An Interpolation Problem for Coefficients of H∞Functions

H∞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 {nk}∞ k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, th...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1974-02, Vol.42 (2), p.402-408
Main Author: Fournier, John J. F.
Format: Article
Language:English
Subjects:
Coefficients
Fourier coefficients
Fourier series
Integers
Interpolation
Mathematical functions
Mathematical theorems
Polynomials
Series convergence
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Summary:H∞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 {nk}∞ k = 0is a sequence of nonnegative integers with$n_{k + 1} > (1 + \delta)n_k$for all k, and if$\sum^\infty_{k = 0} |v_k|^2 < \infty$, then there is a function g in H∞with$\hat g(n_k) = v_k$for all k. Previous proofs of this fact have not indicated how to construct such H∞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.
ISSN:0002-9939
1088-6826
DOI:10.2307/2039516