SEVERAL ELEMENTS OF THE MULTI-RESOLUTION WAVELET ANALYSIS. Part 1: BASICS OF THE WAVELETS AND THE MULTI-RESOLUTION WAVELET ANALYSIS

Part 1 of this paper represents an introduction into the multi-resolution wavelet analysis. The wavelet-based analysis is an exciting new problem-solving tool used by mathematicians, scientists and engineers. In the paper, the authors try to present the fundamental elements of the multi-resolution wavelet analysis in a way that is accessible to an engineer, a scientist and an applied mathematician both as a theoretical approach and as a potential practical method of solving problems (particularly, boundary problems of structural mechanics and mathematical physics). The main goal of the contemporary wavelet research is to generate a set of basic functions (or general expansion functions) and transformations that will provide an informative, efficient and useful description of a function or a signal. Another central idea is that of multi-resolution whereby decomposition of a signal represents the resolution of the detail. The multi-resolution decomposition seems to separate components of a signal in a way that is superior to most other methods of analysis, processing or compression. Due to the ability of the discrete wavelet transformation technique to decompose a signal at different independent scaling levels and to do it in a very flexible way, wavelets can be named "the microscopes of mathematics". Indeed, the use of the wavelet analysis and wavelet transformations requires a new point of view and a new method of interpreting representations