The idea of a series expansion of a function is to express the function as an infinite series of simpler terms, which follow a pattern. Truncating such an expansion allows us to approximate the function.
Depending on the subsequent goal of the expansion (e.g. finding the limit of a quotient of functions, or performing frequency analysis), we usually have to choose two things: the form that the terms of the expansion take, and desired properties of the truncated approximation, the latter almost always being a property which expresses some notion of "closeness" between the approximation and the function.
What are probably the two most common examples of series expansions illustrate this quite well.
To find the limit of a function at a point, we mainly need our approximation to be locally close to the function being approximated. To do so, we can attempt to match their derivatives at that point up to a certain order. This is exactly what the Taylor expansion does, which is why we use it to compute limits. On top of that, the terms of the Taylor series are powers, making it a power series, which is nice.
To perform frequency analysis on a signal, we mainly need our approximation to have terms which depend on multiples of a frequency. This is exactly what the Fourier expansion does, which is why we use it for frequency analysis. Unlike in the previous example, we don't mind large local deviations, but do however mind large global deviations. To compute the Fourier expansion, then, we consider the terms to be a basis of sinusoids and project the function on that basis.
This leads me to my two questions related to series expansions:
- Are there any expansions which use terms which aren't powers or sinusoids?
- Are there any expansions where the coefficients of the terms aren't computed to obtain a locally close or globally close approximation, but rather another desirable property of the approximation?