25, 42A20, 47B38

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869410-0

MathSciNet review:
869410

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Abstract: We consider the problem of using functions to form biorthogonal expansions in the spaces , for various values of . The work of Paley and Wiener and of Levinson considered conditions of the form which insure that is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for . In this paper, rather than imposing an explicit growth condition, we assume that is a multiplier sequence on . Conditions are given insuring that inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, and are shown to be the eigenvalues and eigenfunctions of an unbounded operator which is closely related to a differential operator, generates a strongly continuous group and generates a strongly continuous semigroup. Half-range expansions, involving or on are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0869410-0

Article copyright:
© Copyright 1987
American Mathematical Society