On gibb's phenomenon for sampling series in wavelet subspaces

Shim, Hong-Tae; Kim, Hong Oh

doi:10.1080/00036819608840447

Cited by 2 publications

(1 citation statement)

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“…That is, let the scaling function be Shannon scaling function, f be a square integrable bounded function with a jump discontinuity, there exists a Gibbs phenomenon at discontinuous points. The Gibbs phenomenon for wavelets expansions is further studied in [6][7][8][9][10][11][12][13]. In 1991, Richards [7] showed a Gibbs phenomenon for periodic spline functions.…”

Section: Introductionmentioning

confidence: 99%

Gibbs phenomenon for p-ary subdivision schemes

2019

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When a Fourier series is used to approximate a function with a jump discontinuity, the Gibbs phenomenon always exists. This similar phenomenon exists for wavelets expansions. Based on the Gibbs phenomenon of a Fourier series and wavelet expansions of a function with a jump discontinuity, in this paper, we consider that a Gibbs phenomenon occurs for the p-ary subdivision schemes. Similar to the method of (Appl. Math. Lett. 76:157-163, 2018), we generalize the results about the stationary binary subdivision schemes in (Appl. Math. Lett. 76:157-163, 2018) to the case of p-ary subdivision schemes. By considering the masks of subdivision schemes, we obtain a sufficient condition to determine whether there exists a Gibbs phenomenon for p-ary subdivision schemes in the limit function close to the discontinuous point. This condition consists of the positivity of the partial sums of the values of the masks. By applying this condition, we can avoid the Gibbs phenomenon for p-ary subdivision schemes near discontinuity points. Finally, some examples in classical subdivision schemes are given to illustrate the results in this paper.

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