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Register a new userJacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator
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The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrodinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrodinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.
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The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as more encompassing characterisation of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree $ell_{mathcal D}ge1$, the three term recurrence relations are broken. The extra $ell_{mathcal D}$ `lower degree polynomials, which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree $mathcal{N}$. The discrete orthogonality relations are shown to hold for another group of `new orthogonal polynomials called Krein-Adler polynomials based on the Hermite, Laguerre and Jacobi polynomials.
Let ${mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$|f|_A := sup_{x in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A subset {Bbb R}$. Let $$V_a^b(f) := int_a^b{|f^{prime}(x)| , dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 frac nkleq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{|P(1)|}} leq c_2 left( frac nk + 1 right)$$ for all integers $n geq 1$ and $k geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac nkright)^{1/2} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{|P(1)|}} leq c_2 left(frac nk + 1right)^{1/2}$$ for all integers $n geq 1$ and $k geq 1$.
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
We show that for a Jacobi operator with coefficients whose (j+1)th moments are summable the jth derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant case.
The Cholesky factorization of the moment matrix is considered for the generalized Charlier, generalized Meixner and generalized Hahn of type I discrete orthogonal polynomials. For the generalized Charlier we present an alternative derivation of the Laguerre-Freud relations found by Smet and Van Assche. Third order and second order order nonlinear ordinary differential equations are found for the recursion coefficient $gamma_n$. Laguerre-Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche. Finally, the generalized Hahn of type I discrete orthogonal polynomials are studied as well, and Laguerre-Freud equations are found and the differences with the equations found by Dominici and by Filipuk and Van Assche are given.
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