A Characterization of Sparse Nonstationary Gabor Expansions

Abstract

We investigate the problem of constructing sparse time–frequency representations with flexible frequency resolution, studying the theory of nonstationary Gabor frames (NSGFs) in the framework of decomposition spaces. Given a painless NSGF, we construct a compatible decomposition space and prove that the NSGF forms a Banach frame for the decomposition space. Furthermore, we show that the decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients and we prove an upper bound on the approximation error occurring when thresholding the frame coefficients for signals belonging to the decomposition space.

Appendix: Proof of Theorem 2.1

Proof

To simplify notation we let $D^s_{p,q}:=D(\mathcal {Q},\,L^p,\,\ell ^q_{\omega ^s}).$ Let us first prove Theorem 2.1(1). Allowing the extension $q=\infty ,$ and repeating the arguments from the proof of [4, Proposition 5.7], we can show that

$$\begin{aligned} D^{s+\varepsilon }_{p,\infty }\hookrightarrow D^s_{p,q}\hookrightarrow D^s_{p,\infty },\quad \varepsilon >d/q, \end{aligned}$$

for any $s\in \mathbb {R}$ and $0<p<\infty$ using Definition 2.2(4). It therefore suffice to show that $\mathcal {S}(\mathbb {R}^d)\hookrightarrow D^s_{p,\infty } \hookrightarrow \mathcal {S}^{\prime }(\mathbb {R}^d)$ for any $s\in \mathbb {R}$ and $0<p<\infty .$ For $N\in \mathbb {N},$ we define semi-norms on $\mathcal {S}(\mathbb {R}^d)$ by

$$\begin{aligned} p_N(g):=\sup _{\xi \in \mathbb {R}^d}\left\{ u(\xi )^N\sum _{\left| {\beta }\right| \le N}\left| {\partial ^\beta \hat{g}(\xi )}\right| \right\} ,\quad g\in \mathcal {S}\left( \mathbb {R}^d\right) , \end{aligned}$$

with $u(\xi )=1+\Vert \xi \Vert _2$ as usual. Following the approach in [4, p. 149], and applying Proposition 2.1(4), we get

$$\begin{aligned} \left\| {f}\right\| _{D^s_{p,\infty }}\le Cp_N(f)\quad \text {and}\quad \left\| {f}\right\| _{D^s_{1,1}}\le C^{\prime }p_{N^{\prime }}(f), \end{aligned}$$

for sufficiently large N and $N^{\prime }.$ This proves that $\mathcal {S}(\mathbb {R}^d)\hookrightarrow D^s_{p,\infty }$ and $\mathcal {S}(\mathbb {R}^d)\hookrightarrow D^s_{1,1}.$ To show that $D^s_{p,\infty } \hookrightarrow \mathcal {S}^{\prime }(\mathbb {R}^d)$ we need to take a different approach than in [4]. Setting $\widetilde{\psi }_T:=\sum _{T^{\prime }\in \widetilde{T}}\psi _{T^{\prime }},$ we first note that for $f\in D^s_{p,\infty }$ and $\varphi \in \mathcal {S}(\mathbb {R}^d),$

$$\begin{aligned} \left| {\left\langle {f,\,\varphi }\right\rangle }\right| \le \sum _{T\in \mathcal {T}}\left\| {\psi _T(D)f\widetilde{\psi }_T(D)\varphi }\right\| _{L^1}\le \sum _{T\in \mathcal {T}}\left\| {\psi _T(D)f}\right\| _{L^\infty }\left\| {\widetilde{\psi }_T(D)\varphi }\right\| _{L^1}. \end{aligned}$$

Using Lemma 8.1 below (with $g=\mathcal {F}^{-1}\{\psi _T\hat{f}(T\xi )\}$) we thus get

$$\begin{aligned} \left| {\left\langle {f,\,\varphi }\right\rangle }\right|\le & {} C_1 \sum _{T\in \mathcal {T}}\left| {T}\right| ^{1/p}\left\| {\psi _T(D)f}\right\| _{L^p}\left\| {\widetilde{\psi }_T(D)\varphi }\right\| _{L^1} \nonumber \\\le & {} C_1\left\| {f}\right\| _{D^s_{p,\infty }}\sum _{T\in \mathcal {T}}\left| {T}\right| ^{1/p}\omega _T^{-s}\left\| {\widetilde{\psi }_T(D)\varphi }\right\| _{L^1}\nonumber \\\le & {} C_2\left\| {f}\right\| _{D^s_{p,\infty }}\left\| {\left\{ \left\| {\widetilde{\psi }_T(D)\varphi }\right\| _{L^1}\right\} _{T\in \mathcal {T}}}\right\| _{\ell ^1_{\omega ^{\gamma /p-s}}}, \end{aligned}$$

(7.3)

since $|T|=|Q|^{-1}|Q_T|\le |Q|^{-1}\omega _T^\gamma$ according to Definition 2.2(5). Applying (2.2) we may continue on (7.3) and write

$$\begin{aligned} \left| {\left\langle {f,\,\varphi }\right\rangle }\right| \le C_3\left\| {f}\right\| _{D^s_{p,\infty }}\left\| {\varphi }\right\| _{D^{\gamma /p-s}_{1,1}}\le C_4\left\| {f}\right\| _{D^s_{p,\infty }}p_N(\varphi ), \end{aligned}$$

for sufficiently large N since $\mathcal {S}(\mathbb {R}^d)\hookrightarrow D^s_{1,1}.$ We conclude that $D^s_{p,\infty }\hookrightarrow \mathcal {S}^{\prime }(\mathbb {R}^d)$ which proves Theorem 2.1(1).

The proof of Theorem  2.1(2) follows directly from Theorem  2.1(1) and the arguments in [4, p. 150].

