The weighted \(L_ p\)-norms of orthonormal polynomials for Freud weights
The paper is concerned with orthogonal polynomials for weights on the real line. The weight functions are of the form \(W : = e^{ - Q}\), where \(Q : \mathbb{R} \to \mathbb{R}\) is even, continuous on \(\mathbb{R}\), \(Q''\) is continuous in \((0, \infty)\), and \(Q' > 0\) in \((0, \infty)\), while for some \(A,B>1\), \[ A \leq {d \over dx} \bigl( xQ'(x) \bigr)/Q'(x) \leq B, \quad x \in (0, \infty), \] has to hold. For example, the Freud weights \(W(x) = \exp ( - | x |^ \alpha)\), \(\alpha > 1\), satisfy these hypotheses. Let \(p_ n (W^ 2,x)\) denote the \(n\)th orthonormal polynomial for the weight function \(W^ 2\), \(n \geq 1\). For instance in case of \(W(x) = \exp ( - x^ 2)\) the \(p_ n (W^ 2, x)\) are the classical Hermite polynomials (with normalization in \(L^ 2 (W, \mathbb{R}))\). Let further \(a_ n = a_ n (Q)\) denote the \(n\)th Mhaskar-Rakhmanov-Saff number for \(Q\). It is shown in the paper that for \(0 < p < \infty\), and \(n \geq 2\), \[ \bigl \| p_ n (W^ 2, \cdot) W (\cdot) \bigr \|_{L^ p (\mathbb{R})} \sim a_ n^{1/p - 1/2} \begin{cases} 1, & p < 4, \\ (\log n)^{1/4}, & p = 4, \\ (n^{ - 2/3})^{1/p - 1/4}, & p > 4. \end{cases} . \]
scientific article; zbMATH DE number 597635