Frames Adapted to a Phase-Space Cover

Abstract

We construct frames adapted to a given cover of the time–frequency or time-scale plane. The main feature is that we allow for quite general and possibly irregular covers. The frame members are obtained by maximizing their concentration in the respective regions of phase-space. We present applications in time–frequency, wavelet, and Gabor analysis.

Appendices

Appendix 1: Proof of Theorem 3.5

In this appendix, we prove Theorem 3.5. The proof is essentially contained in [47], but is not explicitly stated in the required generality. We therefore show how to derive Theorem 3.5 from some technical lemmas in [47].

Remark 6.7

We quote simplified versions of some statements in [47]. The article [47] considers a technical variant of the amalgam space ${W_R(L^\infty ,L^1_w)}({\mathcal {G}})$, called the weak amalgam space ${W_R^\mathrm{weak}(L^\infty ,L^1_w)}$ (see [47, Sect. 2.4]), which we do not wish to introduce here. By [47, Proposition 1], $L^1({\mathcal {G}}) \hookrightarrow {W_R^\mathrm{weak}(L^\infty ,L^1_w)}({\mathcal {G}}) \hookrightarrow {W_R(L^\infty ,L^1_w)}({\mathcal {G}})$. Some results from [47] that we quote assume that a certain function $g$ belongs to ${W_R(L^\infty ,L^1_w)}({\mathcal {G}})$ and are proved in [47] under the weaker assumption: $g \in {W_R^\mathrm{weak}(L^\infty ,L^1_w)}$.

We quote the following estimate:

Lemma 6.8

[25, Lemma 3.8], [47, Lemma 2] Let $E$ be a solid, translation invariant BF space, let $w$ be an admissible weight for it, and let $\varGamma \subseteq {\mathcal {G}}$ be a relatively separated set. Then for every $f \in E$ and $g \in W_R(C_0,L^1_w)$, the sequence $(\left<f,L_\lambda g\right>)_{\lambda \in \varLambda }$ belongs to ${E_\mathrm{d}}(\varLambda )$ and

$$\begin{aligned} ||(\left<f,L_\lambda g\right>)_\lambda ||_{{E_\mathrm{d}}} \lesssim ||f||_E||g||_{W_R(L^\infty ,L^1_w)}. \end{aligned}$$

The implicit constants depend on the spreadness $\rho (\varGamma )$—cf. (15).

Suppose that Assumptions (A1) and (A2) from Sect. 2.3.2 hold.

For a solid, translation invariant BF space $E$, we consider an ${L^2}$-valued version of ${E_\mathrm{d}}(\varGamma )$,

$$\begin{aligned} {E_{\mathrm{d},{L^2}}}={E_{\mathrm{d},{L^2}}}(\varGamma ) := \left\{ \, (f_\gamma )_{\gamma \in \varGamma } \in ({L^2}({\mathcal {G}}))^\varGamma \, \Big | \, (||f_\gamma ||_{L^2})_{\gamma \in \varGamma } \in {E_\mathrm{d}}(\varGamma ) \, \right\} , \end{aligned}$$

and endow it with the norm $||(f_\gamma )_{\gamma \in \varGamma }||_{{E_{\mathrm{d},{L^2}}}} := ||(||f_\gamma ||_{{L^2}})_{\gamma \in \varGamma }||_{{E_\mathrm{d}}}$.

Let $\left\{ \, {T_\gamma } \, \Big | \, \gamma \in \varGamma \, \right\}$ be a well-spread family of operators—cf. Sect. 3. Let $U \subseteq {\mathcal {G}}$ be a relatively compact neighborhood of the identity. Consider the operators $C_T$ and $S_U$ formally defined by

$$\begin{aligned} C_T(f)&:= ({T_\gamma }(f))_{\gamma \in \varGamma }, \quad f \in S_E, \end{aligned}$$

(36)

$$\begin{aligned} S_U((f_\gamma )_{\gamma \in \varGamma })&:= \sum _{\gamma \in \varGamma } P(f_\gamma ) 1_{\gamma U}, \quad f_\gamma \in {L^2}({\mathcal {G}}), \end{aligned}$$

(37)

where $1_{\gamma U}$ denotes the characteristic function of the set $\gamma U$. These operators satisfy the following mapping properties.

