1 Introduction

This paper is concerned with specific problems in image/signal analysis that have been treated very intensively in the last years. In many applications, the detection of directional information in images is important. Several approaches to deal with this problem have been suggested, e.g., ridgelets (Candès and Donoho 1999a), curvelets (Candès and Donoho 1999b), contourlets (Do and Vetterli 2005), and shearlets (Guo et al. 2006), just to name a few.

For applications in geosciences, in particular the shearlet approach has been quite successfully applied, and is by now already established as a powerful tool. Indeed, shearlets have with great success been used for the analysis of seismic data, in particular for channel boundary detection (Karbalaali et al. 2017). Using complex shearlets, the results could even be improved (Karbalaali et al. 2017). In the meantime, the results have also been generalized to the three-dimensional case (Karbalaali et al. 2018). Moreover, the identification of shallow geoharzard channels by means of shearlets is possible (Karbalaali et al. 2018). For reasons that will be explained a little bit later, all these investigations have been performed by cone-adapted shearlets - and this is exactly the topic we are concerned within this paper. We are convinced that the results obtained so far clearly indicate the potential of shearlet schemes for the solving problems in Earth sciences.

Among all these above mentioned methods to detect directional information, the shearlet transformation stands out since it stems from a square-integrable representation of a certain group, the full shearlet group (Dahlke et al. 2008). This is very similar to the classical wavelet setting. This fundamental property of shearlets also paves the way to the application of the very important coorbit theory that has been derived by Feichtinger and Gröchenig (1988, 1989a, 1989b, 1992). Given a square-integrable group representation that is also integrable, the coorbit theory gives rise to associated smoothness spaces, the coorbit spaces, where smoothness is measured by the decay of the so-called voice tranform. Moreover, this theory gives rise to a very natural discretization method which yields atomic decompositions and even Banach frames for the coorbit spaces.

In several studies (Dahlke et al. 2009, 2010, 2011, 2013, 2016), it has been shown that all the conditions needed for the application of the coorbit theory can be satisfied in the shearlet setting. This gives rise to new families of smoothness spaces, the shearlet coorbit spaces.

Although this is all nice and promising, a certain drawback of the shearlet transform remains. Due to the involved shearing operation, the shearlet transform has a directional bias, which means that the detection of edges in critical directions is difficult. This problem can be ameliorated by the so-called cone adapted shearlets. Then, several mother shearlets are used. They are associated with a cone-like partition of the frequency plane. In recent studies (Dahlke et al. 2011, 2013 also coorbit versions of cone-adapted shearlets have been established. These spaces are defined by requiring that the frames coefficients are only nonzero if the shearing part of the transform is not too large. Here. ‘too large’ refers to the scaling, see Sect. 3. The cone-adapted approach has the advantage that trace theorems and embeddings into Besov spaces can be established (Dahlke et al. 2011, 2013). For these purposes, the original shearlet coorbit spaces would be ‘too big’.

In recent studies, it has turned out that the detection of geometric information by means of the coorbit version of the shearlet transform can be performed very similar to the classical approach. Indeed, in Dahlke et al. (2010), Sect. 4 it has been shown that, e.g., directional information can be extracted by examining the decay of the continuous shearlet transform as the scaling parameter tends to zero, just as for the classical case. We also refer to the conclusion for additional information.

Now everybody would probably believe that these cone-adapted shearlet spaces are rich enough for a reasonable analysis. However, a rigorous proof of this fact is still missing. In this note, we establish some results in this direction. To this end, it is advantageous to work with a slightly modified definition based on the Banach frame property. Then, reasonable conditions on the nontriviality of these spaces can be derived.

This paper is organized as follows. In Sect. 2 we briefly recall the basic setting of coorbit theory and introduce the shearlet coorbit spaces. Then, in Sect. 3 we discuss the cone-adapted spaces and prove a general result concerning nontriviality. We also investigate the relations to the previous definition of cone-adapted spaces. Examples that illustrate our findings are also given.

2 Shearlet coorbit theory

In what follows, we repeat the basic notation and facts on the shearlet group and its square integrable representations including the corresponding admissible functions, the so-called mother shearlets and supply the basic concepts on shearlet coorbit spaces, see (Dahlke et al. 2011) for further details. But instead of applying the concept to shearlet with compact support, we focus here on the bandlimited case.

