Short wavelets and matrix dilation equations

Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a two-band orthogonal filter bank). For “multifilters” those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust (see J. Approx. Theory, vol.78, p.373-401, 1994) constructed two scaling functions that have extra properties not previously achieved. The functions Φ₁ and Φ₂ are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function Φ, apart from Haar´s piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2×2 matrix coefficients while retaining orthogonality of the multiwavelets. This note derives the properties of Φ₁ and Φ₂ from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions. The properties were derived by Geronimo, Hardin, and Massopust from the iterated interpolation that led to Φ1 and Φ₂. One pair of wavelets was found earlier by direct solution of the orthogonality conditions (using Mathematica). Our construction is in parallel with recent progress by Hardin and Geronimo, to develop the underlying algebra from the matrix coefficients in the dilation equation-in another language, to build the 4×4 paraunitary polyphase matrix in the filter bank. The short support opens new possibilities for applications of filters and wavelets near boundaries