Proposition 6.9

Assume (A1) and (A2), and let $\left\{ \, {T_\gamma } \, \Big | \, \gamma \in \varGamma \, \right\}$ be a well-spread family of operators. Then the operators $C_T$ and $S_U$ in (36) and (37) satisfy the following:

• (a) The analysis operator $C_T$ maps $S_E$ boundedly into ${E_{\mathrm{d},{L^2}}}(\varGamma )$.

• (b) For every relatively compact neighborhood of the identity $U$, and every sequence $F \equiv (f_\gamma )_\gamma \in {E_{\mathrm{d},{L^2}}}$, the series defining $S_U(F)$ converge absolutely in ${L^2}({\mathcal {G}})$ at every point. Moreover, the operator $S_U$ maps ${E_{\mathrm{d},{L^2}}}(\varGamma )$ boundedly into $E$ (with a bound that depends on U).

Proof

Part (b) is proved in [47, Proposition 4 (b)] under a weaker hypothesis. Part (a) is a slight variant of [47, Proposition 4 (a)]; for completeness we give a full argument. Let $(\varGamma , \varTheta , g)$ be an envelope for $\left\{ \, {T_\gamma } \, \Big | \, \gamma \in \varGamma \, \right\}$.

Let $f \in S_E$. Since $\eta _\gamma$ is bounded, $f \eta _\gamma \in E$. By the definition of well-spread family (cf. 20),

$$\begin{aligned} \left| {T_\gamma }f(x) \right|&\le \int \limits _{\mathcal {G}}\left| f(y) \right| g(\gamma ^{-1}y) \varTheta (y^{-1}x) \mathrm{d}y = \left( \left| f \right| L_\gamma g \right) * \varTheta (x). \end{aligned}$$

By Young’s inequality $L^1* L^2 \hookrightarrow L^2$, we have

$$\begin{aligned} ||{T_\gamma }f||_2 \le ||\varTheta ||_2 \int \limits _{\mathcal {G}}\left| f(y) \right| g(\gamma ^{-1}y) \mathrm{d}y \lesssim ||\varTheta ||_{W(L^\infty ,L^1_w)} \int \limits _{\mathcal {G}}\left| f(y) \right| g(\gamma ^{-1}y) \mathrm{d}y. \end{aligned}$$

Now Lemma 6.8 yields $||C_T(f)||_{{E_{\mathrm{d},{L^2}}}}\lesssim ||f||_{E} ||g||_{W_R(L^\infty ,L^1_w)}$, as desired. $\square$

Remark 6.10

Note that in the last proof, the use of the $L^2$ norm is somewhat arbitrary; a number of other function norms could have been used instead (cf. [47, Proposition 4]).

Now we prove the key approximation result (cf. [47, Theorem 1]).

Theorem 6.11

Assume (A1) and (A2), and let $\left\{ \, {T_\gamma } \, \Big | \, \gamma \in \varGamma \, \right\}$ be a well-spread family of operators. Given $\varepsilon >0$, there exists $U_0$, a relatively compact neighborhood of e such that for all $U \supseteq U_0$,

$$\begin{aligned} \left\| \sum _{\gamma \in \varGamma } T_\gamma f \!-\! S_UC_Tf\right\| _E \le \varepsilon ||f||_{E}, \quad f \in S_E. \end{aligned}$$

(38)

Remark 6.12

The neighborhood $U_0$ can be chosen uniformly for any class of spaces $E$ having the same weight $w$ and the same constant $C_{E,w}$ (cf. (16)).

Concerning the parameters in Assumptions (A1) and (A2) and (20), the choice of $U_0$ only depends on $||K||_{W(L^\infty ,L^1_w)}$, $||K||_{W_R(L^\infty ,L^1_w)}$, $||\varTheta ||_{W(L^\infty ,L^1_w)}$, $||\varTheta ||_{W_R(L^\infty ,L^1_w)}$, $||g||_{{W_R(L^\infty ,L^1_w)}}$, and $\rho (\varGamma )$ (cf. (15)).

Proof of Theorem 6.11

Let $f \in S_E$, and let $U$ be a relatively compact neighborhood of e. Because of the inclusion $S_E\hookrightarrow W(L^\infty ,E)$ in Proposition 2.6, it suffices to dominate the left-hand side of (38) by $\varepsilon ||f||_{W(L^\infty ,E)}$.

