Orthogonal Polynomials

  title={Orthogonal Polynomials},
  author={Vilmos Totik},
  • V. Totik
  • Published 18 December 2005
  • Mathematics
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed. 
On zeros of regular orthogonal polynomials on the unit circle
A new approach to the study of zeros of orthogonal polynomials with respect to an Hermitian and regular linear functional is presented. Some results concerning zeros of kernels are given.
Riemann-Hilbert Methods in the Theory of Orthogonal Polynomials
In this paper we describe various applications of the Riemann-Hilbert method to the theory of orthogonal polynomials on the line and on the circle.
On generating symmetric orthogonal polynomials
In this article, we show how to generate symmetric sequences of orthogonal polynomials whose moments are given. The advantage of this method is that only one Hankel determinant must be calculated.
Complex versus real orthogonal polynomials of two variables
Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of
Fourier Transforms of Some Special Functions in Terms of Orthogonal Polynomials on the Simplex and Continuous Hahn Polynomials
In this paper, Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained using the Parseval’s identity and
On Fractional Orthonormal Polynomials of a Discrete Variable
A fractional analogue of classical Gram or discrete Chebyshev polynomials is introduced. Basic properties as well as their relation with the fractional analogue of Legendre polynomials are presented.


Orthogonal Polynomials of Several Variables
Results parallel to the theory of orthogonal polynomials in one variable are established using a vectormatrix notation, which reports on the recent development on the general theory of hospitalisation in several variables.
A Note on the Asymptotics of Orthogonal Polynomials on a Complex Arc
The asymptotics of orthogonal polynomials are studied. The measure is concentrated on a complex are and has masses in the region exterior to the are, and we obtain strong asymptotics for the
Characterization Theorems for Orthogonal Polynomials
We survey in this paper characterization theorems dealing with polynomial sets which are orthogonal on the real line.
Asymptotics for Orthogonal Polynomials
Orthogonal polynomials on a compact set.- Asymptotically periodic recurrence coefficients.- Probabilistic proofs of asymptotic formulas.- Orthogonal polynomials on unbounded sets.- Zero distribution
General orthogonal polynomials
In this chapter the theory of general orthogonal polynomials will be studied independently of its application to Pade approximants. This theory is not new; it can, for example, be found in a book by
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
Riemann-Hilbert problems Jacobi operators Orthogonal polynomials Continued fractions Random matrix theory Equilibrium measures Asymptotics for orthogonal polynomials Universality Bibliography.
Sieved orthogonal polynomials. VII. Generalized polynomial mappings
Systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second-order difference equations are studied in detail. Applications are given to the theory
Analytic aspects of Sobolev orthogonal polynomials revisited ( A . Mart
This paper surveys some recent achievements in the analytic theory of polynomials orthogonal with respect to inner products involving derivatives. Asymptotic properties, zero location, approximation