Time and Frequency

Abstract

The amount of work needed to compute a transformation of a function depends quite heavily on the way it is represented by the computer. There are many advantages to using combination of more basic functions. In the early 19th century, Jean-Baptiste Joseph Fourier chose sines and cosines as building blocks because he could obtain easy formulas for their derivatives. His work, augmented by many other researchers, showed that any given smooth function can be approximated arbitrarily well by a finite linear combination of sines and cosines. The number of components depends only on the smoothness of the target function and the desired degree of approximation. Such expansions provide compact descriptions of complicated functions and simplify the transmission and display of multimedia information.