Haar Bases forL2(imagen) and Algebraic Number Theory Original Research Article

Gröchenig and Madych showed that a Haar-type orthonormal wavelet basis ofL2(imagen) can be constructed from the characteristic functionχQof a setQif and only ifQis an affine image of an integral self-affine tileTwhich tiles imagenusing the integer lattice imagen. An integral self-affine tileT=T(A, image) is the attractor of an iterated function systemT=union or logical summi=1 A−1(T+di) where Aset membership, variantMn(image) is an expandingn×ninteger matrix and the digit set image={d1, d2, …, dm}subset of or equal toimagenhasm=det(A), provided that the Lebesgue measureμ(T)>0. Two necessary conditions forT(A, image) to tile imagenwith the integer lattice imagenare that image be a complete set of coset representatives of imagen/A(imagen) and that image[A, image]=imagen, where image[A, image] is the smallest A-invariant lattice containing all {di−dj:i≠j}. These two conditions are necessary and sufficient in the special case that det(A)=2. We study these two conditions for an arbitrary matrix Aset membership, variantMn(image). We prove that a digit set image satisfying the two conditions exists whenever det(A)greater-or-equal, slantedn+1. When det(A)=2 there are number-theoretic obstructions to the existence of such image. Using these we exhibit a (non-expanding) Aset membership, variantM2(image) for which no digit set has image[A, image]=image2. However we show that for all expanding integer matrices A in dimensions 2 and 3, there exists some digit set image that satisfies the two conditions. Could this be true for all expanding integer matrices in dimensionsngreater-or-equal, slanted4? A necessary condition is that the (non-Galois) field[formula]have class number one for allngreater-or-equal, slanted4.