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Uniqueness for non-harmonic
trigonometric series


Author: Kaora Yoneda
Journal: Proc. Amer. Math. Soc. 124 (1996), 1795-1800
MSC (1991): Primary 42A63
DOI: https://doi.org/10.1090/S0002-9939-96-03427-2
MathSciNet review: 1328382
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Abstract: When $\lambda _{n} > 0$, $\lambda _{n} \uparrow \infty $ and

\begin{equation*}\frac {1}{2}\left |a_{0}\right | +\sum _{n=1}^{\infty }\frac {\left |a_{n}\right |+\left |b _{n}\right |}{\lambda _{n}^{2}} < \infty , \end{equation*}

if

\begin{equation*}\frac {1}{2}a_{0}+\sum _{n=1}^{\infty }(a_{n}\cos \lambda _{n}x+b_{n}\sin \lambda _{n}x) = 0 \text {\quad everywhere $(-\infty , \infty )$}, \end{equation*}

then

\begin{equation*}a_{0}=a_{1}=b_{1}=\dots =a_{n}=b_{n}=\dots =0. \end{equation*}

More generalized results are given.

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

  • 1. N. K. Bary, A treatise on trigonometric series. Vols. I, II, Authorized translation by Margaret F. Mullins. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0171116
  • 2. A. Zygmund, Über die Beziehungen der trigonometrisch en Reihen und Integrale, Mat. Anal. 99 (1928), 562--589.
  • 3. A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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Additional Information

Kaora Yoneda
Affiliation: Department of Mathematics and Information Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 593, Japan
Email: yoneda@mathsun.cias.osakafu-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-96-03427-2
Keywords: Uniqueness, trigonometric series
Received by editor(s): March 30, 1994
Received by editor(s) in revised form: November 29, 1994
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society