"Quasi-Random Delays yield Shift-Invariant Wavelet Decompositions in Coorbit Spaces"Holighaus, NickiPrior constructions of wavelet decompositions depend on sampling the wavelet transform in a non-uniform pattern, usually considering dilation by integer powers $2^{\delta k}$ with $\delta>0$, and scale-dependent step size $2^{\delta k}b$, with $b>0$, realizing a generalized shift-invariant system. The variable step size leads to significant complications in the frame-theoretic analysis of sampled wavelet systems and severely impacts computational methods for coefficient manipulation and reconstruction. We present a novel sampling paradigm that leads to shift-invariant wavelet decompositions across wavelet-type coorbit spaces. Specifically, considering some uniform step size $b\in \mathbb{R}^+$ and a countable number of scales $(a_k)_{k\in\mathbb{Z}}\subset \mathbb{R}^+$, we show that the sampled wavelet transform is norm-preserving and boundedly invertible. The proposed sampling scheme relies crucially on the introduction of a quasi-random shift at each scale. Here, we consider Kronecker-type quasi-random sequences specifically, for which the proposed sampling scheme coincides with a time-frequency lattice for all scales below a chosen threshold. I will present the current state of our work-in-progress investigation of shift-invariant wavelet systems with quasi-random delays on the real line, in both theoretical results and numerical studies. This is joint work with Günther Koliander, Clara Hollomey, and Friedrich Pillichshammer. |
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