Bases

Abstract

The classical subject of Fourier series is about approximating periodic functions by sines and cosines, specifically, about expressing an arbitrary periodic function as an infinite series of sines and cosines. (Any function that vanishes outside some interval can be viewed as a periodic function on Rby merely extending it periodically.) The sines and cosines are the “basic” periodic functions in terms of which we express all others. To use — chemical analogy, the sines and cosines are the atoms; the other functions are the molecules. Unlike the physical situation, however, there can be other atoms, other functions, that can serve as the “basic” functions just as effectively as sines and cosines.