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.
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It sounds like you want a unitary integral transform, which has been discussed here. The only alternative to Fourier that comes to mind is Hartley. – J.G. Mar 14 '21 at 17:26
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You might be interested by Wavelets? – LL 3.14 Mar 15 '21 at 21:24