On the Smallest and Largest Zeros of Müntz-Legendre Polynomials

Canadian mathematical bulletin

Úlfar F. Stefánsson¹ 

[1] School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

关键词: Müntz polynomials;    Müntz-Legendre polynomials;   

DOI  :  10.4153/CMB-2011-135-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Müntz-Legendre polynomials $L_n(Lambda;x)$ associated with asequence $Lambda={lambda_k}$ are obtained by orthogonalizing thesystem $(x^{lambda_0}, x^{lambda_1}, x^{lambda_2}, dots)$ in$L_2[0,1]$ with respect to the Legendre weight. If the $lambda_k$'sare distinct, it is well known that $L_n(Lambda;x)$ has exactly $n$zeros $l_{n,n}lt l_{n-1,n}lt cdots lt l_{2,n}lt l_{1,n}$ on $(0,1)$. First we prove the following global bound for the smallest zero, $$expiggl(-4sum_{j=0}^n frac{1}{2lambda_j+1}iggr) lt l_{n,n}.$$An important consequence is that if the associated Müntz space isnon-dense in $L_2[0,1]$, then $$inf_{n}x_{n,n}geqexpiggl({-4sum_{j=0}^{infty} frac{1}{2lambda_j+1}}iggr)gt 0,$$sothe elements $L_n(Lambda;x)$ have no zeros close to 0. Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,$$lim_{nightarrowinfty} vert log l_{k,n}vert sum_{j=0}^n(2lambda_j+1)= Bigl(frac{j_k}{2}Bigr)^2, $$where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$.

【 授权许可】

Unknown