For \(a\in \mathbb {R}^*:= \mathbb {R} \setminus \{0\}\) and \(s \in \mathbb {R}\), let

$$ A_a:= \begin{pmatrix} a &{} 0\\ 0 &{} {\text {sgn}} \,(a) \sqrt{|a|} \end{pmatrix} \quad \textrm{and} \quad S_s:= \begin{pmatrix} 1 &{} s\\ 0 &{} 1 \end{pmatrix} $$

denote the parabolic scaling matrix and the shear matrix, respectively, where \({\text {sgn}} \,(a)\) denotes the sign of a. The (full) shearlet group \(\mathbb {S}\) is defined to be the set \(\mathbb {R}^* \times \mathbb {R}\times \mathbb {R}^2\) endowed with the group operation

$$\begin{aligned} (a,s,t) \, (a',s',t') = (a a', s+s'\sqrt{|a|}, t + S_s A_a t'). \end{aligned}$$

The left and right invariant Haar measures on \(\mathbb {S}\) are given by

$$ \mu _{\mathbb {S},l} = \frac{da}{|a|^3} \, ds \, dt \quad \textrm{and} \quad \mu _{\mathbb {S},r} = \frac{da}{|a|} \, ds \, dt,$$

respectively and the modular function of \(\mathbb {S}\) by \(\triangle (a,s,t) = 1/|a|^2\). In the following, we use the left-invariant Haar measure \(\mu _\mathbb {S}= \mu _{\mathbb {S},l}\). Let \(L_x,R_x\) denote the left and right translations by \(x \in \mathbb {S}\), i.e., \(L_xF(y):= F(x^{-1}y)\) and \(R_x F(y):= F(yx)\).

Recall that a unitary representation of a locally compact group G with the left–invariant Haar measure \(\mu _G\) on a Hilbert space \(\mathcal {H}\) is a homomorphism \(\pi \) from G into the group of unitary operators \(\mathcal {U}(\mathcal {H})\) on \(\mathcal {H}\) which is continuous with respect to the strong operator topology. For the shearlet group the mapping \(\pi : \mathbb {S}\rightarrow \mathcal {U}(L_2(\mathbb {R}^2))\) defined by

$$\begin{aligned}{} & {} \pi (a,s,t) \, \psi (x):= |a|^{-\frac{3}{4}} \, \psi (A_a^{-1}S_s^{-1}(x-t))\nonumber \\{} & {} = |a|^{-\frac{3}{4}} \, \psi \left( \frac{1}{a} \left( x_1-t_1 - s(x_2-t_2) \right) , \frac{\textrm{sgn} \, a}{\sqrt{|a|}} \, (x_2-t_2) \right) \end{aligned}$$
(1)

is a unitary representation of \(\mathbb {S}\), see (Dahlke et al. 2008, 2009). In the following, we use the abbreviation \( \psi _{a,s,t}:= \pi (a,s,t) \, \psi \). Let the Fourier transform of a function \(f\in L_1(\mathbb {R}^2)\) be defined by

$$ \mathcal {F}f(\omega ) = \hat{f}(\omega ) = \int \limits _{\mathbb {R}^2} f(x) e^{-2 \pi i \left\langle \omega ,x\right\rangle }\, dx. $$

It is well-known that this transform admits an extension to \(L_2(\mathbb {R}^2)\), see e.g. Hackbusch (2003, Theorem 6.2.21). Then straightforward computation yields

$$\begin{aligned} \hat{\psi }_{a,s,t} (\omega ) = |a|^{\frac{3}{4}} e^{-2\pi i t \omega } \hat{\psi } \left( A_a^\textrm{T}S_s^\textrm{T}\omega \right) = |a|^{\frac{3}{4}} e^{-2\pi i t \omega } \hat{\psi } \left( a\omega _1,{\text {sgn}} \,(a) \sqrt{|a|} (s\omega _1 + \omega _2) \right) . \end{aligned}$$
(2)

A nonzero function \(\psi \in L_2(\mathbb {R}^2)\) is called admissible, if

$$\begin{aligned} \int \limits _{\mathbb {S}} |\langle \pi (g)\psi , \psi \rangle |^2 d\mu _{\mathbb {S}}(g) < \infty . \end{aligned}$$

If a unitary representation \(\pi \) is irreducible and there exists at least one admissible function \(\psi \in L_2(\mathbb {R}^2)\) then \(\pi \) is called square integrable.

