Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 0330469
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A80, 30A78

Retrieve articles in all journals with MSC: 30A80, 30A78

Additional Information

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