Given a function $f(x)$ we can use the Fourier Transform to find $F(w)$, which represents how we can build up the signal by $\sin$/$\cos$ as we find the coefficients, frequencies and phases. (I realise this is very simplified I just wanted to fit it in a sentence)
My question is could we have represented this image by any set of orthogonal basis? I know we can have a $\operatorname{sinc}$ transform. If we can how would one go about deriving the transfer function?
Thank you in advance! :)
Yes, there are various sets of orthogonal functions that can be used for such an expansion provided that their inner product (as an integral of the product of any two) gives Kronecker's delta (for orthonormal) or a scalar multiple of it (for orthognal). There are many such functions, particularly polynomial functions. You get the transfer function as the response (convolution integral) to a Dirac impulse for the input, as usual. Whether you will get a simple relation for the frequency-domain transfer function as for Fourier analysis depends on the mathematical properties of these new transforms. Most likely, this will much more complicated than when using complex exponentials as in F.T.
