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On the Smallest and Largest Zeros of Müntz-Legendre Polynomials

  Published:2011-06-29
 Printed: Mar 2013
  • Úlfar F. Stefánsson,
    School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
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Abstract

Müntz-Legendre polynomials $L_n(\Lambda;x)$ associated with a sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in $L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s are 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, $$ \exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}. $$ An important consequence is that if the associated Müntz space is non-dense in $L_2[0,1]$, then $$ \inf_{n}x_{n,n}\geq \exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0, $$ so the elements $L_n(\Lambda;x)$ have no zeros close to 0. Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed, $$ \lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n (2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2, $$ where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$.
Keywords: Müntz polynomials, Müntz-Legendre polynomials Müntz polynomials, Müntz-Legendre polynomials
MSC Classifications: 42C05, 42C99, 41A60, 30B50 show english descriptions Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
None of the above, but in this section
Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]
Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX]
42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
42C99 - None of the above, but in this section
41A60 - Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]
30B50 - Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX]
 

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