Title: Chebyshev systems and Sturm oscillation theory for discrete polynomials Show Chinese title

Title: 切比雪夫系统和离散多项式的斯特姆振动理论

Abstract: We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $\Phi_n=\{\varphi_k\}_{k=1}^n$ 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 $\Phi_n$ 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 $\sum_{k=m}^{n}a_k\varphi_k$ 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. Show Chinese abstract

Abstract: 我们证明了Chebyshev交替定理在区间$[0,q]_{\mathbb{Z}}=[0,q]\cap \mathbb{Z}$上线性无关离散函数$\Phi_n=\{\varphi_k\}_{k=1}^n$的类似情况。特别地，我们建立了一个离散函数$f$的最佳一致逼近多项式当且仅当$\Phi_n$是一个Chebyshev$T_{\mathbb{Z}}$-系统时，其具有长度为$n+1$的Chebyshev交替集。 此外，我们得到了斯特姆振动定理的离散版本，根据该定理，多项式$\sum_{k=m}^{n}a_k\varphi_k$的离散零点个数不少于$m-1$且不超过$n-1$。 这表明$\Phi_n$是一个$T_{\mathbb{Z}}$-系统，并且有效的离散斯特姆-赫尔维茨谱间隙定理。 作为应用，我们研究了去除最大零点的正交多项式。 我们建立了此类多项式傅里叶展开系数的单调性性质，从而加强了 H. Cohn 和 A. Kumar 的结果。 我们将此应用于解决具有谱间隙的多项式的尤丁型极值问题。