Chebyshev systems and Sturm oscillation theory for discrete polynomials

We prove an analogue of Chebyshev’s alternation theorem for linearly independent discrete functions ${\mathrm{\Phi }}_n=\{{\phi }_k{\}}_{k=1}^n$ 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 $f$ admits a Chebyshev alternance set of length $n+1$ if and only if ${\mathrm{\Phi }}_n$ is a Chebyshev $T_{\mathbb{Z}}$-system. Also, we 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 ${\mathrm{\Phi }}_n$ 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.