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

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.