On a theorem of Ingham on nonharmonic Fourier series
HTML articles powered by AMS MathViewer
- by Robert M. Young
- Proc. Amer. Math. Soc. 92 (1984), 549-553
- DOI: https://doi.org/10.1090/S0002-9939-1984-0760944-9
- PDF | Request permission
Abstract:
A well-known result due to Ingham [3] shows that the system of complex exponentials $\{ {e^{i\lambda _{n}t}}\}$ is a basic sequence in ${L^2}( - \pi ,\pi )$ whenever ${\lambda _{n + 1}} - {\lambda _n} \geqslant \gamma > 1$. In this note, we show that the system need not be basic if ${\lambda _{n + 1}} - {\lambda _n} > 1$.References
- E. T. Copson, Asymptotic expansions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 55, Cambridge University Press, New York, 1965. MR 0168979
- S. V. Hruščëv, N. K. Nikol′skiĭ, and B. S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 214–335. MR 643384
- A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z. 41 (1936), no. 1, 367–379. MR 1545625, DOI 10.1007/BF01180426
- Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR 0003208
- Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
- Raymond M. Redheffer, Completeness of sets of complex exponentials, Advances in Math. 24 (1977), no. 1, 1–62. MR 447542, DOI 10.1016/S0001-8708(77)80002-9
- Raymond M. Redheffer and Robert M. Young, Completeness and basis properties of complex exponentials, Trans. Amer. Math. Soc. 277 (1983), no. 1, 93–111. MR 690042, DOI 10.1090/S0002-9947-1983-0690042-8
- Laurent Schwartz, Étude des sommes d’exponentielles. 2ième éd, Publications de l’Institut de Mathématique de l’Université de Strasbourg, V. Actualités Sci. Ind., Hermann, Paris, 1959 (French). MR 0106383
- Ivan Singer, Bases in Banach spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154, Springer-Verlag, New York-Berlin, 1970. MR 0298399 E. C. Titchmarsh, A class of trigonometrical series, J. London Math. Soc. 3 (1928), 300-304.
- Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684
Bibliographic Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 549-553
- MSC: Primary 42C15
- DOI: https://doi.org/10.1090/S0002-9939-1984-0760944-9
- MathSciNet review: 760944