Abstract
The Gibbs’ phenomenon in the classical Fourier series is well-known. It is closely related with the kernel of the partial sum of the series. In fact, the Dirichlet kernel of the Fourier series is not positive. The poisson kernel of Cesaro summability is positive. As the consequence of the positiveness, the partial sum of Cesaro summability does not exhibit the Gibbs’ phenomenon. Most kernels associated with wavelet expansions are not positive. So wavelet series is not free from the Gibbs’ phenomenon. Because of the excessive oscillation of wavelets, we can not follow the techniques of the Fourier series to get rid of the unwanted quirk. Here we make a positive kernel for Meyer wavelets and as the result the associated summability method does not exhibit Gibbs’ phenomenon for the corresponding series.
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This paper was supported by the KRF Research Grant, 2000-015-DP0049
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Shim, HT., Jung, K.H. A summability for Meyer wavelets. Korean J. Comput. & Appl. Math. 9, 487–496 (2002). https://doi.org/10.1007/BF03021556
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DOI: https://doi.org/10.1007/BF03021556