Short wavelets and matrix dilation equations

Abstract

Scaling functions and orthogonal wavelets are created from the coe cients of a lowpass and highpass lter (in a two-band orthogonal lter bank). For \multi lters" those coe cients are matrices. This gives a new block structure for the lter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust constructed two scaling functions that have extra properties not previously achieved. The functions 1 and 2 are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function , apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2 by 2 matrix coe cients while retaining orthogonality. This note derives the properties of 1 and 2 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 in 1] from the iterated interpolation that led to 1 and 2. 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 coe cients in the dilation equation | in another language, to build the 4 by 4 paraunitary polyphase matrix in the lter bank. The short support opens new possibilities for applications of lters and wavelets near boundaries.

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