Note that since ${T_\gamma }f \in S_E$, $S_UC_Tf(x) = \sum _{\gamma \in \varGamma } {T_\gamma }f(x) 1_{\gamma U}(x)$. Hence, using (20), let us estimate

$$\begin{aligned} \left| \sum _{\gamma \in \varGamma } {T_\gamma }f(x) - S_UC_Tf(x) \right|&= \left| \sum _{\gamma \in \varGamma } 1_{\gamma ({\mathcal {G}}\setminus U)}(x) {T_\gamma }(f)(x) \right| \\&\le \sum _{\gamma \in \varGamma } \int \limits _{{\mathcal {G}}} \left| f(y) \right| g(\gamma ^{-1}y) \varTheta (y^{-1}x) 1_{\gamma ({\mathcal {G}}\setminus U)}(x) \mathrm{d}y. \end{aligned}$$

The rest of the proof is carried out exactly as in [47, Theorem 1]. Indeed, the proof there only depends on the estimate just derived.² (The definition of well-spread family of operators was tailored so that the proof in [47, Theorem 1] would still work.) $\square$

Finally, we can prove Theorem 3.5.

Proof of Theorem 3.5

Let $\left\{ {T_\gamma }: \gamma \in \varGamma \right\}$ be a well-spread family of operators, and suppose that the operator $\sum _\gamma {T_\gamma }: S_E\rightarrow S_E$ is invertible. We have to show that for $f \in S_E$, $||f||_E\approx ||C_T(f)||_{{E_{\mathrm{d},{L^2}}}(\varGamma )}$. The estimate $||C_T(f)||_{{E_{\mathrm{d},{L^2}}}(\varGamma )} \lesssim ||f||_E$ is proved in Proposition 6.9 (a). To establish the second inequality, consider the operator $PS_UC_T: S_E\rightarrow S_E$. Then for $f \in S_E$,

$$\begin{aligned} \left\| \sum _{\gamma \in \varGamma } {T_\gamma }f \!-\! P S_UC_Tf\right\| _E\!=\! \left\| P \sum _{\gamma \in \varGamma } {T_\gamma }f \!-\!PS_UC_Tf \right\| _E\lesssim \left\| \sum _{\gamma \in \varGamma } {T_\gamma }f \!-\!S_UC_Tf\right\| _E. \end{aligned}$$

This estimate, together with Theorem 6.11, implies that $||\sum _\gamma {T_\gamma }- P S_UC_T||_{S_E\rightarrow S_E} \rightarrow 0$ as $U$ grows to ${\mathcal {G}}$. Hence, there exists $U$ such that $PS_UC_T$ is invertible on $S_E$. Consequently, for $f \in S_E$, $||f||_E\approx ||PS_UC_Tf||_E\lesssim ||C_T(f)||_{{E_{\mathrm{d},{L^2}}}(\varGamma )}$. Here we have used the boundedness of $S_U$—contained in Proposition 6.9 (b)—and the boundedness of $P:E\rightarrow W(L^\infty ,E) \hookrightarrow E$—contained in Proposition 2.6. $\square$

Appendix 2: Pseudodifferential Operators and Proof of Proposition 4.3

The Weyl transform of a distribution $\sigma \in \mathcal {S}'({{\mathbb R}^d}\times {{\mathbb R}^d})$ is an operator $\sigma ^w$ that is formally defined on functions $f:{{\mathbb R}^d}\rightarrow {\mathbb C}$ as

$$\begin{aligned} \sigma ^w (f)(x) := \int \limits _{{{\mathbb R}^d}\times {{\mathbb R}^d}} \sigma \left( \frac{x+y}{2},\xi \right) e^{2\pi i(x-y)\xi } f(y) \mathrm{d}y \mathrm{d}\xi , \quad x \in {{\mathbb R}^d}. \end{aligned}$$

The fundamental results in the theory of pseudodifferential operators provide conditions on $\sigma$ for the operator $\sigma ^w$ to be well defined and bounded on various function spaces. We now quote some results about pseudodifferential operators acting on modulation spaces—cf. Sect. 4.1.

In [34, 35], it was shown that modulation spaces on ${{\mathbb R}^{2d}}$ serve as symbol classes to study pseudodifferential operators acting on modulation spaces on ${{\mathbb R}^d}$, recovering and extending classical results from Sjöstrand [50, 51]. We quote the following simplified version of [35, Theorems 4.1, 4.6, and Corollaries 3.3, 4.7]. (The GRS-condition for admissible weights in Sect. 4.1 is important here.)

