Interpolation for entire functions of exponential type and a related trigonometric moment problem
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- by Robert M. Young PDF
- Proc. Amer. Math. Soc. 56 (1976), 239-242 Request permission
Abstract:
A classical theorem of Hausdorff-Young shows that when $1 < p < 2$, the system of equations $\hat \varphi (n) = {c_n}( - \infty < n < \infty )$ admits a solution $\varphi$ in ${L^q}( - \pi ,\pi )$ whenever $\{ {c_n}\} \in {l^p}$. Here, as usual, $\hat \varphi$ denotes the complex Fourier transform of $\varphi$ and $q$ is the conjugate exponent given by ${p^{ - 1}} + {q^{ - 1}} = 1$. The purpose of this note is to show that if a set $\{ {\lambda _n}\}$ of real or complex numbers is “sufficiently close” to the inte gers, then the corresponding system $\hat \varphi ({\lambda _n}) = {c_n}$ is also solvable for $\varphi$ whenever $\{ {c_n}\} \in {l^p}$. The proof is accomplished by establishing a similar interpolation theorem for a related class of entire functions of exponential type.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 56 (1976), 239-242
- MSC: Primary 30A80
- DOI: https://doi.org/10.1090/S0002-9939-1976-0409832-3
- MathSciNet review: 0409832