The following result from Dahlke et al. (2009) shows that the unitary representation \(\pi \) defined in (1) is a square-integrable representation of \(\mathbb {S}\).

Theorem 2.1

A function \(\psi \in L_2(\mathbb {R}^2)\) is admissible if and only if it fulfills the admissibility condition

$$\begin{aligned} C_\psi := \int \limits _{\mathbb {R}}\int \limits _{\mathbb {R}} \frac{|\hat{\psi }(\omega _1,\omega _2)|^2}{\omega _1^2} \, d\omega _2 \, d\omega _1 < \infty . \end{aligned}$$
(3)

Then, for any \(f \in L_2(\mathbb {R}^2)\), the following equality holds true:

$$\begin{aligned} \int \limits _\mathbb {S}|\langle \psi _{a,s,t},f \rangle |^2 \, d \mu _{\mathbb {S}} (a,s,t) = C_\psi \, \left\Vert f\right\Vert _2^2. \end{aligned}$$
(4)

In particular, the unitary representation \(\pi \) is irreducible and hence square-integrable.

A function \(\psi \in L_2(\mathbb {R}^2)\) fulfilling the admissibility condition (3) is called a continuous shearlet. The transform \(\mathcal {S}\mathcal {H}_\psi : L_2(\mathbb {R}^2) \rightarrow L_2(\mathbb {S})\) defined by

$$\begin{aligned} \mathcal {S}\mathcal {H}_\psi f(a,s,t) = \langle \psi _{a,s,t},f \rangle \end{aligned}$$
(5)

is called Continuous Shearlet Transform. The admissibility condition is important, since it implies a resolution of identity that allows the reconstruction of a function \(f \in L_2(\mathbb {R}^2)\) from its Continuous Shearlet Transform.

In order to invent Shearlet Coorbit Spaces, let w be real-valued, continuous, submultiplicative weight on \(\mathbb {S}\), i.e., \(w(gh) \le w(g)w(h)\) for all \(g,h \in \mathbb {S}\). Furthermore, we will always assume that the weight function w satisfies all the coorbit-theory conditions as stated in Gröchenig (1991, Section 2.2).

Let now m be a w-moderate weight on \(\mathbb {S}\) which means that \(m(xyz) \le w(x)m(y)w(z)\) for all \(x,y,z \in \mathbb {S}\). For \(1 \le p \le \infty \), let

$$\begin{aligned} L_{p,m} (\mathbb {S}):= \{F \; \textrm{measurable}: Fm \in L_p(\mathbb {S}) \}. \end{aligned}$$

To define our coorbit spaces we need the set

$$\begin{aligned} {\mathcal {A}}_w:= \{\psi \in L_2(\mathbb {R}^2): \mathcal {S}\mathcal {H}_\psi (\psi ) = \langle \pi (\cdot ) \psi ,\psi \rangle \in L_{1,w}(\mathbb {S})\} \end{aligned}$$

of analyzing vectors. In particular, we assume that our weight is symmetric with respect to the modular function, i.e., \(w(g) = w(g^{-1}) \triangle (g ^{-1})\). Starting with an ordinary weight function w, its symmetric version can be obtained by \(w^{\#} (g):= w(g) + w(g^{-1}) \triangle (g ^{-1})\). Moreover, it was proved in Lemma 2.4 of Feichtinger and Gröchenig (1988) that \({\mathcal {A}}_w = {\mathcal {A}}_{w^{\#}}\). To make the whole machinery work, it is important to ensure that the set \({\mathcal {A}}_w\) is nontrivial. In Dahlke et al. (2009), it has been shown that under certain quite natural conditions this is indeed the case for the shearlet case. Moreover, it was shown in Dahlke et al. (2011), that \({\mathcal {A}}_w\) contains under certain conditions on the weight w shearlets with compact support.

