The Gibbs' phenomenon for Fourier-Bessel series

Fay, Temple H.; Kloppers, P. Hendrik

doi:10.1080/0020739021000053936

Cited by 10 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, oscillation at the edges of an interval occurs when polynomials of a high degree over a set of equispaced interpolation points are used. The phenomenon is similar to the Gibbs phenomenon in Fourier series approximations [25]. According to Runge's phenomenon, the magnitude of the n-th order derivatives of this function grows quickly for a large degree and the equidistance between points, which leads to a larger Lebesgue constant for large N.…”

Section: Remark 1: On Approximation With Polynomialsmentioning

confidence: 57%

Review and Improvement of the Finite Moment Problem

Hjouj¹,

Jouini²

2020

TOCENGJ

View full text Add to dashboard Cite

Background: This paper reviews the Particle Size Distribution (PSD) problem in detail. Mathematically, the problem faced while recovering a function from a finite number of its geometric moments has been discussed with the help of the Spline Theory. Undoubtedly, the splines play a major role in the theory of interpolation and approximation in many fields of pure and applied sciences. B-Splines form a practical basis for the piecewise polynomials of the desired degree. A high degree of accuracy has been obtained in recovering a function within the first ten to fifteen geometric moments. The proposed approximation formula has been tested on several types of synthetic functions. This work highlights some advantages, such as the use of a practical basis for the approximating space, the exactness of computing the moments of these basis functions and the reduction of the size along with an appropriate transformation of the resulting linear system for stability. Objective: The aim is to recover a function from a finite number of its geometric moments. Methods: The main tool is the Spline Theory. Undoubtedly, the role of splines in the theory of interpolation and approximation in many fields of pure and applied sciences has been well- established. B-Splines form a practical basis for the piecewise polynomials of the desired degree. Results: A high degree of accuracy has been obtained in recovering a function within the first ten to fifteen geometric moments. The proposed approximation formula is tested on several types of synthetic functions. Conclusion: This work highlights some advantages, such as the use of a practical basis for the approximating space, the exactness of computing the moments of these basis functions and the reduction of the size along with the data transformation of the resulting linear system for stability.

show abstract

Section: Remark 1: On Approximation With Polynomialsmentioning

confidence: 57%

Review and Improvement of the Finite Moment Problem

Hjouj¹,

Jouini²

2020

TOCENGJ

View full text Add to dashboard Cite

show abstract

“…Gibbs-Wilbraham Constant. On the one hand, Gibbs' constant for a jump discontinuity for Fourier-Bessel partial sums seems to be numerically equal to that for ordinary Fourier partial sums (a proof is not given in [505]). On the other hand, the analog of (2/π)G corresponding to de la Vallée Poussin sums is…”

Section: 31mentioning

confidence: 99%

Errata and Addenda to Mathematical Constants

Finch¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In the authors' article [4], Fourier-Bessel series are considered and the Gibbs constant is calculated numerically for three examples. This constant appeared to the same as that obtained using Fourier series.…”

Section: Introductionmentioning

confidence: 99%

The Gibbs phenomenon for series of orthogonal polynomials

Fay

Kloppers

2006

International Journal of Mathematical Education in Science and

Self Cite

View full text Add to dashboard Cite

This note considers the four classes of orthogonal polynomials -Chebyshev, Hermite, Laguerre, Legendre -and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same as that for Fourier series expansions. Each class of polynomials has features which are interesting numerically. Finally a plausibility argument is included showing that this phenomenon for the Gibbs constants should not have been unexpected. These findings suggest further investigations suitable for undergraduate research projects or small group investigations.

show abstract

The Gibbs' phenomenon for Fourier-Bessel series

Cited by 10 publications

References 2 publications

Review and Improvement of the Finite Moment Problem

Review and Improvement of the Finite Moment Problem

Errata and Addenda to Mathematical Constants

The Gibbs phenomenon for series of orthogonal polynomials

Contact Info

Product

Resources

About