Wavelet series vs. Fourier series for approximation
Shaun
I'm new to the wavelet series and almost as new to the Fourier series
so please bear with me if this question seems stupid. Does a Wavelet
series approximation of a function minimize any kind of approximation
error?
I'm working with a system which needs to numerically approximate any
discrete input function with finite 'energy' (i.e. the definite
integral from - to + infinity of the square of the real part of these
functions is < +infinity). From what I've read, these functions are
elements of the Lp space (p=2) and the 'energy' of these functions is
the L2 norm of the L2 vector space. The Fourier and wavelet series
both seem like good ways to approximate functions in L2 and I've found
that a Fourier series approximation minimizes the Mean Squared Error
(MSE) of the approximation (i.e. the L2 norm of the difference between
the original function and its FS approximation is minimized). I
haven't seen anything about error minimization for wavelet series
approximations, however.
From a signal processing standpoint, the discrete wavelet transform
seems much more flexible than the DFT since it gives you finer control
over time/frequency resolution within the constraints of the
uncertainty principle and allows for multi-resolution analysis of the
discrete input function. However, accuracy of the approximation is
also very important and it would be good to know if the error in
approximation had a upper-bound when using wavelets. It would help a
lot with my choice of algorithms if this were true or if there were
certain types of mother wavelets I could use which made this true.
Hope y'all can help.
Cheers,
Shaun