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.

A new characterization of Riemann-integrable functions

2006 ◽  
Vol 139 (4) ◽  
pp. 6708-6714 ◽  
Author(s):  
V. K. Zakharov ◽  
A. A. Seredinskii
Keyword(s):  
Integrable Functions ◽  
Riemann Integrable Functions

scholarly journals The generalized method of exhaustion

2002 ◽  
Vol 31 (6) ◽  
pp. 345-351 ◽  
Author(s):  
Anthony A. Ruffa
Keyword(s):  
Velocity Distribution ◽  
Group Velocity ◽  
Geometric Approach ◽  
Integration Formula ◽  
Usual Procedure ◽  
Integrable Functions ◽  
Algebraic Generalization ◽  
Riemann Integrable Functions ◽  
Simple Integration

The method of exhaustion is generalized to a simple integration formula that is valid for the Riemann integrable functions. Both a geometric approach (following the usual procedure for the method of exhaustion) and an independent algebraic generalization approach are provided. Applications provided as examples include use of the formula to generate new series for common functions as well as computing the group velocity distribution resulting from waves diffracted from an aperture.

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