Wavelet transformA new mathematical microscope

In the last decade, a newmathematical microscope has allowed scientists and engineers to view the details of time varying and transient phenomena, in a manner hitherto not possible through conventionaltools. This invention, which goes by the name of wavelet transform, has created revolutionary changes in the areas of signal processing, image compression, not to speak about the basic sciences. This novel procedure enables one to achieve the so calledtime-frequency localization andmultiscale resolution, by suitably focussing and zooming around the neighborhood of one’s choice. Wavelets are of very recent origin; their construction, properties and applications are subjects of intense current research. In this article, we explain with illustrations the working of this transform and its advantages vis-a-vis the Fourier transform. In two companion articles, we describe the procedure to construct wavelet basis sets and their applications to data analysis and image compression.