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An interpolation problem for coefficients of $ H\sp{\infty }$ functions


Author: John J. F. Fournier
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
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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 \vert{v_k}{\vert^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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0330469-7
Keywords: Interpolation by Fourier coefficients, $ {H^\infty }$ function, Rudin-Shapiro polynomials, Hadamard set, $ \Lambda (2)$ set, Riesz product
Article copyright: © Copyright 1974 American Mathematical Society

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