Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn upper bound on Jacobi polynomials
109
0
0.0
(
0
)
Authors
Ilia Krasikov
Ask ChatGPT about the research
No Arabic abstract
Let ${bf P}_k^{(alpha, beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality begin{equation*} max_{x in [delta_{-1},delta_1]}sqrt{(x- delta_{-1})(delta_1-x)} (1-x)^{alpha}(1+x)^{beta} ({bf P}_{k}^{(alpha, beta)} (x))^2 < frac{3 sqrt{5}}{5}, end{equation*} where $delta_{-1}<delta_1$ are appropriate approximations to the extreme zeros of ${bf P}_k^{(alpha, beta)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erd{e}lyi, A.P. Magnus and P. Nevai conjecture [Erd{e}lyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614], by proving that begin{equation*} max_{x in [-1,1]}(1-x)^{alpha+{1/2}}(1+x)^{beta+{1/2}}({bf P}_k^{(alpha, beta)} (x))^2 < 3 alpha^{1/3} (1+ frac{alpha}{k})^{1/6}, end{equation*} in the region $k ge 6, alpha, beta ge frac{1+ sqrt{2}}{4}.$
rate research
Read More
We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Greens functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.
T. Erd{e}lyi, A.P. Magnus and P. Nevai conjectured that for $alpha, beta ge - {1/2} ,$ the orthonormal Jacobi polynomials ${bf P}_k^{(alpha, beta)} (x)$ satisfy the inequality begin{equation*} max_{x in [-1,1]}(1-x)^{alpha+{1/2}}(1+x)^{beta+{1/2}}({bf P}_k^{(alpha, beta)} (x) )^2 =O (max left{1,(alpha^2+beta^2)^{1/4} right}), end{equation*} [Erd{e}lyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. Here we will confirm this conjecture in the ultraspherical case $alpha = beta ge frac{1+ sqrt{2}}{4},$ even in a stronger form by giving very explicit upper bounds. We also show that begin{equation*} sqrt{delta^2-x^2} (1-x^2)^{alpha}({bf P}_{2k}^{(alpha, alpha)} (x))^2 < frac{2}{pi} (1+ frac{1}{8(2k+ alpha)^2} ) end{equation*} for a certain choice of $delta,$ such that the interval $(- delta, delta)$ contains all the zeros of ${bf P}_{2k}^{(alpha, alpha)} (x).$ Slightly weaker bounds are given for polynomials of odd degree.
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.
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.
We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2times 2$ introduced by C. Calderon et al. in arXiv:1810.08560, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three term recurrence relation and the Christoffel-Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the algebra of second order differential operators associated with the weight matrix.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
