Generalized Rudin-Shapiro Systems

Abstract

The classical Rudin–Shapiro construction produces a sequence of polynomials with ±1 coefficients such that on the unit circle each such polynomial P satisfies the "flatness" property ||P||_∞ ≤ √2||P||₂. It is shown how to construct blocks of such flat polynomials so that the polynomials in each block form an orthogonal system. The construction depends on a fundamental generating matrix and a recursion rule. When the generating matrix is a multiple of a unitary matrix, the flatness, orthogonality, and other symmetries are obtained. Two different recursion rules are examined in detail and are shown to generate the same blocks of polynomials although with permuted orders. When the generating matrix is the Fourier matrix, closed-form formulas for the polynomial coefficients are obtained. The connection with the Hadamard matrix is also discussed.