Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. (English) Zbl 1405.33012

The Legendre polynomials satisfy the Bernstein’s inequality \((1-x^2)^{1/4}|P_n(x)|\leq 2/\sqrt{\pi(2n+1)}\), for any \(x\in[-1,1]\). The authors are mainly interested in obtaining uniform estimates for certain expressions of the form \((-x)^a(1+x)^b|P_n^{(\alpha,\beta)}(x)|\) on \([-1,1]\), where \(P_n^{(\alpha,\beta)}\) are the Jacobi polynomials. The obtained estimates allowed them to investigate the dispersive estimates of certain Schrödinger equations whose Hamiltonian is a discrete Laguerre operator \(H_\alpha\) acting in \(\ell^2(\{0,1,2,\ldots\})\). This is based on the fact that the kernel of the evolution group \(\text{e}^{\text{i}tH_\alpha}\) can be expressed by means of the Jacobi polynomials.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
81U30 Dispersion theory, dispersion relations arising in quantum theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics


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