Chebyshev systems and Sturm oscillation theory for discrete polynomials

Abstract

We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions ${\Phi }_n=\{{\phi }_k{\}}_{k=1}^n$ on the interval $[0,q{]}_{\mathbb{Z}}=[0,q]\cap \mathbb{Z}$. In particular, we establish that the polynomial of best uniform approximation of a discrete function admits a Chebyshev alternance set of length $n+1$ if and only if ${\Phi }_n$ is a Chebyshev $T_{\mathbb{Z}}$-system. We also obtain a discrete version of Sturm's oscillation theorem, according to which the number of discrete zeros of the polynomial ${\sum }_{k=m}^n{a}_k{\phi }_k$ is no less than $m-1$ and no more than $n-1$. This implies that ${\Phi }_n$ is a $T_{\mathbb{Z}}$-system and a discrete Sturm-Hurwitz spectral gap theorem is valid. As applications, we study the orthogonal polynomials with removed largest zeros. We~establish the monotonicity property of coefficients in the Fourier expansions of such polynomials, thereby strengthening the results of H.~Cohn and A.~Kumar. We apply this to solve a Yudin-type extremal problem for polynomials with spectral gap.

31 pages