A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley.
In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to Ingrid Daubechies.
Assuming the scaling function has compact support, then implies that there is a finite sequence of coefficients for , and for , such that
Defining another function, known as mother wavelet or just the wavelet
one can show that the space , which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to inside . Or put differently, is the orthogonal sum (denoted by ) of and . By self-similarity, there are scaled versions of and by completeness one has
thus the set
is a countable complete orthonormal wavelet basis in .