Optimal Wavelet Representation Ofsignals and the Wavelet

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 15, 12, 32, 17]. 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 23]. 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 eecient 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 scalelimited signals and shows that bandlimited signals are essentially scalelimited, and gives the wavelet sampling theorem, which states that the scaling function expansion coeecients of a function with respect to a M-band wavelet basis, at a certain scale (and above) completely specify a bandlimited signal (i.e., behave like Nyquist (or higher) rate samples).