Speaker: Wayne Lawton

Title: A new result about refinable functions with dilations by PV numbers.

Abstract: A PV number is an algebraic integer all of whose Galois conjugates have modulus less than 1. Erdos proved that if \(u\) is a PV number and \(f\) is a nonzero generalized function satisfying \(f(x) = f(ax) + f(ax-1)\) then the Fourier transform \(F\) of \(f\) does not tend to 0 at infinity so \(f\) is not integrable. Dai, Feng and Wang extended this to \(f\) satisfying \(f(x) = sum c(k) f(ax - d(k))\) where \(d(k)\) are integers. We further extend this to the case where \(d(j)\) are in \(Z[a]\) so proving that integrable multiresolutions on quasilattices are not possible using translations of a single refinable function. Our proof uses the dynamics of hyperbolic toral automorphisms and the distributions of zeros of nonharmonic trigonometric polynomials.