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On the Gibbs–Wilbraham Phenomenon for Sampling and Interpolatory Series

Part of: Approximations and expansions Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  12 July 2019

Keaton Hamm
Keaton Hamm*
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA ([email protected])
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Abstract

We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot characteristic of the phenomenon for certain cardinal functions, and characterize the existence of an overshoot for sampling series.

Keywords

Gibbs phenomenongeneralized sampling kernelscardinal functionssampling expansions

MSC classification

Primary: 41A05: Interpolation
Secondary: 41A30: Approximation by other special function classes 42C15: General harmonic expansions, frames
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1163 - 1171
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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Footnotes

To N. Sivakumar

References

1.Atreas, N. and Karanikas, C., Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis, J. Fourier Anal. Appl. 5(6) (1999), 575–588.Google Scholar
2.Atreas, N. and Karanikas, C., Reducing Gibbs ripples for some wavelet sampling series in Advances in the Gibbs phenomenon (ed. Jerri, A.), Chapter 11, pp. 335–362 (Sampling Publishing, Potsdam, NY, 2007).Google Scholar
3.Baxter, B. J. C. and Sivakumar, N, On shifted cardinal interpolation by Gaussians and multiquadrics, J. Approx. Theory 87(1) (1996), 36–59.Google Scholar
4.Buhmann, M. D., Multivariate cardinal interpolation with radial-basis functions, Constr. Approx. 6(3) (1990), 225–255.Google Scholar
5.Butzer, P. L., Ries, S. and Stens, R. L., Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx Theory 50(1) (1987), 25–39.Google Scholar
6.Fornberg, B. and Flyer, N., The Gibbs phenomenon for radial basis functions in Advances in the Gibbs phenomenon (ed. Jerri, A.), Chapter 6, pp. 197–217 (Sampling Publishing, Potsdam, NY, 2007).Google Scholar
7.Foster, J. and Richards, F. B., Gibbs–Wilbraham splines, Constr. Approx. 11(1) (1995), 37–52.Google Scholar
8.Gibbs, J. W., Fourier's series, Nature 59(1522) (1898), 200.Google Scholar
9.Gibbs, J. W., Fourier's series, Nature 59(1539) (1899), 606.Google Scholar
10.Hamm, K. and Ledford, J., Cardinal interpolation with general multiquadrics, Adv. Comput. Math. 42(5) (2016), 1149–1186.Google Scholar
11.Hamm, K. and Ledford, J., Cardinal interpolation with general multiquadrics: convergence rates. Adv. Comput. Math. 44(4) (2018), 1205–1233.Google Scholar
12.Hewitt, E. and Hewitt, R. E., The Gibbs–Wilbraham phenomenon: an episode in Fourier analysis, Arch. Hist. Exact Sci. 21(2) (1979), 129–160.Google Scholar
13.Huang, D. and Zhang, Z., Asymptotic behavior of Gibbs functions for M-band wavelet expansions, Acta Mat. Sin. (Engl. Ser.) 15(2) (1999), 165–172.Google Scholar
14.Jerri, A. (ed.), Advances in The Gibbs phenomenon (Sampling Publishing, Potsdam, NY, 2007).Google Scholar
15.Kelly, S. E., Gibbs phenomenon for wavelets, Appl. Comput. Harmon. Anal. 3(1) (1996), 72–81.Google Scholar
16.Ledford, J., On the convergence of regular families of cardinal interpolators, Adv. Comput. Math. 41(2) (2015), 357–371.Google Scholar
17.Richards, F. B., A Gibbs phenomenon for spline functions, J. Approx. Theory 66(3) (1991), 334–351.Google Scholar
18.Schoenberg, I. J., Cardinal spline interpolation (SIAM, Philadelphia, 1973).Google Scholar
19.Shannon, C. E., Communication in the presence of noise, Proc. IRE 37(1) (1949), 10–21.Google Scholar
20.Walter, G. G. and Shim, H.-T., Gibbs' phenomenon for sampling series and what to do about it, J. Fourier Anal. Appl. 4(3) (1998), 357–375.Google Scholar
21.Wilbraham, H., On a certain periodic function, Cambridge and Dublin Math. J. 3 (1848), 198–201.Google Scholar