Alternatives to the Fourier transform

Question

I am searching for methods which can be used to represent bounded real-valued functions defined on a finite interval. More specifically, I am interested in approximations of such functions which have a finite number of controllable parameters or equivalently, approximations of such functions which can be represented in a finite-dimensional function space. Furthermore, it would be very desirable if an order could be associated with the basis functions of such a function space with respect to some metric.