### Smooth Wavelet Tight Frames with Zero Moments

#### Abstract

This paper considers the design of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. In particular, this paper takes up the design of systems that are analogous to Daubechies orthonormal wavelets - that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. Gröbner bases are used to obtain the solutions to the nonlinear design equations.

Reprint: Smooth Wavelet Tight Frames with Zero Moments. Applied and Computational Harmonic Analysis (ACHA), Vol. 10, No. 2, pp. 163-181, March 2001.

Note: It turns out these wavelet frames can be obtained with methods that are simpler than Grobner bases. See the double-density DWT.

#### Examples from the paper

Minimal-length wavelet tight frames having a prescribed number
of zeros at z=-1 and z=1 (specified by the values K_{i}).