Distributional Weight Functions for Orthogonal Polynomials

1978 ◽  
Vol 9 (4) ◽  
pp. 604-626 ◽  
Author(s):  
Robert D. Morton ◽  
Allan M. Krall
Keyword(s):  
Orthogonal Polynomials ◽  
Weight Functions ◽  
Distributional Weight

Complex Weight Functions for Classical Orthogonal Polynomials

1991 ◽  
Vol 43 (6) ◽  
pp. 1294-1308 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
David R. Masson ◽  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Weight Functions ◽  
Classical Orthogonal Polynomials ◽  
The Real ◽  
Complex Functions ◽  
Wilson Polynomials ◽  
Distributional Weight

AbstractWe give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

A New Composite Quadrature Rule

2013 ◽  
Vol 5 (04) ◽  
pp. 595-606
Author(s):  
Weiwei Sun ◽  
Qian Zhang
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Quadrature Rule ◽  
Numerical Results ◽  
Weight Functions

AbstractWe present a new composite quadrature rule which is exact for polynomials of degree 2N+K– 1 withNabscissas at each subinterval andKboundary conditions. The corresponding orthogonal polynomials are introduced and the analytic formulae for abscissas and weight functions are presented. Numerical results show that the new quadrature rule is more efficient, compared with classical ones.

Polynomial inequalities in regions with corners in theweighted Lebesgue spaces

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5647-5670 ◽  
Author(s):  
Fahreddin Abdullayev
Keyword(s):  
Orthogonal Polynomials ◽  
Lebesgue Space ◽  
Weight Functions ◽  
Weighted Lebesgue Space ◽  
Algebraic Polynomials ◽  
Lebesgue Spaces ◽  
Polynomial Inequalities

In this work, we investigate the order of the growth of the modulus of orthogonal polynomials over a contour and also arbitrary algebraic polynomials in regions with corners in a weighted Lebesgue space, where the singularities of contour and the weight functions satisfy some condition.

scholarly journals Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space

2004 ◽  
Vol 2004 (52) ◽  
pp. 2761-2772 ◽  
Author(s):  
Fred Brackx ◽  
Nele De Schepper ◽  
Frank Sommen
Keyword(s):  
Euclidean Space ◽  
Orthogonal Polynomials ◽  
Clifford Algebra ◽  
Unit Ball ◽  
Real Axis ◽  
Jacobi Polynomials ◽  
Orthogonality Relation ◽  
Weight Functions ◽  
Open Unit ◽  
Open Unit Ball

A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clifford algebra-valued functions. The method consists in transforming the orthogonality relation on the open unit ball into an orthogonality relation on the real axis by means of the so-called Clifford-Heaviside functions. Consequently, appropriate orthogonal polynomials on the real axis give rise to Clifford algebra-valued orthogonal polynomials in the unit ball. Three specific examples of such orthogonal polynomials in the unit ball are discussed, namely, the generalized Clifford-Jacobi polynomials, the generalized Clifford-Gegenbauer polynomials, and the shifted Clifford-Jacobi polynomials.

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