离散多项式的切比雪夫系统与施图姆振荡理论
Chebyshev systems and Sturm oscillation theory for discrete polynomials
AI总结：
该研究建立了离散函数族的切比雪夫交错定理与离散施图姆振荡定理，验证了相关系统性质，并应用于带谱隙的正交多项式，强化了已有结果且解决了一类极值问题。
摘要：
我们证明了定义在离散区间$[0,q]_{\scriptscriptstyle\mathbb{Z}}=[0,q]\cap \mathbb{Z}$上的线性无关离散函数族$\Phi_n=\{\varphi_k\}_{k=1}^n$的切比雪夫交错定理的离散类比。具体而言，我们证明了离散函数的最佳一致逼近多项式存在长度为$n+1$的切比雪夫交错点集，当且仅当$\Phi_n$是切比雪夫$T_{\scriptscriptstyle\mathbb{Z}}$-系统。我们还得到了施图姆振荡定理的离散版本：多项式$\sum_{k=m}^{n}a_k\varphi_k$的离散零点个数不小于$m-1$且不大于$n-1$。这一结论表明$\Phi_n$是$T_{\scriptscriptstyle\mathbb{Z}}$-系统，且离散施图姆-赫尔维茨谱隙定理成立。作为应用，我们研究了去掉最大零点的正交多项式，证明了这类多项式的傅里叶展开系数具有单调性，从而强化了H. Cohn与A. Kumar的结果，并将其用于求解带谱隙多项式的尤丁型极值问题。
英文摘要：
We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $Φ_n=\{φ_k\}_{k=1}^n$ on the interval $[0,q]_{\scriptscriptstyle\mathbb{Z}}=[0,q]\cap \mathbb{Z}$. 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 $Φ_n$ is a Chebyshev $T_{\scriptscriptstyle\mathbb{Z}}$-system. We also obtain a discrete version of Sturm's oscillation theorem, according to which the number of discrete zeros of the polynomial $\sum_{k=m}^{n}a_kφ_k$ is no less than $m-1$ and no more than $n-1$. This implies that $Φ_n$ is a $T_{\scriptscriptstyle\mathbb{Z}}$-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
