An interpolation problem for coefficients of $H^{\infty }$ functions
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- by John J. F. Fournier
- Proc. Amer. Math. Soc. 42 (1974), 402-408
- DOI: https://doi.org/10.1090/S0002-9939-1974-0330469-7
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Abstract:
${H^\infty }$ denotes the space of all bounded functions g on the unit circle whose Fourier coefficients $\hat g(n)$ are zero for all negative n. It is known that, if $\{ {n_k}\} _{k = 0}^\infty$ is a sequence of nonnegative integers with ${n_{k + 1}} > (1 + \delta ){n_k}$ for all k, and if $\sum _{k = 0}^\infty |{v_k}{|^2} < \infty$, then there is a function g in ${H^\infty }$ with $\hat g({n_k}) = {v_k}$ for all k. Previous proofs of this fact have not indicated how to construct such ${H^\infty }$ functions. This paper contains a simple, direct construction of such functions. The construction depends on properties of some polynomials similar to those introduced by Shapiro and Rudin. There is also a connection with a type of Riesz product studied by Salem and Zygmund.References
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Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 402-408
- MSC: Primary 30A80; Secondary 30A78
- DOI: https://doi.org/10.1090/S0002-9939-1974-0330469-7
- MathSciNet review: 0330469