Markov-type inequalities and extreme zeros of orthogonal polynomials

Abstract

Let {pm}m=0∞ be an orthogonal system in a Hilbert space H, πn=Span{p0,p1,…,pn}, and L a linear operator defined on H. Under certain assumptions on L, we prove that {cn−2(L)}n∈N, the reciprocals of the squared sharp constants {cn(L)}n∈N in the inequality ‖Lf‖≤cn(L)‖f‖,f∈πn, are equal to the smallest eigenvalues of certain positive definite Jacobi matrices or, equivalently, to the smallest zeros of certain polynomials, orthogonal with respect to a measure supported on R+. Hence, the problem of estimating the sharp constants cn(L) is transformed to the problem of estimating the extreme eigenvalues (zeros) of these Jacobi matrices (orthogonal polynomials). By applying Gershgorin’s Circle Theorem we obtain new upper bounds for the sharp constants in the Markov L2 inequalities with the Laguerre and Gegenbauer weights, and examine the asymptotic behavior of these constants as the parameters in the weights tend to infinity.