Multi-dimensional Summability Theory and Continuous Wavelet Transform

Abstract

The connection between multi-dimensional summability theory and continuous wavelet transform is investigated. Two types of \(\theta \)-summability of Fourier transforms are considered, the circular and rectangular summability. Norm and almost everywhere convergence of the \(\theta \)-means are shown for both types. The inversion formula for the continuous wavelet transform is usually considered in the weak sense. Here, the inverse wavelet transform is traced back to summability means of Fourier transforms. Using the results concerning the summability of Fourier transforms, norm and almost everywhere convergence of the inversion formula are obtained for functions from the \(L_p\) and Wiener amalgam spaces.