Weighted Laudau-type inequalities
The motivation for this paper was the inequality by \textit{A. K. Varma} [Acta Math. Hung. 49, 169-172 (1987; Zbl 0612.33008)], \[ |P'|\leq {1\over 2(2n- 1)} |P''|^2+ {2n^2\over 2n- 1} |P|^2 \] (where \(|f|:= (\int_{\mathbb{R}} |f(x)|^2 e^{- x^2} dx)^{1/2}\) and \(P\) is a polynomial of degree at most \(n\)) in which the equality is attained only if \(P= cH_n\), where \(H_n\) is the Hermite polynomial of degree \(n\), orthogonal to all polynomials of degree at most \(n- 1\) with respect to the weighted function \(\exp(- x^2)\) on \(\mathbb{R}\). The main result of the paper is the inequality of Landau-type [cf. \textit{E. Landau}, Proc. Lond. Math. Soc., II. Ser. 13, 43-49 (1913)], \[ |wf'|_X\leq L\cdot \sqrt{|wf|_X\cdot |wf''|_X}, \] where \(X\) is an arbitrary rearrangement invariant Banach function space with the norm \(|\cdot|_X\), \(w\) is the Freud-type weight [\textit{H. N. Mhaskar}, J. Approximation Theory 63, No. 2, 238-254 (1990; Zbl 0716.42020)], \(f: \mathbb{R}\to \mathbb{R}\) is a twice iterated integral of \(f''\) such that \(wf''\in X\) and \(f(0)= f'(0)= 0\), \(L\) is a positive constant independent of \(f\). In the case of Hermite weight \(w\) and the \(L^2\)-norm considered by Varma, the author's approach enables simpler characterization of Hermite polynomials.
scientific article; zbMATH DE number 770357