You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We show that the homogeneous approximation property and the comparison theorem hold for arbitrary coherent frames. This observation answers some questions about the density of frames that are not covered by the theory of Balan, Casazza, Heil, and Landau. The proofs are a variation of the method developed by Ramanathan and Steger.'
address: |
    Faculty of Mathematics\
    University of Vienna\
    Nordbergstrasse 15\
    A-1090 Vienna, Austria
author:
- Karlheinz Gröchenig
title: The Homogeneous Approximation Property and the Comparison Theorem for Coherent Frames
---

[^1]

Frames provide redundant non-orthogonal expansions in Hilbert space, intuitively they should therefore be “denser” than orthonormal bases. The first fundamental result is Landau’s necessary condition for sets of sampling of [band-limited]{}  functions [@landau67]. A second fundamental idea is the homogeneous approximation property and the comparison theorem of Ramanathan and Steger [@RS95] in the context of Gabor frames. Since then many contributions have investigated the density of frames and varied and applied the method of Ramanathan and Steger. See [@CDH99; @FNP07; @GR96; @sch98] for a sample of papers. The approach in [@RS95] culminates in the deep investigation of Balan, Casazza, Heil, and Landau [@BCHL06; @BCHL06a] who have found a density theory for a general class of frames that are labeled by a discrete Abelian group.

However, the BCHL theory is not universally applicable, for instance, the density of wavelet frames does not fall under this general theory, and a separate investigation is necessary, as was shown in a sequence of papers by Heil and Kutyniok [@HK03; @HK07] and by Sun [@Sun07; @SunZ04; @Sun07b]. Their work uses quite explicit and long calculations in the group of affine transformations.

In this note we show that the approach of Ramanathan and Steger yields the homogeneous approximation property and a comparison theorem for the class of coherent frames. These frames arise as subsets of the orbit of a square-integrable group representation. In particular, wavelet frames possess this structure. The advantage of the abstract point of view is the conciseness and simplicity of the proofs and the much more general range of validity. The additional insight is that the homogeneous approximation property and the comparison theorem are a consequence of structure alone, we do not need to assume additional localization properties as in [@BCHL06a].

We will work in the context of a locally compact group $G$ with Haar measure $dx$. We write $|U| = \int _G \chi _U(x) \, dx $ for the measure of a set $U\subseteq G$ and $K^c$ for the complement of $K$ in $G$.

*Amalgam spaces.* Let $U=U\inv \subseteq G $ be a symmetric compact neighborhood of the identity element $e\in G $ and let $$f^\sharp (x) = \sup _{y\in xU} |f(y)|$$ be the local maximum function of $f$. Then the left amalgam space $W(L^\infty , L^2)$ is defined by the norm $$\label{eq:1}
\|f \|_{W(L^\infty , L^2)} =   \|f^\sharp \|_2 \, .$$

Recall that a set $\cX \subseteq G $ is called relatively (left) separated if $$\label{eq:3}
  \sup _{x\in G} \mathrm{card}\, (\cX \cap xU) <\infty \, .$$ for some, hence all, compact neighborhood(s) $U$ of $e$. Equivalently, for any compact set $K\subseteq G$ the sum $\sum _{x\in \cX } \chi _{xK} $ is uniformly bounded in $G$.

\[lem:1\] Let $\cX = \{x_j\}$ be a relatively (left) separated set in $G$ and $K\subseteq G $ a compact set. Then $$\label{eq:2}
  \sum _{x_j \not \in K} |f(x_j)|^2 \leq C \, \int _{K^c U } |{f^\sharp}(x)
  |^2 \, dx \, .$$

Set $C_0 = \|\sum _{j\in J} \chi _{x_j U } \|_\infty  $. By this constant is finite. First we note that $|f(x_j)| \leq {f^\sharp}(x)$ whenever $ x_j \in xU$ or $ x\in x_jU$ (because $U$ is symmetric). Consequently $$|f(x_j)|^2 \leq \frac{1}{|U|}\int _{x_j U} {f^\sharp}(x)^2 \, dx \, .$$ So after summing over $j$, we obtain $$\begin{aligned}
  \sum _{x_j \not \in K} |f(x_j)|^2 &\leq &  \frac{1}{|U|} \sum _{x_j
    \not \in K} \int _{x_j U} {f^\sharp}(x)^2 \, dx \\
&\leq & \frac{1}{|U|} C_0 \, \int _{K^c U} {f^\sharp}(x) ^2 \, dx \, ,\end{aligned}$$ because $\sum _{x_j \not \in K } \chi _{x_j U } (x) \leq  C_0 \chi
_{K^cU} (x)$ for all $x\in G$.

