(A) weighted $L^2$-norm inequality for semi-classical weights

Abstract

Consider an $L^2$-norm on the space of polynomials with complex coefficients given by ◁수식 삽입▷(원문을 참조하세요) such that $w(x)＞0$ on $(a,b)$ and all moments $w_n:=\∫_a^bx^nω(x)dx, n≥ 0$, are finite. Then, using the orthonormal polynomial system ${P_n(x)}_{n=0}^∞$ with respect to positive measure $w(x)dx$, Mirsky showed that there exists a constant $\gamma_n=\gamma_n(a,b;w)$ such that ◁수식 삽입▷(원문을 참조하세요) where $p(x)$ is an arbitrary polynomial of degree at most $n$. Furthermore, he found an upper bound of the constant $\gamma_n$ using the orthonormal polynomial systems. In this paper, we showed that Markov type inequalites hold for any positive distributional weight satisfying Pearson type functional equation including classical weight are given and then we also found a best constants $Γ_n$ of $\gamma_n$ for polynomials with non-classical weight which is not treated yet in $L^2$-spaces. This result generalizes that of Agarwal and Milovanovi? to semi-classical orthogonal polynomials.