Theorem 6.13

Let $w$ be an admissible TF weight—cf. Definition 4.1—and let $\varphi \in M^1_w({{\mathbb R}^d})$ be nonzero. Let us denote $\widetilde{w}(z_1,z_2) = w(-z_2,z_1)$. Then the following statements hold true:

• (i) If $\sigma \in M^{\infty ,1}_{\widetilde{w}}({{\mathbb R}^{2d}})$, then $\sigma ^w$ is bounded on $M^p_v({{\mathbb R}^d})$ for all $w$-moderated weights $v$ and all $p \in [1,+\infty ]$.

• (ii) If $\sigma \in M^{\infty ,1}_{\widetilde{w}}({{\mathbb R}^{2d}})$ and $\sigma ^w$ is invertible as an operator on ${{L^2}({\mathbb R}^d)}$, then $\sigma ^w$ is invertible as an operators on $M^p_v({{\mathbb R}^d})$ for all $w$-moderated weights $v$ and all $p \in [1,+\infty ]$.

• (iii) Let $T:\mathcal {S}({{\mathbb R}^d})\rightarrow \mathcal {S}'({{\mathbb R}^d})$ be a linear and continuous operator. For $(x,\xi ) \in {{\mathbb R}^d}\times {{\mathbb R}^d}$ let us denote $\varphi _{(x,\xi )}(t):=e^{2\pi i \xi t}\varphi (t-x)$. If there exists a function $H \in L^1_w({{\mathbb R}^{2d}})$ such that

  $$\begin{aligned} \left| \left<T(\varphi _{(x,\xi )}),\varphi _{(x',\xi ')}\right> \right| \le H(x'-x,\xi '-\xi ), \quad (x,\xi ),(x',\xi ') \in {{\mathbb R}^d}\times {{\mathbb R}^d}, \end{aligned}$$

  then there exists $\sigma \in M^{\infty ,1}_{\widetilde{w}}({{\mathbb R}^{2d}})$ such that $T=\sigma ^w$ on $\mathcal {S}({{\mathbb R}^d})$.

As an application of Theorem 6.13, we now prove Proposition 4.3.

Proof of Proposition 4.3

With the notation of Theorem 6.13, we use (23) to estimate

$$\begin{aligned} \left| \left<T(\varphi _{(x,\xi )}),\varphi _{(x',\xi ')}\right> \right|&=\left| V_\varphi T(\varphi _{(x,\xi )})(x',\xi ') \right| \\&\le \int \limits _{{{\mathbb R}^{2d}}} \left| V_\varphi \varphi _{(x,\xi )}(z'') \right| H((x',\xi ')-z'') \mathrm{d}z''\\&= \int \limits _{{{\mathbb R}^{2d}}} \left| V_\varphi \varphi (z''-(x,\xi )) \right| H((x',\xi ')-z'') \mathrm{d}z''\\&= (H*\left| V_\varphi \varphi \right| )((x',\xi ')-(x,\xi )). \end{aligned}$$

Since $\varphi \in {M^1_w}({{\mathbb R}^d})$ and $H \in L^1_w({{\mathbb R}^{2d}})$, we deduce that $H*\left| V_\varphi \varphi \right| \in L^1_w({{\mathbb R}^{2d}})$.

Hence, Theorem 6.13 implies that there exists $\sigma \in M^{\infty ,1}_{\widetilde{w}}({{\mathbb R}^{2d}})$ such that $T \equiv \sigma ^w$ on $\mathcal {S}({{\mathbb R}^d})$. Since both operators are bounded on $L^2({{\mathbb R}^d})$, it follows that $T \equiv \sigma ^w$ on ${{L^2}({\mathbb R}^d)}$. By hypothesis, $T=\sigma ^w: {{L^2}({\mathbb R}^d)}\rightarrow {{L^2}({\mathbb R}^d)}$ is invertible. A new application of Theorem 6.13 implies that $\sigma ^w: M^p_v({{\mathbb R}^d}) \rightarrow M^p_v({{\mathbb R}^d})$ is invertible. It is tempting to conclude that then $T:M^p_v({{\mathbb R}^d}) \rightarrow M^p_v({{\mathbb R}^d})$ is invertible because it “is” $\sigma ^w$. If $p<+\infty$, that conclusion is indeed correct because both operators coincide on the dense space $\mathcal {S}$. The case $p=+\infty$ requires some care. We now discuss this in detail.

