In the field of signal processing a popular problem is to try and estimate how something has moved as compared to a previous point in time.
In attacking this problem a long standing popular approach has been to use the phase shift property of the Fourier transform.
$$\mathcal{F}\{g(t-a)\}(f) = \mathcal{F}\{g(t)\}(f) \cdot e^{-2\pi afi}$$
This value $a$, we can then try to estimate from the phase spectrums of Fast (Discrete) Fourier Transforms.
However, let us say that an FFT implementation would be too unwieldy and we would rather use simpler functions, such as polynomials to try and calculate the same thing. How could this be done?
My own experiments have been limited to building polynomial ON bases on $t\in[-1,1]$
As we can see, the polynomials (except perhaps for constant and linear term) roughly vary as slow frequency sines, except for a sharper behavior at the edges of the interval.
How could I turn the information captured by a transform made up of such a basis into phase information which I can use for position estimation?
If I had a property like quadrature as we know we do with sines and cosines in the FFT, $\sin(tN)^2 + \cos(tN)^2 = 1$, I expect this to be easy, but since I have polynomials, I expect that I will lack such a property.
Maybe I even made a mistake constructing an ON basis. Would it be wiser to try and use the binomial identities $(t+a)^N = \sum{N \choose k}t^{N-k}a^{k}$ somehow ?