For an analyzing vector \(\psi \) we can consider the space

$$\begin{aligned} \mathcal {H}_{1,w} := \{f \in L_2(\mathbb {R}^2): \mathcal {S}\mathcal {H}_\psi (f) = \langle \pi (\cdot )\psi ,f\rangle \in L_{1,w}(\mathbb {S})\}, \end{aligned}$$
(6)

with norm \( \Vert f\Vert _{\mathcal {H}_{1,w}}:= \Vert \mathcal {S}\mathcal {H}_\psi f\Vert _{L_{1,w}(\mathbb {S})} \) and its anti-dual \(\mathcal {H}_{1,w}^{\sim }\), the space of all continuous conjugate-linear functionals on \(\mathcal {H}_{1,w}\). The spaces \(\mathcal {H}_{1,w}\) and \(\mathcal {H}_{1,w}^{\sim }\) are \(\pi \)-invariant Banach spaces with continuous embedding \( \mathcal {H}_{1,w}\hookrightarrow L_2(\mathbb {R}^2) \hookrightarrow \mathcal {H}_{1,w}^{\sim }. \) Then the inner product on \(L_2(\mathbb {R}^2) \times L_2(\mathbb {R}^2)\) extends to a sesquilinear form on \(\mathcal {H}_{1,w}\times \mathcal {H}_{1,w}^{\sim }\), therefore for \(\psi \in \mathcal {H}_{1,w}\) and \(f \in \mathcal {H}_{1,w}^{\sim }\) the extended representation coefficients

$$\mathcal {S}\mathcal {H}_{\psi }(f)(a,s,t):=\langle \pi (a,s,t)\psi ,f \rangle _{\mathcal {H}_{1,w}\times \mathcal {H}_{1,w}^{\sim }}$$

are well-defined.

We are interested in the following Banach spaces which are called shearlet coorbit spaces

$$\begin{aligned} \mathcal {S}\mathcal {C}_{p,m}:= \{f \in \mathcal {H}_{1,w}^{\sim } :\, \mathcal {S}\mathcal {H}_{\psi }(f) \in L_{p,m}(\mathbb {S}) \}, \quad \Vert f\Vert _{\mathcal {S}\mathcal {C}_{p,m}}:= \Vert \mathcal {S}\mathcal {H}_{\psi }f\Vert _{L_{p,m}(\mathbb {S})}. \end{aligned}$$
(7)

Note that the definition of \(\mathcal {S}\mathcal {C}_{p,m}\) is independent of the analyzing vector \(\psi \) and of the weight w in the sense that \(\tilde{w}\) with \(w(g) \le C \tilde{w}(g)\) for all \(g \in \mathbb {S}\) and \({\mathcal {A}}_{\tilde{w}} \not = \{ 0 \}\) give rise to the same space see Feichtinger (1988, Theorem 4.2).

In applications, one may start with some sub-multiplicative weight m and use the symmetric weight \(w:= m^{\#}\) for the definition of \({\mathcal {A}}_w\). Obviously, we have that m is w-moderate.

In order to discretize the shearlet coorbit spaces, we consider a (countable) family \(X = \{ (a_i,s_i,t_i): i \in {\mathcal {I}} \}\) in \(\mathbb {S}\). X is said to be U-dense if \(\cup _{i \in {\mathcal {I}}} (a_i,s_i,t_i) U= \mathbb {S}\), and separated if for some compact neighborhood Q of unit element e we have \((a_i,s_i,t_i) Q \cap (a_j,s_j,t_j) Q=\emptyset , i \not =j\), and relatively separated if X is a finite union of separated sets. Let \(\alpha > 1\) and \(\sigma ,\tau > 0\) be defined such that

$$\begin{aligned} \left[ \frac{1}{\alpha },\alpha \right) \times \left[ -\sigma ,\sigma \right) \times {\left[ -\tau /2,\tau /2 \right) } \subset U. \end{aligned}$$
(8)

Then it was shown in Dahlke et al. (2009) that the set

$$\begin{aligned} X := \{ (\epsilon \alpha ^{-j}, \sigma \alpha ^{-j/2} k, S_{\sigma \alpha ^{-j/2} k } A_{\alpha ^{-j}} \tau l) : j \in \mathbb {Z}, k \in \mathbb {Z}, l \in \mathbb {Z}^2, \, \epsilon \in \{-1,1\}\} \end{aligned}$$
(9)

is U-dense and relatively separated.

3 Non-triviality of cone-adapted shearlet coorbit spaces

It was shown in Dahlke et al. (2011, 2013) that for shearlet coorbit spaces trace theorems and embeddings into Besov spaces can be established. However, it turned out that the original shearlet coorbit spaces are ‘too big’. To this end, restrictions of \(\mathcal {S}\mathcal {C}_{p,m}\) to so-called cone-adapted shearlet coorbit spaces were considered. In Dahlke et al. (2011, 2013), the definition of the cone-adapted shearlet spaces was due to the concept of atomic decompositions. Now, in what follows, a Banach frame based setting will be the starting point for the definition of these space. The essential theorem that allows for such a framework is Theorem U (here specified for shearlet coorbit spaces) which can be found in Gröchenig (1991, page 27).

