Continuum Wavelets and Distributions

Abstract

The purpose of this work is to obtain a wavelet expansion of information flows, which are distribution flows (in the terminology of Schwartz). The concept of completeness is introduced for a family of abstract functions. Using the mentioned families, nested spaces of distribution flows are constructed. The projection of the enclosing space onto the nested space generates a wavelet expansion. Decomposition and reconstruction formulas for the above expansion are derived. These formulas can be used for wavelet expansion of the original information flow coming from the analog device. This approach is preferable to the approach in which the analog flow is converted into a discrete numerical flow using quantization and digitization. The fact is that quantization and digitization lead to significant loss of information and distortion. This paper also considers the wavelet expansion of a discrete flow of distributions using the Haar type functions.