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

A non-MRA $$C^r$$ Frame Wavelet with Rapid Decay