Characterization of Orthogonal Polynomials in Norm-Attainable Classes

Abstract

In this note, we investigate the norm-attainability of classical orthogonal polynomials, including Chebyshev, Hermite, Laguerre, and Legendre polynomials, within specific weight functions and intervals. It establishes the conditions under which these polynomials can achieve norm-attainment in their respective Hilbert spaces. The study demonstrates the norm-attainability of Chebyshev polynomials under the weight function (1 − x 2 ) −1/2 on the interval [−1, 1], proves the norm-attainment of Hermite polynomials under a normal distribution weight function, establishes the norm-attainability of Laguerre polynomials with a gamma distribution weight function on the positive real line, and verifies the norm-attainment of Legendre polynomials with a weight function equal to 1 on the interval [−1, 1].