Wavelets and Applications

Abstract

In Section 2.5 we defined the sum function associated to a given family of functions f₀, f_l,... defined on the same set. However, in practice sum functions frequently appear in a different way: a certain class of functions is given, and we want to find “simple functions” f₀, f_l,... such that each function f in the class has an expansion

$$f\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}} {f_n}\left( x \right)$$

(4.1)

for some coefficients a _n . We note that this idea is similar to what we have seen in the context of power series and Fourier series: these cases correspond to the functions f _n being polynomials or trigonometric functions, respectively.