A note on a trigonometric moment problem
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- 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
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 49 (1975), 411-415
- MSC: Primary 42A80
- DOI: https://doi.org/10.1090/S0002-9939-1975-0367548-5
- MathSciNet review: 0367548