Fractional Wavelets, Derivatives, and Besov Spaces

We show that a multi-dimensional scaling function of order γ (possibly fractional) can always be represented as the convolution of a polyharmonic B-spline of order γ and a distribution with a bounded Fourier transform which has neither order nor smoothness. The presence of the B-spline convolution factor explains all key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multi-scale differentiation property, and smoothness of the basis functions. The B-spline factorization also gives new insights on the stability of wavelet bases with respect to differentiation. Specifically, we show that there is a direct correspondence between the process of moving a B-spline factor from one side to another in a pair of biorthogonal scaling functions and the exchange of fractional integrals/derivatives on their wavelet counterparts. This result yields two “eigen-relations” for fractional differential operators that map biorthogonal wavelet bases into other stable wavelet bases. This formulation provides a better understanding as to why the Sobolev/Besov norm of a signal can be measured from the $l _{ p }$ -norm of its rescaled wavelet coefficients. Indeed, the key condition for a wavelet basis to be an unconditional basis of the Besov space $B _{ q } ^{ s }$ $(L _{ p }$ $(R ^{ d }$ )) is that the s-order derivative of the wavelet be in $L _{ p }$ .