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On the pointwise convergence of a class of nonharmonic Fourier series


Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 89 (1983), 65-73
MSC: Primary 42C15; Secondary 42C30
DOI: https://doi.org/10.1090/S0002-9939-1983-0706513-7
MathSciNet review: 706513
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Abstract: Extending a classical theorem of Levinson [1, Theorem XVIII], we show that when the numbers $ \left\{ {{\lambda _n}} \right\}$ are given by $ {\lambda _n} = n + \tfrac{1}{4}(n > 0)$, $ {\lambda _0} = 0$, and $ {\lambda _{ - n}} = - {\lambda _n}(n > 0)$, each function $ f$ in $ {L^2}( - \pi ,\pi )$ has a unique nonharmonic Fourier expansion $ f(t) \sim \sum\nolimits_{ - \infty }^\infty {{c_n}{e^{i{\lambda _n}t}}} $, which is equiconvergent with its ordinary Fourier series, uniformly on each closed subinterval of $ ( - \pi ,\pi )$.


References [Enhancements On Off] (What's this?)

  • [1] N. Levinson, Gap and density theorems, Amer. Math. Soc. Colloq. Publ., vol. 26, Amer. Math. Soc., Providence, R.I., 1940. MR 0003208 (2:180d)
  • [2] R. M. Redheffer and R. M. Young, Completeness and basis properties of complex exponentials, Trans. Amer. Math. Soc. 277 (1983), 93-111. MR 690042 (84c:42047)
  • [3] R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, 1980. MR 591684 (81m:42027)
  • [4] -, On complete biorthogonal systems, Proc. Amer. Math. Soc. 83 (1981), 537-540. MR 627686 (84c:42048)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0706513-7
Keywords: Biorthogonal system, equiconvergence, nonharmonic Fourier series
Article copyright: © Copyright 1983 American Mathematical Society

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