Gibbs Phenomenon in Wavelet Analysis
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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
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