Gibbs Phenomenon in Wavelet Analysis
- 55 Downloads
We prove Dirichlet-type pointwise convergence theorems for the wavelet transforms and series of discontinuous functions and we examine the Gibbs ripples close to the jump location. Examples are given of wavelets without ripples, and an example (the Mexican hat) shows that the Gibbs ripple in continuous wavelet analysis can be 3.54% instead of 8.9% of the Fourier case. For the discrete case we show that there exist two Meyer type wavelets the first one has maximum ripple 3.58% and the second 9.8%. Moreover we describe several examples and methods for estimating Gibbs ripples both in continuous and discrete cases. Finally we discuss how a wavelet transform generates a summability method for the Fourier case.
Mathematical Subject Classification42C99 42A50 94A12
Key wordsGibbs Phenomenon
Unable to display preview. Download preview PDF.
- Chui, C.K., An Introduction to Wavelets. Academic Press, New York, 1991.Google Scholar
- Holshneider, M., Wavelets an Analysis Tool. Oxford Science Publication, Oxford, 1995.Google Scholar
- Jerri, A.J., Error Analysis in Applications of Generalization of the Sampling Theorem. In: Advanced Topics in Shannon Sampling, R.J. Marks II (ed), Springer, New York, 1993.Google Scholar
- Jerri, A.J., Integral and Discrete Transforms With Applications and Error Analysis. Marcel Decker, New York, 1992.Google Scholar
- Jerri, A.J., The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Kluwer Academic Publishers, (1997 Preprint).Google Scholar
- Papoulis, A., The Fourier Integrals and its Applications. McGraw Hill, New York, 1962.Google Scholar
- Tichmarsh, E.C., Introduction to the Theory of Fourier Integrals. Oxford University Press, Oxford, 1948.Google Scholar
- Walter, G. G., Wavelets and other orthogonal systems with applications. CRC, 1994.Google Scholar
- Zygmund, A., Trigonometric Series I and II. 2nd edition, Cambridge Univ. Press, Cambridge, 1959.Google Scholar