Wavelet matrices and the representation of discrete functions

Abstract . This paper includes a formulation of the notion of wavelet matrices and their fundamental invariants. These arise as coefficient systems which define compactly supported wavelet systems which satisfy m -scale scaling equations, for m ≥ 2. A Haar wavelet matrix is a square wavelet matrix, and there is a characteristic Haar mapping from the class of all wavelet matrices to Haar wavelet matrices. The Haar wavelet matrices are equivalent to a unitary group of a specific dimension, while the wavelet matrices admit a unitary action and the characteristic Haar mapping is an equivariant mapping under this action. Moreover, on the fibres of this mapping there is an infinite dimensional Lie group defined, and in specific cases, this allows for a complete classification of all wavelet matrices. In particular, a given wavelet matrix consists of a first row (the scaling vector or “lowpass” filter), and the remaining rows (the wavelet vectors or “high-pass” filters). An algorithm for determining the wavelet vectors in terms of the scaling vectors is given. In analogy with the representation of L 2 functions by means of a wavelet system (an orthonormal expansion), there is a representation of arbitrary discrete functions in terms of a wavelet matrix used as a discrete orthogonal system.