Next we consider the setup for coorbit theory and coherent frames. Let $(\pi , \cH )$ be a unitary, *square-integrable* [representation]{} of $G$ on a Hilbert space $\cH $. For fixed $g\in \cH , g\neq 0$, we set $V_gf =  \langle f, \pi (x) g\rangle$; this representation coefficient is the analog of the wavelet transform or [short-time Fourier transform]{} for the group $G$.

We investigate frames of the form $\{\pi (x_j) g : j\in J\}$, so-called coherent frames. This means that there exist positive constants $A,B >0$, the frame bounds, such that $$\label{eq:4a}
A\|f\|^2 \leq \sum _{j} |\langle f, \pi (x_j)g\rangle |^2 = \sum _{j}
|V_gf(x_j) |^2 \leq B\|f\|^2$$ for all $f\in \cH $. From it is clear that the sampling of the representation coefficients with respect to $g$ on the set $\cX $ must be in $\ell ^2$. This is close, but not equivalent, to saying the $V_gf \in W(L^\infty , L^2)$. Thus the frame condition motivates the following definition.

A vector (wavelet) $g\in \cH $ is “nice”, if the mapping $V_g$ (the generalized wavelet transform) maps $\cH $ into $W(L^\infty
  ,L^2)$.

In most situations the set of nice vectors is dense in $\cH $ and examples can be easily constructed (see Remark 1). In the following we assume that $\{\pi (x_j) g : j\in J\}$ is a frame for $\cH $ and that $\{ h_j : j\in J\}$ is a dual frame. We may take any dual frame, it need not be the canonical dual. We remark that $\{ x_j\}$ must be relatively separated in this case (otherwise the upper frame bound would not exist).

The intuition behind coherent frames is that the frame vector $\pi
(x_j)g$ lives near the point $x_j$ in the “$G$-plane”. Consequently, a vector of the form $\sum _{x_j \in K} c_j \pi (x_j)g$ should live on $K$.

To formalize this intuition, we introduce the following subspaces of $\cH $. Let $L\subseteq G$ be a compact subset of $G$ and $y\in
G$, then $V_{L,y}$ is the subspace of $\cH $ spanned by the vectors of the dual frame $h_j , x_j \in yL$. Formally, $$\label{eq:4}
  V(yL) = \mathrm{span}\, \{ h_j : x_j \in yL\} \, ,$$ and we let $P_{L,y}$ be the orthogonal projection from $\cH $ onto $V(yL)$.

The following homogeneous approximation property is a weak form of localization of a frame. The proof for coherent frames follows closely the proof for bandlimited functions in [@GR96].

\[hap\] Assume that $g$ is nice and that $\{\pi (x_j) g : j\in J\}$ is a frame for $\cH $ and that $\{ h_j : j\in J\}$ is a dual frame. Then for every $f\in \cH $ and $\epsilon >0$ there exists a compact set $L\subseteq G$ (depending on $f$ and $\epsilon $) such that $$\label{eq:6}
   \sup _{x\in yK} \| \pi (x)
  f - P_{KL,y} \pi (x) f \| < \epsilon \,$$ holds for *all* compact sets $K\subseteq G$ and all $y\in G$.

Since $g$ is nice, the representations coefficient $V_gf $ is in $W(L^\infty , L^2)$ for all $f\in L^2$. Therefore for any $\delta >0$ there is a compact set $L\subseteq G$, such that on $L^cU$, an open neighborhood of the complement of $L$, the maximal function $(V_gf)^\sharp $ is small, precisely, $$\label{eq:42}
  \int _{L^cU} (V_gf)^\sharp (x) \, dx < \delta \, .$$

We will use the frame expansion

$$\label{eq:43}
  \pi (x) f = \sum _j \langle \pi (x) f, \pi (x_j ) g \rangle h_j \, .$$

Recall the following property of orthogonal projections: If $P_{KL,y} h = \sum
_{j: x_j \in yKL} c_j h_j$, then $$\label{eq:7}
  \|h-P_{KL,y}y\| \leq \|h-\sum _{j: x_j \in yKL} d_j h_j \| \, ,$$ for any choice of coefficients $d_j$. Choosing the special coefficients of the frame expansion , we obtain that $$\begin{aligned}
\|\pi (x) f-P_{KL,y}\pi (x)f\|^2 &\leq & \|\pi (x) f -  \sum _{j: x_j
  \in yKL} \langle \pi (x)f, \pi (x_j )g\rangle    h_j \|^2 \\
&= &  \|\sum _{j: x_j
  \not \in yKL} \langle \pi (x)f, \pi (x_j) g\rangle    h_j \|^2 \\
&\leq & A\inv  \sum _{j: x_j   \not \in yKL} |\langle \pi (x)f, \pi (x_j)
g\rangle |^2 \\
&=& A\inv  \sum _{j: x_j   \not \in yKL} |\langle f, \pi (x \inv x_j) 
g\rangle |^2 \, .\end{aligned}$$ The last inequality follows from the fact that $\{ h_j\} $ is a frame and obeys the Bessel inequality with constant $A\inv $, where $A $ is the lower frame bound of $\{\pi (x_j)g\}$ in .

