Applications of function series? What applications exist for function series?
Preferably "applied mathematical" applications, rather than application in pure mathematics.
 A: Function series are probably some of the most well applied objects in mathematics. In general, the I would guess that the most common application of function series in applied math is to approximate a complicated function by a finite sum of simpler functions.
The most prolific example would be Fourier series, which is used to compress data. Fourier series represented a periodic function as a linear combinations of sines and cosines. Often, with a few clever tricks, you can approximate any function locally very well by just a few terms from the Fourier series (and a few really just means finite).
Another example is Taylor series, which can approximate an analytic function locally as a polynomial. This is useful because polynomials are very well understood. Taylor series are often used to find a fit to a finite number of data points.
Another series of functions I know are useful are Hermite, Legendre, and Laguerre polynomials. For instance, the profile of the B2 bomber was optimized through approximations with series of Legendre polynomials. These polynomials are nice because they have a property called orthonormality.
