A note on a trigonometric moment problem

Young, Robert M.

doi:10.1090/S0002-9939-1975-0367548-5

A note on a trigonometric moment problem
HTML articles powered by AMS MathViewer

by Robert M. Young PDF

Proc. Amer. Math. Soc. 49 (1975), 411-415 Request permission

Abstract:

A sequence $\{ {\lambda _n}\} _{n = - \infty }^\infty$ is said to be an interpolating sequence for ${L^2}( - \pi ,\pi )$ if the system of equations \[ {c_n} = \int _{ - \pi }^\pi {f(t)} {e^{i{\lambda _n}t}}dt\quad ( - \infty < n < \infty )\] admits a solution $f$ in ${L^2}( - \pi ,\pi )$ whenever $\{ {c_n}\} \in {l^2}$. If the solution is unique then $\{ {\lambda _n}\}$ is said to be a complete interpolating sequence. It is shown that if the imaginary part of ${\lambda _n}$ is uniformly bounded and if $|\operatorname {Re} ({\lambda _n}) - n| \leq L < 1/4( - \infty < n < \infty )$, then $\{ {\lambda _n}\}$ is a complete interpolating sequence and $\{ {e^{i{\lambda _n}t}}\}$ is a Schauder basis for ${L^2}( - \pi ,\pi )$. It is also shown that this result is sharp in the sense that the condition $|{\lambda _n} - n| < 1/4$ is not sufficient to guarantee that $\{ {\lambda _n}\}$ is an interpolating sequence.

References

Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. MR 47179, DOI 10.1090/S0002-9947-1952-0047179-6
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
M. Ĭ. Kadec′, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR 155 (1964), 1253–1254 (Russian). MR 0162088
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
Robert M. Young, Interpolation in a classical Hilbert space of entire functions, Trans. Amer. Math. Soc. 192 (1974), 97–114. MR 357823, DOI 10.1090/S0002-9947-1974-0357823-6
Robert M. Young, Inequalities for a perturbation theorem of Paley and Wiener, Proc. Amer. Math. Soc. 43 (1974), 320–322. MR 340948, DOI 10.1090/S0002-9939-1974-0340948-4