Abstract
We deal with the maximum Gibbs ripple of the sampling wavelet series of a discontinuous function f at a point t ∈R, for all possible values of a satisfyingf(t)=αf(t−0)+(1−a)f(t+0). For the Shannon wavelet series, we make a complete description of all ripples, for any a in [0,1]. We show that Meyer sampling series exhibit Gibbs Phenomenon for α<0.12495 and α>0.306853. We also give Meyer sampling formulas with maximum overshoots shorter than Shannon's for several α in [0,1].
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References
Butzer, P.L., Ries, S., and Stens, R.L. (1987). Approximation of Continuous and Discontinuous Functions by Generalized Sampling Series,J. Approx. Theory,50, 25–39.
Butzer, P.L., Stens, R.L., and Splettstöβer, W. (1988).The Sampling Theorem and Linear Prediction in Signal Analysis. Jber. d. Dt. Math.-Verein,90, 1–70.
Daubechies I. (1992).Ten Lectures on Wavelets. CBMS-NSF Conf. Ser. in Appl. Math. Society for Industrial and Applied Mathematics, Philadelphia, PA.
Fischer, A. (1995). Multiresolution Analysis and Multivariate Approximation of Smooth Signals in CB (Rd).J. Fourier Anal. Appl.,2, 161–180.
Gomes, S. and Cortina, E. (1995). Some Results on the Convergence of Sampling Series based on Convolution Integrals,SIAM J. Math. Anal.,26(5), 1386–1402.
Helmberg, G. (1994). The Gibbs Phenomenon for Fourier Interpolation.J. Approx. Theory,78, 41–63.
Helmberg, G. and Wagner, P. (1997). Manipulating Gibbs' Phenomenon for Fourier Interpolation,J. Approx. Theory,89, 308–320.
Jerri, A. (1993). Error Analysis in Applications of Generalizations of the Sampling Theorem.Advanced Topics in Shannon Sampling and Interpolation Theory. Marks II, R.J., Ed., Springer-Verlag, New York.
Jerri, A. (1992).Integral and Discrete Transforms with Applications and Error Analysis. Marcel-Dekker, New York.
Jerri, A. (1998).The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Mathematics and its Applications, 446. Kluwer Academic Publishers, Dordrecht, The Netherlands.
Karanikas, C. (1998). Gibbs phenomenon in wavelet analysis,Result. Math.,34, 330–341.
Walter, G.G. (1994).Wavelets and Other Orthogonal Systems with Applications, CRC Press, Boca Raton. FL.
Walter, G.G. and Shim, H.T. (1998). Gibbs Phenomenon for Sampling Series and What to do About it,J. Fourier Anal. Appl.,4(3), 357–375.
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Communicated by John J. Benedetto
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Atreas, N., Karanikas, C. Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis. The Journal of Fourier Analysis and Applications 5, 575–588 (1999). https://doi.org/10.1007/BF01257192
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DOI: https://doi.org/10.1007/BF01257192