Wavelets became a necessary mathematical tool in many investigations. They are used in those cases when the result of the analysis of a particular signal (i.e. processes, objects, functions etc.) should contain not only the list of its typical frequencies (scales) but also the knowledge of the definite local coordinates where these properties are important, e.g. where the drastic changes in this or that process characteristics happen. Wavelets form a complete orthonormal system of functions with a finite support by using dilations and translations. That is why by changing a scale (dilations) they can distinct the local characteristics of a signal at various scales, and by translations they cover the whole region in which it is studied. Due to the completeness of the system, they also allow for the inverse transform to be done. The locality property of wavelets leads to their substantial superiority over Fourier transform which provides us only with the knowledge of global frequencies (scales) of the object under investigation because the system of functions used (sine, cosine) is defined on the infinite interval.
The literature devoted to wavelets is very voluminous, and one can easily get a lot of references by sending the corresponding request to Internet web sites. Mathematical problems are treated in many monographs in detail. The programs exploiting wavelet transform are widely used now not only for scientific research but for commercial purposes as well. At the same time, the direct transition from pure mathematics to computer programming is non-trivial and asks often for individual approach to the problem under investigation and for a specific choice of wavelets used.
The discrete wavelets look as strangers to all those accustomed to analytical calculations because they can not be represented by analytical expressions (except of a simplest one) or by solutions of some differential equations, and instead are given numerically as solutions of definite functional equations containing rescaling. Moreover, in practical calculations their direct form is not required even, and the numerical values of the coefficients of the functional equation are only used. Thus wavelets are defined by the iterative algorithm of the dilation and translation of a single function. This leads to very important procedure called multiresolution analysis which gives rise to the multiscale local analysis of the signal and fast numerical algorithms. Each scale contains an independent non-overlapping set of information about the signal. In combination, they provide its complete analysis and diagnosis of the underlying processes.
After such an analysis was done, one can compress the resulting data by omitting some inessential part of the encoded information. This is done with the help of the so-called quantization procedure. Usually, it gives rise to a substantial reduction of the required computer storage memory and transmission facilities, and, consequently, to a lower expenditure. Unfortunately, it introduces unavoidable systematic errors. Nevertheless, when one tries to reconstruct the initial signal, the inverse transformation (synthesis) happens to be rather stable and reproduces its most important characteristics if proper methods are applied. One needs good mathematical background, experience and intuition to proceed with wavelet applications.
Because of their unique properties, wavelets were used in functional analysis in mathematics, in studies of (multi)fractal properties and singularities of functions, for solving some differential equations, for investigation of inhomogeneous processes involving widely different scales of interacting perturbations, for pattern recognition, for image and sound compression, for solving many problems of physics, biology, medicine, technique etc. Some practical applications are described here.
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