To prove Theorem  2.1(3) we let $f\in D^s_{p,q}$ and choose $I\in C^\infty _c(\mathbb {R}^d)$ with $0\le I(\xi )\le 1$ and $I(\xi )\equiv 1$ in a neighbourhood of $\xi =0.$ Also, we define $(\widetilde{f})^{\widehat{}}:=I\hat{f}$ and

$$\begin{aligned} \widetilde{f}_\varepsilon :=\mathcal {F}^{-1}\left\{ \varphi _\varepsilon \,*\, (\widetilde{f})^{\widehat{}}\right\} \in \mathcal {S}\left( \mathbb {R}^d\right) , \end{aligned}$$

with $\varphi _\varepsilon (\xi ):=\varepsilon ^{-d}\varphi (\xi /\varepsilon )$ and $\varphi$ being a compactly supported mollifier. Since ${{\mathrm{supp}}}(I)$ is compact, we may choose a finite subset $T^*\subset \mathcal {T},$ such that ${{\mathrm{supp}}}(I)\subset \cup _{T\in T^*}Q_T$ and $\sum _{T\in T^*}\psi _T(\xi )\equiv 1$ on ${{\mathrm{supp}}}(I).$ Using Lemma 8.2 below we obtain

$$\begin{aligned} \Vert \widetilde{f}\Vert _{L^p}= & {} \left\| {\mathcal {F}^{-1}I\mathcal {F}\left( \mathcal {F}^{-1}\left( \sum _{T\in T^*}\psi _T\cdot \hat{f}\right) \right) }\right\| _{L^p}\\\le & {} C\sum _{T\in T^*}\left\| {\mathcal {F}^{-1}I}\right\| _{L^{\tilde{p}}}\left\| {\psi _T(D)f}\right\| _{L^p}<\infty , \end{aligned}$$

with $\tilde{p}=\min \{1,\,p\}.$ The dominated convergence theorem thus yields

$$\begin{aligned} \left\| {\widetilde{f}-\widetilde{f}_\varepsilon }\right\| _{D^s_{p,q}}\le C\left\| {\left\{ \left\| {\widetilde{f}-\widetilde{f}_\varepsilon }\right\| _{L^p}\right\} _{T\in \mathcal {T}}}\right\| _{\ell ^q_{\omega ^s}}\rightarrow 0,\quad \text {as}\,\varepsilon \rightarrow 0, \end{aligned}$$

so the proof is done if we can show that $\Vert f-\widetilde{f}\Vert _{D^s_{p,q}}$ can be made arbitrary small by choosing $\widetilde{f}$ appropriately. To show this, we define the set $T_{\circ }:=\{T\in \mathcal {T}~|~I(\xi )\equiv 1 \text { on}\,{{\mathrm{supp}}}(\psi _T)\}.$ Denoting the complement $T_{\circ }^c,$ Lemma  8.2 below yields

$$\begin{aligned} \left\| {f-\widetilde{f}}\right\| _{D^s_{p,q}}^q= & {} \sum _{T\in T_{\circ }^c}\omega _T^{sq}\left\| {\mathcal {F}^{-1}\left( \psi _T(\hat{f}-I\hat{f})\right) }\right\| _{L^p}^q\\\le & {} C_1\sum _{T\in T_{\circ }^c}\omega _T^{sq}\left( \left\| {\psi _T(D)f}\right\| _{L^p}+\left\| {\mathcal {F}^{-1}I\mathcal {F}\left( \psi _T(D)f\right) }\right\| _{L^p}\right) ^q\\\le & {} C_2\sum _{T\in T_{\circ }^c}\omega _T^{sq}\left\| {\psi _T(D)f}\right\| _{L^p}^q. \end{aligned}$$

Finally, since $f\in D^s_{p,q}$ we can choose ${{\mathrm{supp}}}(I)$ large enough, such that $\Vert f-\widetilde{f}\Vert _{D^s_{p,q}}<\varepsilon ,$ for any given $\varepsilon >0.$ This proves Theorem  2.1(3). $\square$

In the proof of Theorem 2.1 we used the following two lemmas. A proof of Lemma 8.1 can be found in [3, Lemma 3] and a proof of Lemma  8.2 can be found in [42, Proposition 1.5.1].

Lemma 8.1

Let $g\in L^p(\mathbb {R}^d)$ and ${{\mathrm{supp}}}(\hat{g})\subset \Gamma ,$ with $\Gamma \subset \mathbb {R}^d$ compact. Given an invertible affine transformation T,  let $\hat{g}_T(\xi ):=\hat{g}(T^{-1}\xi ).$ Then for $0<p\le q\le \infty ,$

$$\begin{aligned} \left\| {g_T}\right\| _{L_q}\le C\left| {T}\right| ^{1/p-1/q}\left\| {g_T}\right\| _{L_p}, \end{aligned}$$

for a constant C independent of T.

Lemma 8.2

Let $\Omega$ and $\Gamma$ be compact subsets of $\mathbb {R}^d.$ Let $0<p\le \infty$ and $\tilde{p}=\min \{1,\,p\}.$ Then there exists a constant C such that

$$\begin{aligned} \left\| {\mathcal {F}^{-1}M\mathcal {F}f}\right\| _{L^p}\le C \left\| {\mathcal {F}^{-1}M}\right\| _{L^{\tilde{p}}}\left\| {f}\right\| _{L^p}, \end{aligned}$$

for all $f\in L^p(\mathbb {R}^d)$ with ${{\mathrm{supp}}}(\hat{f})\subset \Omega$ and all $\mathcal {F}^{-1}M\in L^{\tilde{p}}(\mathbb {R}^d)$ with ${{\mathrm{supp}}}(M)\subset \Gamma .$