Approximation of multiple integrals by length of <i>α</i>‐dense curves

Benabidallah, A.; Cherruault, Y.; Mora, Gaspar

doi:10.1108/03684920210436381

Cited by 2 publications

(3 citation statements)

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“…We recall that in Benabidallah et al (2002) and Mora et al (2002), the following result has been proved. Theorem 4.…”

Section: Gaussian Rulementioning

confidence: 89%

“…The theory of a-dense curves developed by Cherruault (1998Cherruault ( , 1999, and has permitted us to approximate multiple integrals by simple integrals of oscillatory functions. In Benabidallah et al (2001), the one-variable functions obtained by using the reducing transformations are periodic but discontinuous, while in Benabidallah et al (2002) the integrands are just continuous and not periodic.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of multiple integrals by simple integrals involving periodic functions

2004

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In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using a-dense curves and approximating multiple integrals. Using first, periodic and regular a-dense curves we propose a trapezoidal formula for calculating the periodic integrand obtained. Then, we consider the simple integrals as integrals with weight. We propose a method to evaluate the moments of the weight function. This allows us to build a recurrent formula for the orthogonal polynomials family and to use a Gaussian rule to estimate the simple integral. Finally, we adapt the Filon's method, consisting in evaluating the Fourier coefficients of a function, to the oscillatory integrand obtained by using reducing transformations.

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“…We recall that in Benabidallah et al (2002) and Mora et al (2002), the following result has been proved. Theorem 4.…”

Section: Gaussian Rulementioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Approximation of multiple integrals by simple integrals involving periodic functions

2004

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, it seems reasonable to think that the α-dense curves could be used to reduce the integral (1.1) to a single variable one, because as we have pointed out above, these curves generalize the space-filling curves. In fact, this idea has been successfully used in [3,4,8,10,[19][20][21] to obtain, under suitable conditions, the approximation stated in (1.2).…”

Section: Example 22mentioning

confidence: 99%