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 .[1] 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 .