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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Nonharmonic Fourier series and spectral theory
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by Harold E. Benzinger PDF
Trans. Amer. Math. Soc. 299 (1987), 245-259 Request permission

Abstract:

We consider the problem of using functions ${g_n}(x): = exp(i{\lambda _n}x)$ to form biorthogonal expansions in the spaces ${L^p}( - \pi , \pi )$, for various values of $p$. The work of Paley and Wiener and of Levinson considered conditions of the form $\left | {{\lambda _n} - n} \right | \leq \Delta (p)$ which insure that $\{ {g_n}\}$ is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for $p = 2$. In this paper, rather than imposing an explicit growth condition, we assume that $\{ {\lambda _n} - n\}$ is a multiplier sequence on ${L^p}( - \pi , \pi )$. Conditions are given insuring that $\{ {g_n}\}$ inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, ${\lambda _n}$ and ${g_n}$ are shown to be the eigenvalues and eigenfunctions of an unbounded operator $\Lambda$ which is closely related to a differential operator, $i\Lambda$ generates a strongly continuous group and $- {\Lambda ^2}$ generates a strongly continuous semigroup. Half-range expansions, involving ${\text {cos}}{\lambda _n}x$ or ${\text {sin}}{\lambda _n}x$ on $(0, \pi )$ 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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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