
On the Smallest and Largest Zeros of MÃ¼ntzLegendre Polynomials
MÃ¼ntzLegendre
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 $l_{n,n}\lt l_{n1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ 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 MÃ¼ntz space is
nondense 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:MÃ¼ntz polynomials, MÃ¼ntzLegendre polynomials Categories:42C05, 42C99, 41A60, 30B50 