Recurrence relations, continued fractions, and orthogonal polynomials

1984 ◽  
Vol 49 (300) ◽  
pp. 0-0 ◽  
Author(s):  
Richard Askey ◽  
Mourad Ismail
Keyword(s):  
Orthogonal Polynomials ◽  
Continued Fractions ◽  
Recurrence Relations

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory

2009 ◽  
Vol 52 (1) ◽  
pp. 95-104 ◽  
Author(s):  
L. Miranian
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Classical Theory ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
The Matrix ◽  
Classical Subject ◽  
First Time

AbstractIn the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

An Application of Separate Convergence for Continued Fractions to Orthogonal Polynomials

1992 ◽  
Vol 35 (3) ◽  
pp. 381-389
Author(s):  
William B. Jones ◽  
W. J. Thron ◽  
Nancy J. Wyshinski
Keyword(s):  
Distribution Function ◽  
Asymptotic Formula ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Continued Fractions ◽  
Polynomial Sequence ◽  
Convergence Results ◽  
Compact Subsets

AbstractIt is known that the n-th denominators Qn (α, β, z) of a real J-fractionwhereform an orthogonal polynomial sequence (OPS) with respect to a distribution function ψ(t) on ℝ. We use separate convergence results for continued fractions to prove the asymptotic formulathe convergence being uniform on compact subsets of

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