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# A Complete Family of Scaling Functions: The (α, τ)-Fractional Splines

## T. Blu, M. Unser

### Proceedings of the Twenty-Eighth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'03), Hong Kong SAR, People's Republic of China, April 6-10, 2003, vol. VI, pp. 421-424.

We describe a new family of scaling functions, the (α, τ)-fractional splines, which generate valid multiresolution analyses. These functions are characterized by two real parameters: α, which controls the width of the scaling functions; and τ, which specifies their position with respect to the grid (shift parameter). This new family is complete in the sense that it is closed under convolutions and correlations.

We give the explicit time and Fourier domain expressions of these fractional splines.

We prove that the family is closed under generalized fractional differentiations, and, in particular, under the Hilbert transformation. We also show that the associated wavelets are able to whiten 1⁄ƒλ-type noise, by an adequate tuning of the spline parameters.

A fast (and exact) FFT-based implementation of the fractional spline wavelet transform is already available. We show that fractional integration operators can be expressed as the composition of an analysis and a synthesis iterated filterbank.

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