Vector-valued nonstationary Gabor frames

Approximately dual frames of vector‐valued nonstationary Gabor frames and reconstruction errors

“…The mixed frame operator $$$ {S}_{\mathbf{g},\mathbf{h}} $$$ defined as in () is written as $$$$ {S}_{\mathbf{g},\mathbf{h}}\mathbf{f}={U}_{\mathbf{h}}{U}_{\mathbf{g}}^{\ast}\mathbf{f}=\sum \limits_{m,n\in \mathrm{\mathbb{Z}}}\left\langle \mathbf{f},{\mathbf{M}}_{m{b}_n}{\mathbf{g}}^{(n)}\right\rangle {\mathbf{M}}_{m{b}_n}{\mathbf{h}}^{(n)} $$$$ for $$$ \mathbf{f}\in {L}^2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}^K\right) $$$. Walnut's representation of $$$ {S}_{\mathbf{g},\mathbf{h}} $$$ has recently been rigorously proven for VVNSG systems $$$ \mathcal{G}\left(\mathbf{G},\mathbf{b}\right) $$$ and $$$ \mathcal{G}\left(\mathbf{H},\mathbf{b}\right) $$$, under the assumption that all components of window functions are in Wiener space $$$ W\left(\mathrm{\mathbb{R}}\right) $$$ [11]. $$$ W\left(\mathrm{\mathbb{R}}\right) $$$…”

“…State‐of‐the‐art results on NSG frames for $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$ are collected in [1, 2, 8–12] and [13]. In [2], the authors determined a simple sufficient condition for an NSG system with compactly supported window functions to constitute a frame for $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$.…”

“…Ottosen and Nielsen in [12] and [13] considered sparseness properties of adaptive time‐frequency representations obtained using NSG frames and provided a characterization of sparse NSG expansions. Recently, Lian and Song in [11] generalized the notion of NSG frames from $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$ to the vector‐valued Hilbert space $$$ {L}^2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}^K\right) $$$ and investigated the existence of vector‐valued NSG (VVNSG) frames $$$$ {\left\{{\left({M}_{m{b}_n}{g}_1^{(n)},{M}_{m{b}_n}{g}_2^{(n)},\dots, {M}_{m{b}_n}{g}_K^{(n)}\right)}^t\right\}}_{m,n\in \mathrm{\mathbb{Z}}} $$$$ with fast decaying window functions $$$ {\left({g}_1^{(n)},{g}_2^{(n)},\dots, {g}_K^{(n)}\right)}^t $$$ in $$$ {L}^2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}^K\ri...$…”

Approximately dual frames of nonstationary Gabor frames for l2(Z) and reconstruction errors

“…12 State-of-the-art results on NSG frames for L 2 (R) are collected in previous studies. [12][13][14][15][16] In Balazs et al, 13 the authors determined a simple sufficient condition for an NSG system with compactly supported window functions to constitute a frame for L 2 (R). The condition guarantees that the frame operator is a multiplication operator and thus easily inverted.…”

“…Holighaus 12 investigated the structural properties of dual systems for NSG frames and showed that whenever an NSG system, composed of compactly supported window functions with moderate overlap and sufficiently small frequency shift parameters, constitutes a frame, the canonical dual frame is not too different in structure. Lian and Song 16 generalized the notion of NSG frames from L 2 (R) to the vector-valued Hilbert space L 2 (R, C L ) and showed the existence of vector-valued NSG frames with fast decaying window functions.…”