Orthogonal Polynomials on the Unit Circle Associated with the Laguerre Polynomials
نویسنده
چکیده
Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.
منابع مشابه
Bi-orthogonal Polynomials on the Unit Circle, Regular Semi-classical Weights and Integrable Systems
Abstract. The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has a...
متن کاملThe Operational matrices with respect to generalized Laguerre polynomials and their applications in solving linear dierential equations with variable coecients
In this paper, a new and ecient approach based on operational matrices with respect to the gener-alized Laguerre polynomials for numerical approximation of the linear ordinary dierential equations(ODEs) with variable coecients is introduced. Explicit formulae which express the generalized La-guerre expansion coecients for the moments of the derivatives of any dierentiable function in termsof th...
متن کاملOrthogonal Polynomials on the Unit Circle with Verblunsky Coefficients Defined by the Skew-shift
I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost-every Aleksandrov measure is pure point. Furthermore, I show in the case of the two dimensional skew-shift the zeros of para-orthogonal polynomials obey the same statistics as an appro...
متن کاملApplication of Laguerre Polynomials for Solving Infinite Boundary Integro-Differential Equations
In this study, an efficient method is presented for solving infinite boundary integro-differential equations (IBI-DE) of the second kind with degenerate kernel in terms of Laguerre polynomials. Properties of these polynomials and operational matrix of integration are first presented. These properties are then used to transform the integral equation to a matrix equation which corresponds t...
متن کاملA Note on Summability of Multiple Laguerre Expansions
A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an `1-radial function f0(|x|) is shown to be an `1-radial function that coincides with the Laguerre polynomial expansion of f0, which allows us to settle the problem of summa...
متن کامل