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Uniqueness for non-harmonic trigonometric series

Author(s): Kaora Yoneda
Journal: Proc. Amer. Math. Soc. 124 (1996), 1795-1800.
MSC (1991): Primary 42A63
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Abstract: When $\lambda _{n} > 0$ , $\lambda _{n} \uparrow \infty$ and

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

$\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:

1.: N. K. Bary, Treatise on trigonometric series, Pergamon Press, New York, 1964. MR 30:1347
2.: A. Zygmund, Über die Beziehungen der trigonometrisch en Reihen und Integrale, Mat. Anal. 99 (1928), 562--589.
3.: ------, Trigonometric series, vol. 1, Cambridge University Press, New York, 1959. MR 21:6498


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