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The generalized Shannon system in wavelet space

Published online by Cambridge University Press:  17 February 2009

Hong Oh Kim  and
Jong Ha Park
Hong Oh Kim
Affiliation:
Department of Mathematics, KAIST, Kusong Dong 373–1, Yusong-Gu, Taejon 305–701, Korea.
Jong Ha Park
Affiliation:
Department of Mathematics, KAIST, Kusong Dong 373–1, Yusong-Gu, Taejon 305–701, Korea.
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Abstract

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The Shannon system is generalized and the expansion of a function in the generalized Shannon system is considered. No study of a wavelet expansion exists without the assumption of ‘fast’ decay of wavelets. The wavelet ψ which is associated with the generalized Shannon system has a ‘slow’ decay. The expansion of a function in the system is shown to converge at a point which satisfies the Lipschitz condition of order α > 0. On the other hand, there is a continuous function whose wavelet expansion in the generalized Shannon system diverges. An observation of Gibbs' phenomenon is also given.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 3 , January 2000 , pp. 386 - 400
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Chui, C. K., An introduction to wavelet (Academic Press, New York, 1992).CrossRefGoogle Scholar
[2]Daubechies, I., Ten lectures on wavelets SIAM-NSF Regional Conference Series 61 (SIAM,Philadelphia, 1992).Google Scholar
[3]Gibbs, J. W., “Letter to the editor”, Nature 59 (1899) 606.Google Scholar
[4]Katznelson, Y., An introduction to Harmonic Analysis (John Wiley & Sons, New York, 1968).Google Scholar
[5]Kelly, S., “Pointwise convergence for wavelet expansions”, Ph.D. Thesis, Washington University, St. Louis (1992).Google Scholar
[6]Kim, H. O. and Park, J. H., “Unimodular wavelets and scaling functions”, Comm. Korean Math. Soc. 13 (1998) 289–305.Google Scholar
[7]Madych, W. R., “Some elementary properties of multiresolution analyses of L2(Rn)”, in Wavelets: A Tutorial in Theory and Applications (ed. Chui, C. K.), (Academic Press, New York, 1992), pp. 259–294.Google Scholar
[8]Mallat, S., “Multiresolution approximations and wavelet orthonormal bases of L2(R)”, Trans. Amer. Math. Soc. 315 (1989) 69–88.Google Scholar
[9]Meyer, Y., Ondelettes et opérateurs (Hermann, Paris, 1990).Google Scholar
[10]Michelson, A. A., “Letter to the editor”, Nature 58 (1898) 544–45.Google Scholar
[11]Richards, F. B., “A Gibbs phenomenon for spline functions”, J. Approx. Theory 66 (1996) 334–351.CrossRefGoogle Scholar
[12]Shannon, C. E., “Communication in the presence of noise”, Proc. IRE 37 (1949) 10–21.Google Scholar
[13]Shim, H.-T., “The Gibbs' phenomenon for wavelets”, Ph.D. Thesis, University of Wisconsin-Milwaukee (1994).Google Scholar
[14]Walter, G. G., “Pointwise convergence of wavelet expansions”, J. Approx. Theory 80 (1995).CrossRefGoogle Scholar
[15]Walter, G. G., Wavelets and Other Orthogonal Systems with Applications (CRC Press, Boca Raton, 1994).Google Scholar
[16]Zygmund, A., Trigonometric Series (Cambridge University Press, New York, 1957).Google Scholar