A non-MRA $C^r$ frame wavelet with rapid decay

A generalized filter construction is used to build an example of a non-MRA normalized tight frame wavelet for dilation by 2 in $L^2(\mathbb{R})$. This example has the same multiplicity function as the Journ'e wavelet, yet has a $C^{\mathrm{\infty }}$ Fourier transform and can be made to be $C^r$ for any fixed positive integer $r$.