Abstract:

Meyer constructed a dyadic wavelet basis of functions on the line whose Fourier transforms are $C^\infty$ and supported in $[-8\pi/3, -2\pi/3] \cup [2\pi/3, 8\pi/3]$. This construction was generalized to rational dilations by Auscher; in particular to arbitrary integer dilations. In this talk, I will describe a sufficient geometric condition on expansive, integer dilations for there to exist these Meyer type wavelets in $R^n$. Using this condition, I will describe how one can show that for all expansive $2\times 2$ integer matrices, there exists Meyer type wavelet bases. Open problems relating to both the geometric condition on the matrices and the general existence question of Meyer type wavelet bases in $R^n$ will also be given.