| This dissertation presents a complete characterization of low-pass filters for multivariable wavelets. We use probabilistic methods to prove this characterization, giving a probabilistic interpretation of the low-pass filter and the scaling function; and extending a characterization given by Gundy for the case of dilation by 2 in one dimension. Our results required the development of a generalization of radix representations for integer vectors. We show that if all singular values of a dilation matrix are strictly bigger than 2n , then the matrix yields a radix representation for every vector in Zn , with a given digit set. We present a weaker representation, called a pseudodigit representation, that we may use in proving the main characterization theorem, as well.;With radix representations, we are able to extend some results of Strichartz and Lagarias and Wang relating to which dilation matrices generate a self-affine tiling of Rn , and, correspondingly, which dilation matrices admit a Haar-like scaling function. Finally, we present some examples and supplementary material. We show that the characterization of scaling functions given by Hernandez and Weiss holds in the multivariable case. We also present a Mathematica program for finding digits and pseudodigits of dilation matrices. |