We note that $M^p_v({{\mathbb R}^d}) \hookrightarrow M^\infty _{1/w}({{\mathbb R}^d})$ and use the facts that $M^\infty _{1/w}({{\mathbb R}^d})$ can be identified with the dual-space of the (separable) Banach space $M^1_w({{\mathbb R}^d})$ and that $\mathcal {S}$ is dense in $M^\infty _{1/w}({{\mathbb R}^d})$ with respect to the weak* topology (see [33, Chap. 11]). To conclude that $T=\sigma ^w$ on $M^p_v({{\mathbb R}^d})$, we show that both operators are continuous with respect to the weak* topology of $M^\infty _{1/w}$.

Let $f \in M^p_v({{\mathbb R}^d})$, and let us show that $T(f)=\sigma ^w(f)$. Let $\left\{ f_k: k \in {\mathbb N} \right\} \subseteq \mathcal {S}({{\mathbb R}^d})$ be a sequence such that $f_k \longrightarrow f$ in the weak* topology of $M^\infty _{1/w}({{\mathbb R}^d})$. The operator $\sigma ^w:M^\infty _{1/w}({{\mathbb R}^d}) \rightarrow M^\infty _{1/w}({{\mathbb R}^d})$ is weak* continuous because it is the adjoint of the operator $\overline{\sigma }^{w}: M^1_w({{\mathbb R}^d}) \rightarrow M^1_w({{\mathbb R}^d})$. Hence $T(f_k) = \sigma ^w(f_k) \longrightarrow \sigma ^w(f)$ in the weak* topology of $M^\infty _{1/w}({{\mathbb R}^d})$. Let us note that this implies that

$$\begin{aligned} V_\varphi T(f_k) (z) \longrightarrow V_\varphi \sigma ^w(f)(z), \text{ as } k \longrightarrow +\infty , \text{ for } \text{ all } \quad z \in {{\mathbb R}^{2d}}. \end{aligned}$$

(39)

Indeed, if $z=(x,\xi ) \in {{\mathbb R}^d}\times {{\mathbb R}^d}$, the function $\varphi _{(x,\xi )} :=e^{2\pi i \xi \cdot }\varphi (\cdot -x)$ belongs to $M^1_w({{\mathbb R}^d})$ and consequently $V_\varphi T(f_k)(z) = \left<T(f_k),\varphi _{(x,\xi )}\right> \longrightarrow \left<\sigma ^w(f),\varphi _{(x,\xi )}\right>= V_\varphi \sigma ^w(f)(z)$. Similarly, since $f_k \longrightarrow f$ in the weak* topology of $M^\infty _{1/w}$, we know that $V_\varphi f_k (z) \longrightarrow V_\varphi f(z)$ for all $z \in {{\mathbb R}^{2d}}$.

Using the enveloping condition in (23), we estimate for $z \in {{\mathbb R}^{2d}}$:

$$\begin{aligned} \left| V_\varphi T(f)(z)- V_\varphi T(f_k)(z) \right|&\le \int \limits _{{{\mathbb R}^{2d}}} \left| V_\varphi (f-f_k)(z') \right| H(z-z') \mathrm{d}z'. \end{aligned}$$

The integrand in the last expression tends to $0$ pointwise as $k \longrightarrow +\infty$. In order to apply Lebesgue’s dominated convergence theorem, we show that the integrand is dominated by an integrable function. Since $H \in L^1_w({{\mathbb R}^{2d}})$, it suffices to show that $\sup _k ||V_\varphi (f-f_k)||_{L^\infty _{1/w}} < +\infty$. This is true because $||V_\varphi (f-f_k)||_{L^\infty _{1/w}}=||f-f_k||_{M^\infty _{1/w}}$ and weak*-convergent sequences are bounded. Hence, Lebesgue’s dominated convergence theorem can be applied, and we conclude that $V_\varphi T(f_k) (z) \longrightarrow V_\varphi T(f)(z)$ for all $z \in {{\mathbb R}^{2d}}$. Combining this with (39), we conclude that $V_\varphi T(f) \equiv V_\varphi \sigma ^w(f)$. Hence $T(f)= \sigma ^w(f)$, as desired. $\square$