Some gymnastics: Since $x_j    \not \in yKL$, we have $x\inv x_j 
\not \in x\inv  yKL $. Further, $\{x\inv x_j \} $ is relatively separated, and therefore $$\|\sum _j \chi _{x\inv x_j U} \|_\infty = C_0 < \infty \, ,$$ where this constant is independent of $x$.

We apply Lemma \[lem:1\] and obtain $$\begin{aligned}
  \sum _{j: x\inv x_j   \not \in x\inv yKL} |\langle f, \pi (x \inv x_j)
g\rangle |^2 &=&   \sum _{j: x\inv x_j   \not \in x\inv yKL} |V_gf(x \inv x_j
) |^2 \\
 &\leq & C \int _{(x\inv y KL)^c U} \, |V_gf ^\sharp (x) |^2 \, dx  \end{aligned}$$ So for $K\subseteq G $ compact and $x\in yK$ we obtain $$\label{eq:cc2}
    \| \pi (x) f - P_{KL,y} \pi (x)
  f\|^2 \leq   A\inv C \sup _{x\inv y =z \in K\inv } \int
  _{(zKL)^c U} V_gf ^\sharp (x)^2 \, dx \, .$$ If $K\subseteq G $ is compact and $x\in yK$, then $z=x\inv y \in
K\inv$, therefore $L\subseteq zKL$ for all $z\in K\inv $ and $(zKL)^c U
\subseteq L^cU$. Therefore $$\sup _{z\in K\inv} \int _{(zKL)^cU}
V_gf ^\sharp (u)^2 du  \leq \int _{L^cU} (V_g f)^\sharp (u)^2 \, du  \, .$$ If we now choose $L$ such that $CA\inv \int _{L^cU} (V_g f)^\sharp (u)^2 \,
du < \epsilon $, which is possible by , then we have proved that $$\sup _{x\in yK} \| \pi (x)
  f - P_{KL,y} \pi (x) f \| < \epsilon \, ,$$ and the homogeneous approximation property is proved.

[*REMARK:*]{} In most situations, nice vectors $g$ exist in abundance. For instance, if $(\pi , \cH)$ is *irreducible* and square-integrable, then a nice vector can be constructed as follows: Let $g_0 \in \cH $ be admissible, and $k$ be continuous with compact support. Then $g=\pi (k)g_0= \int k(x) \pi (x)g_0 \, dx$ has the property that $V_gf ^\sharp \in W(C, L^2)$ for all $f\in \cH
$. This follows from a convolution equation in [@fg89jfa]. See also [@FG07].

If $\pi $ is the reducible representation by translations and dilations on $\lrd $ defined by $\pi
(x,s)f(t) = |\det e^{-sA/2}| f(e^{-sA} (t-x)$, where $x,t\in \rd,
s\in {{\mathbb{R}}}^+$ and $A $ is a real-valued $d\times d$-matrix with all eigenvalues having positive real part, then every Schwartz function $g$ such that $\operatorname*{supp}\hat{g} \subseteq \{ \omega : 0 < a \leq |\omega
| \leq b\}$ is nice. This follows immediately from the fact that the wavelet transform $V_gg(x,s) = \langle g, \pi (x,s)g\rangle $ has compact support in $s$. See [@GKT92] for more on admissible vectors in this case.

**The Comparison Lemma.** Before we estimate the dimension of the finite-dimensional subspaces defined in , we state a simple estimate for the trace of an operator.

\[trace\] Let $T$ be a positive trace-class operator on a Hilbert space $\cH $ and $\{h_k: k\in J\}$ be a frame with frame bounds $A, B>0$. Then $$\frac{1}{B}\, \sum _{k} \langle T h_k , h_k\rangle \leq  \mathrm{tr}\, T
\leq \frac{1}{A} \sum _{k} \langle T h_k , h_k\rangle \, .$$

By the spectral theorem there exists an orthonormal set ${\varphi}_j $ and positive eigenvalues $\lambda _j$ such that $Tf = \sum _j \lambda _j
\langle f, {\varphi}_j \rangle {\varphi}_j$. Then $$\begin{aligned}
  \sum _{k} \langle T h_k , h_k\rangle &=& \sum _j \sum _k \lambda _j
  |\langle {\varphi}_j , h_k \rangle |^2 \\
&\leq& B \sum _j \lambda _j \|{\varphi}_j \|_2 \\
&=& B \sum _j \lambda _j = B \, \mathrm{tr}\, T \, .\end{aligned}$$ Likewise, by using the lower frame bound, we obtain $\sum _{k} \langle
T h_k , h_k\rangle \geq A \, \mathrm{tr}\, T$.

