COMPUTATION OF CONVOLUTION OF FUNCTIONS WITHIN THE HAAR BASIS

• Main Authors: Mozgaleva Marina Leonidovna, Akimov Pavel Alekseevich
• Format: Article
• Language: English
• Published: Moscow State University of Civil Engineering (MGSU) 2012-10-01
• Series: Vestnik MGSU
• Subjects: convolution; discrete-continual boundary element method; Haar basis; structural analysis; wavelet analysis
• Online Access: http://vestnikmgsu.ru/files/archive/issues/2012/8/ru/14.pdf

The Wavelet analysis, that replaces the conventional Fourier analysis, is an exciting new problem-solving tool employed by mathematicians, scientists and engineers. Recent decades have witnessed intensive research in the theory of wavelets and their applications. Wavelets are mathematical functions that divide the data into different frequency components, and examine each component with a resolution adjusted to its scale. Therefore, the solution to the boundary problem of structural mechanics within multilevel wavelet-based methods has local and global components. The researcher may assess the infl uence of various factors. High-quality design models and reasonable design changes can be made. The Haar wavelet, known since 1910, is the simplest possible wavelet. Corresponding computational algorithms are quite fast and effective. The problem of computing the convolution of functions in the Haar basis, considered in this paper, arises, in particular, within the waveletbased discrete-continual boundary element method of structural analysis. The authors present their concept of convolution of functions within the Haar basis (one-dimensional case), share their useful ideas concerning Haar functions, and derive a relevant convolution formula of Haar functions.