An application of the Jacobi summability to the wavelet approximation
María Moncayo, Rafael J. Yáñez · Integral Transforms and Special Functions · 2008
In this paper, we propose a link between classical and modern tools used in approximation theory. More precisely, we present an application to the approximation by wavelets which is based on the classical Jacobi summability proposed by [R. Askey, Jacobi summability, J. Approx. Theory 5 (1972), pp. 387–392]. This linear method generalizes other ones based on Cesàro and Abel summability, such that the Nörlund means. We prove the rapid rate of convergence of the Jacobi summability method and we obtain a relation which is valid for avoiding the Gibbs phenomenon in intermediate levels of wavelet approximation. The comparison between the numerical results obtained by the Nörlund means and the Jacobi summability method reveals a slight improvement concerning the reduction of the excessive oscillations by using the approach presented here.