scholarly journals Interpolatory product integration for Riemann-integrable functions

1987 ◽  
Vol 29 (2) ◽  
pp. 195-202 ◽  
Author(s):  
Philip Rabinowitz ◽  
William E. Smith
Keyword(s):  
Orthogonal Polynomials ◽  
Integration Interval ◽  
Product Integration ◽  
Zeros Of Orthogonal Polynomials ◽  
Integrable Functions ◽  
Riemann Integrable Functions ◽  
Integration Rules

AbstractConditions are fround for the convergence of intepolatory product integration rules and the corresponding companion rules for the class of Riemann-integrable functions. These condtions are used to prove convergence for several classes of rules based on sets of zeros of orthogonal polynomials possibly augmented by one both of the endpoints of the integration interval.

Hermite and Hermite-Fejer Interpolation and Associated Product Integration Rules on the Real Line: The L1 Theory

1992 ◽  
Vol 44 (3) ◽  
pp. 561-590 ◽  
Author(s):  
D. S. Lubinsky ◽  
P. Rabinowitz
Keyword(s):  
Orthogonal Polynomials ◽  
Real Line ◽  
Rapid Growth ◽  
Hermite Interpolation ◽  
Point Of View ◽  
Product Integration ◽  
The Real ◽  
Highly Oscillatory ◽  
New Feature ◽  
Integration Rules

AbstractWe investigate convergence in a weighted L1 -norm of Hermite-Fejér and Hermite interpolation at the zeros of orthogonal polynomials associated with weights on the real line. The results are then applied to convergences of product integration rules. From the point of view of orthogonal polynomials, the new feature is that Freud and Erdös weights are treated simultaneously and that relatively few assumptions are placed on the weight. From the point of view of product integration, the rules exhibit convergence for highly oscillatory kernels (for example) and for functions of rapid growth at infinity.

Convolutions and zeros of orthogonal polynomials

2011 ◽  
Vol 61 (7) ◽  
pp. 868-878
Author(s):  
Iván Area ◽  
Dimitar K. Dimitrov ◽  
Eduardo Godoy
Keyword(s):  
Orthogonal Polynomials ◽  
Zeros Of Orthogonal Polynomials

scholarly journals Interlacing of zeros of orthogonal polynomials under modification of the measure

2013 ◽  
Vol 175 ◽  
pp. 64-76 ◽  
Author(s):  
Dimitar K. Dimitrov ◽  
Mourad E.H. Ismail ◽  
Fernando R. Rafaeli
Keyword(s):  
Orthogonal Polynomials ◽  
Interlacing Of Zeros ◽  
Zeros Of Orthogonal Polynomials

Mean Convergence of Lagrange Interpolation for Exponential weights on [-1, 1]

1998 ◽  
Vol 50 (6) ◽  
pp. 1273-1297 ◽  
Author(s):  
D. S. Lubinsky
Keyword(s):  
Orthogonal Polynomials ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exponential Weights ◽  
Mean Convergence ◽  
Zeros Of Orthogonal Polynomials ◽  
Necessary And Sufficient

AbstractWe obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on [-1, 1], such asw(x) = exp(-(1 - x2)-α), α > 0orw(x) = exp(-expk(1 - x2)-α), k≥1, α > 0,where expk = exp(exp(. . . exp( ) . . .)) denotes the k-th iterated exponential.

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