Let $\cE _g= \{\pi (x_j ) g : x_j \in \cX \}$ and $\cE _h= \{\pi (y_k
) h : y_k \in {\mathcal{Y}}\}$ be two coherent frames. As above, denote the dual frame of $\cE_g$ by $\{ h_j \}$ and its frame bounds by $A,B$.

For compact subsets $K,L \subseteq G $ and $y\in G $, define the subspaces $$\begin{aligned}
V_g(yKL) &=& \mathrm{span}\, \{ h_j: x_j \in yKL\} \\  
W_h(yK) &=& \mathrm{span}\, \{ \pi (y_j) h : y_k \in yK\} \, ,  \end{aligned}$$ and let $P=P_g(yKL)$ and $Q=Q_h(yK)$ be the orthogonal projections onto the subspaces $V_g$ and $W_h$. Finally let $T=QPQ: W_h \to W_h $. $T$ is a positive operator of finite rank. We estimate its trace from above and below.

To keep track of these subspaces and projections, keep in mind that $\cE _g$ is a *given frame* whose density we want to understand and that $\cE _h $ is the *reference frame*. If possible, $\cE _h$ is chosen to be a Riesz basis or an orthonormal basis. This has been the case in most applications of the comparison theorem so far.

\[comp\] Given $\epsilon >0$, there exists a compact set $L\subseteq G$, such that for all $y\in G$ and all compact sets $K
\subseteq G$ $$\label{eq:68}
\|h\|^2  B\inv (1-\epsilon)   {\mathrm{card}\,}\{k:  y_k \in yK\} \leq {\mathrm{card}\,}\{x_j \in yKL\} \, .$$

Since $T$ is a product of orthogonal projections, its eigenvalues are between $0$ and $1$ and $$\label{eq:65}
    \mathrm{tr}\, T \leq \mathrm{rank}\, P \leq {\mathrm{card}\,}\{x_j \in yKL\}\, .$$ To obtain a lower bound, we use Lemma \[trace\] and obtain $$\begin{aligned}
\mathrm{tr}\, T &\geq & \frac{1}{B} \sum _{k} \langle T \pi
(y_k)h, \pi (y_k)h \rangle  \\
&\geq & \frac{1}{B}  \sum _{k: y_k \in yK} \langle T \pi (y_k)h, \pi
(y_k)h \rangle\\
 &=&   \sum
    _{k: y_k \in yK} \langle QPQ \pi (y_k)h, \pi (y_k)h \rangle \\
&=& \frac{1}{B}  \sum _{k: y_k \in yK} \langle P \pi (y_k)h, \pi (y_k)h \rangle \\
&=& \frac{1}{B}  \sum _{k: y_k \in yK} \Big(\langle  \pi (y_k)h, \pi (y_k)h \rangle +
  \langle (\mathrm{I}-P) \pi (y_k)h, \pi (y_k)h \rangle \\
&=&  \frac{\|h\|^2}{B} {\mathrm{card}\,}\{k: y_k \in yK\}  + (*)  \, .\end{aligned}$$ To estimate $(*)$, we apply the homogeneous approximation property (Proposition \[hap\]). Choose a compact set $L\subseteq G $ such that $$\|\pi (x)h - P_{KL,y}\pi (x)h\| < \epsilon \|h\|$$ for all compact sets $K\subseteq G$ and $x\in yK$. Then $$\begin{aligned}
(*) &\leq & \frac{1}{B}\sum _{k: y_k \in yK} \|\pi (y_k)h - P (\pi
(y_k)h) \| \|\pi (y_k)h\|  \\
&\leq & \frac{1}{B}\sum _{k: y_k \in yK} \epsilon \|h\| \, \|\pi (y_k)h \|
=  \frac{1}{B}{\mathrm{card}\,}\{k: y_k \in yK\}   \epsilon \|h\|^2 \, .\end{aligned}$$ By combining these estimates we obtain a lower bound $$\label{eq:67}
  \sum _{k: y_k \in yK} \langle T \pi (y_k)h, \pi (y_k)h \rangle  \geq  {\mathrm{card}\,}\{k:
  y_k \in yK\} \|h\|^2  B\inv (1-\epsilon) \, .$$ Combining and , the claim is proved.

The comparison theorem is the starting point for genuine density theorems. For instance, Landau’s necessary conditions for set of sampling are a simple corollary of Theorem \[comp\]. Likewise, Ramanathan and Steger derived the necessary density of nonuniform Gabor frames as an immediate corollary of a special case of Theorem \[comp\]. For more general groups and representations, one may define some form of a Beurling density and deduce a density theorem, but the results are not completely satisfying.

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[^1]: K. G. was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154