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APRG Seminar

Title: On translation invariance of the mixed frame operator
Speaker: Anupam Gumber (IISc Mathematics)
Date: 27 January 2021
Time: 4 pm
Venue: Microsoft Teams (online)

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 $L^2(\mathbb{R}^n)$.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 29 Feb 2024