  We study the harmonic analysis of the quadrature mirror filters coming from
multiresolution wavelet analysis of compactly supported wavelets. It is known
that those of these wavelets that come from third order polynomials are
parametrized by the circle, and we compute that the corresponding filters
generate irreducible mutually disjoint representations of of the Cuntz algebra
$ O_{2} $ except at two points on the circle. One of the two exceptional points
corresponds to the Haar wavelet and the other is the unique point on the circle
where the father function defines a tight frame which is not an orthonormal
basis. At these two points the representation decomposes into two and three
mutually disjoint irreducible representations, respectively, and the two
representations at the Haar point are each unitarily equivalent to one of the
three representations at the other singular point.
