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

 

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
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 )$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0706513-7
PII: S 0002-9939(1983)0706513-7
Keywords: Biorthogonal system, equiconvergence, nonharmonic Fourier series
Article copyright: © Copyright 1983 American Mathematical Society