Uncertainty Principle and Wavelet Transforms

Abstract

The method of wavelet transforms, which provides a decomposition of functions in terms of a fixed orthogonal family of functions of constant shape but varying scales and locations, recently acquired broad significance in the analysis of signals and of experimental data from various physical phenomena. Its value for the whole spectrum of problems in many areas of science and engineering, including the study of electromagnetic and turbulent hydrodynamic fields, image reconstruction algorithms, prediction of earthquakes and tsunami waves, and statistical analysis of economic data, is by now quite obvious.

Although the systematic ideas of wavelet transforms have been developed only since the early 1980s, to get the proper intuitions about sources of their effectiveness it is necessary to become familiar with a few more traditional ideas, tools, and methods. One of those is the celebrated uncertainty principle for the Fourier transforms which will be given special attention in this chapter. A close relative of the wavelet transform—the windowed Fourier transform—will also be studied in this context.