Compression of square integrable functions: Fourier vs. wavelets

P/sub /spl alpha// is the class of functions with /spl alpha/-th derivative bounded in L/sub 2/-norm, /spl alpha/>0. Kolmogorov and Tichomirov (1959) have /spl epsiv/-specified any f/spl isin/P/sub /spl alpha// by a O(/spl epsiv//sup -1//spl alpha//) bits length code obtained from the Fourier (trigonometric) spectrum of f. We prove that the code can be derived from f in linear time. We show that wavelets are equivalent to the trigonometric basis with respect to both the length of the code and the time to get it from the spectrum (to within multiplicative constants). On the other hand, some bases of wavelets outperform Fourier's, if we want to find the value of f at some point given the code of f.