Harmonic analysis—also called Fourier Analysis― studies the representation of functions or signals as the superposition of basic waves. Now it is one of the most applicable fields of modern mathematics. Among its many applications are signal processing/image transmission, various electrical and computer engineering applications, probability theory, physics, and many fields of pure and applied mathematics.
Approximation theory considers the problems of best approximating general and possibly complicated functions by simpler and more easily calculated ones. Concepts ''best'', ''simpler'' and ''easily calculated'' will depend on the applications. Approximation theory is an established and developed area of mathematics. On the other hand, this area currently experiences a significant rise due to its wide applications not only in mathematics (e.g., numerical, wavelet analysis) but also in computer science, signal processing, biomedical optics, and geographic information systems. Recent developments in nonlinear approximation theory are aimed at carrying out fundamental mathematical (compress, denoise, etc.) and algorithmic study to significantly increase our ability to process large data sets.
More specific topics of Harmonic analysis and Approximation Theory within CRM Research lines can be found here.