### Interpolating Multiwavelets and the Sampling Theorem

#### Abstract

This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator.

Interpolating Multiwavelets and the Sampling Theorem, IEEE Trans on Signal Processing, vol 47, no 6, pp 1615-1621, June 1999.

#### Examples from the paper

An interpolating 2-balanced orthogonal multiwavelet basis:

• cardbal2.m Interpolating 2-balanced multiwavelet filters. (Matlab)
• gb Groebner basis defining the filter G(z).
• Programs for deriving the Grobner basis and filters:
• setup Generates the characterizing equations (Maple).
• sfile Generates the Grobner basis (Singular).
• result Generates the filter G(z) from the Grobner basis (Maple).

An Interpolating 3-balanced orthogonal multiwavelet basis:

• cardbal3.m Interpolating 3-balanced multiwavelet filters. (Matlab)
• gb Groebner basis defining the filter G(z).

An interpolating 4-balanced orthogonal multiwavelet basis:

• cardbal4.m Interpolating 4-balanced multiwavelet filters. (Matlab)
• gb Groebner basis defining the filter G(z).