Optimal wavelet representation of signals and the wavelet sampling theorem

The wavelet representation using orthonormal wavelet bases has received widespread attention. Recently M-band orthonormal wavelet bases have been constructed and compactly supported M-band wavelets have been parameterized. This paper gives the theory and algorithms for obtaining the optimal wavelet multiresolution analysis for the representation of a given signal at a predetermined scale in a variety of error norms. Moreover, for classes of signals, this paper gives the theory and algorithms for designing the robust wavelet multiresolution analysis that minimizes the worst case approximation error among all signals in the class. All results are derived for the general M-band multiresolution analysis. An efficient numerical scheme is also described for the design of the optimal wavelet multiresolution analysis when the least-squared error criterion is used. Wavelet theory introduces the concept of scale which is analogous to the concept of frequency in Fourier analysis. This paper introduces essentially scale limited signals and shows that band limited signals are essentially scale limited, and gives the wavelet sampling theorem, which states that the scaling function expansion coefficients of a function with respect to an M-band wavelet basis, at a certain scale (and above) completely specify a band limited signal (i.e., behave like Nyquist (or higher) rate samples)