Theorem 3.1

Let all the assumptions hold true that were made in Theorem U in Gröchenig (1991, page 27). Then for any U-dense and relatively separeted set X,

$$\{\pi (g_i)\psi ,~~ g_i=(a_i,s_i,t_i), i\in {\mathcal {I}}\}$$

is both a set of atoms and a Banach frame for \(\mathcal {S}\mathcal {C}_{p,m}\). Moreover, there exists a “dual frame” \(\{e_i,~~i\in {\mathcal {I}}\}\) in \(\mathcal {H}_{1,w}\) s.t.

  1. (i)

    for \(f\in \mathcal {S}\mathcal {C}_{p,m}\),

    $$\begin{aligned} \Vert f\Vert _{\mathcal {S}\mathcal {C}_{p,m}} \simeq \Vert \{\langle e_i,f\rangle \}_{i\in \mathcal {I}}\Vert _{\ell _{p,m}}\simeq \Vert \{\langle \pi (g_i)\psi ,f\rangle \}_{i\in \mathcal {I}}\Vert _{\ell _{p,m}}~, \end{aligned}$$
    (10)
  2. (ii)

    for \(f\in \mathcal {S}\mathcal {C}_{p,m}\),

    $$\begin{aligned} f = \sum _{i\in \mathcal {I}} \langle e_i,f\rangle \pi (g_i) \psi \end{aligned}$$
    (11)

    with norm convergence in \(\mathcal {S}\mathcal {C}_{p,m}\),

  3. (iii)

    for \(f\in \mathcal {S}\mathcal {C}_{p,m}\), the decomposition

    $$\begin{aligned} f = \sum _{i\in \mathcal {I}} \langle \pi (g_i) \psi ,f \rangle e_i \end{aligned}$$
    (12)

    is also valid.

Let \(c(j_i,k_i,l_i)=\langle e_i,f\rangle \) be the canonical coefficients (thanks to Theorem 3.1), and \(\alpha \) an (arbitrary) positive parameter. We now define cone-adapted shearlet coorbit spaces as closed subspaces of \(\mathcal {S}\mathcal {C}_{p,m}\) given by

$$\begin{aligned} \mathcal {SCC}_{p,m} = \{f\in \mathcal {S}\mathcal {C}_{p,m},~ f=\sum _{i\in {\mathcal {I}}} c(j_i,k_i,l_i)\psi _{j_i,k_i,l_i}~\text {with}~|k_i|\le \alpha ^{j_i/2}\}~. \end{aligned}$$

However, a rigorous proof that these subspaces are indeed non-trivial is still missing. Therefore, we now establish some results in this direction. In a first step, we modify the definition of the cone spaces a little bit to obtain ‘more suitable’ coorbit spaces. Property (12) enables us to define

$$\begin{aligned} \mathcal {NSCC}_{p,m}:= \{f\in \mathcal {S}\mathcal {C}_{p,m},~ f=\sum _{i\in {\mathcal {I}}} \langle \psi _{j_i,k_i,l_i},f\rangle e_{j_i,k_i,l_i}~\text {with}~|k_i|\le \alpha ^{j_i/2}\}~. \end{aligned}$$
(13)

In the publications (Dahlke et al. 2011, 2013), where we were concerned with embeddings and trace theorems, it has been conveninent to work with compactly supported shearlets \(\psi \). However, for our purposes, it is more suitable to work with band-limited shearlets.

Theorem 3.2

Suppose now that \(\hat{\psi }\) has compact support. Define

$$\begin{aligned} \Omega _1 := \bigcup _{j,l,|k|\le \alpha ^{j/2}}{\text {supp}}\,\hat{\psi }_{j,k,l}\text { and } \Omega _2 := \bigcup _{j,l,|k|> \alpha ^{j/2}}{\text {supp}}\,\hat{\psi }_{j,k,l}~. \end{aligned}$$

Suppose \(\Omega _1\setminus \Omega _2\) is nontrivial in the sens that \(|\Omega _1\setminus \Omega _2|>0\). Then, \(\mathcal {NSCC}_{p,m}\not =\{0\}\).

