A countable family $\{\psi_n: n \in \mathbb{N}\}$ of elements in a Hilbert space $\mathcal{H}$ constitutes a frame if there are constants $0< A\leq B < \infty$ s.t. $\forall f \in \mathcal{H}$ we have: \begin{equation} A\|f\|^2 \leq \sum\limits_n|\langle f,\psi_n \rangle|^2\leq B \|f\|^2, \end{equation} where $\langle\cdot, \cdot\rangle$ denotes an inner-product in $\mathcal{H}$. Frames were introduced by Duffin and Schaeffer in 1952 to deal with problems in nonharmonic Fourier series, and have been used more recently to obtain signal reconstruction for signals embedded in certain noises.

For a given pair of frames $\{\psi_n\}$ and $\{\varphi_n\}$ in $\mathcal{H}$, the associated mixed frame operator $S: \mathcal{H} \to \mathcal{H}; f \mapsto \sum_n \langle f, \psi_n \rangle\varphi_n$ is a bounded linear operator. The translation invariance of this operator plays a significant role in investigating reproducing formulas for frame pairs $\{\psi_n\}$ and $\{\varphi_n\}$ in $\mathcal{H}$.

In the present talk, we examine necessary and sufficient conditions for $S$ to be invariant under translations on $\mathbb{R}^n$ when $\{\psi_n\}$ and $\{\varphi_n\}$ belong to a special class of structured frame systems in $L^2(\mathbb{R}^n)$.

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Last updated: 11 Dec 2023