Proof

Pick \(f\in L_2\), \(f\not =0\) such that \({\text {supp}}\,\hat{f}\subset \Omega _1\setminus \Omega _2\) and \(f\in \mathcal {S}\mathcal {C}_{p,m}\). A suitable Schwartz function whose Fourier transform is a \(C^\infty \)-function compactly supported in \(\Omega _1\setminus \Omega _2\) would do the job (it is known that the spaces of Schwartz functions is dense in \(\mathcal {S}\mathcal {C}_{p,m}\), see Dahlke et al. 2011).

Then, for \(|k|>\alpha ^{j/2}\), all coefficients \(\langle \hat{\psi }_{j,k,l},\hat{f}\rangle = 0\). Then there must exist at least one index (j, k, l) with \(|k|\le \alpha ^{j/2}\) with \(\langle \hat{\psi }_{j,k,l},\hat{f}\rangle \not =0\). If not, \(\langle \hat{\psi }_{j,k,l},\hat{f} \rangle = 0\) must hold for all indices (j, k, l) and hence, \(\Vert \{\langle \hat{\psi }_{j,k,l},\hat{f}\rangle \}_{(j,k,l)\in \mathcal {I}}\Vert =0\). Thus, by norm equivalence (10) and Plancherel’s theorem,

$$\Vert \{\langle \hat{\psi }_{j,k,l},\hat{f}\rangle \}_{(j,k,l)\in \mathcal {I}}\Vert _{\ell _{p,m}}= \Vert \{\langle \psi _{j,k,l},f\rangle \}_{(j,k,l)\in \mathcal {I}}\Vert _{\ell _{p,m}}\simeq \Vert f\Vert _{\mathcal {S}\mathcal {H}_{p,m}}=0$$

which is a contradiction. \(\square \)

Example 3.3

In what follows we illustrate/visualize the statement of Theorem 3.2. To this end, we use the classical shearlet definition, see Kutyniok (2021Definition 5 in Chapter 1), whose Fourier transform is given by

$$\begin{aligned} \hat{\psi }(\omega _1,\omega _2) =\hat{\psi }_1(\omega _1)\hat{\psi }_2(\omega _2/\omega _1) \end{aligned}$$

leading to

$$\begin{aligned} \hat{\psi }_{a,s,t}(\omega _1,\omega _2)=|a|^{3/4}e^{-2\pi i t \omega }\hat{\psi }_1(a\omega _1)\hat{\psi }_2(\textrm{sgn}(a)\sqrt{|a|}(s\omega _1+\omega _2)/(a\omega _1))~. \end{aligned}$$

We assume \(\psi _1\) to be a wavelet with \({\text {supp}}(\hat{\psi }_1)\subseteq [c_1,c_2]\), where \(0<c_1<c_2<\infty \), and \(\hat{\psi }_2\) a \(C^\infty _0\) function with \({\text {supp}}(\hat{\psi }_2)\subseteq [-d,d]\), where \(0<d<\infty \). These assumptions clearly imply that the support of \(\hat{\psi }_{a,s,t}\) is given by the set

$$\begin{aligned} c_1/a\le \omega _1\le c_2/a~~\text {and}~~-\omega _1(d\sqrt{|a|}+s)\le \omega _2\le \omega _1(d\sqrt{|a|}-s)~. \end{aligned}$$

Based on the U-dense and relatively separated set X in (9) we set

$$\begin{aligned} a_i = \epsilon \alpha ^{-j_i},~s_i=\sigma \alpha ^{-j_i/2}k_i~~\text {with}~~j_i\in \mathbb {Z},~k_i\in \mathbb {Z}~, \end{aligned}$$

where we have chosen in our particular case \(\alpha =2\) and \(\sigma =2.5\). We restrict the visualisation to the case \(\epsilon =1\) and specialize the support of \(\hat{\psi }\) to \(c_1=1\), \(c_2=2.2\) and \(d=4\). Fig. 1 clearly displays that these choices for \(\alpha \) and \(\sigma \) indeed lead to a U-dense and relatively seperated set. In Fig. 2, \({\text {supp}}(\hat{\pi }(g_i)\hat{\psi })\) for various indices \(i\in \mathcal {I}\) on the \((\omega _1,\omega _2)\)-plane is illustrated. The green framed patches filled with light gray represent \(\Omega _2\). The red framed patches represent \(\Omega _1\). The red framed patches filled with light gray indicate the overlap area and the red framed patches filled with dark gray represent \(\Omega _1\setminus \Omega _2\).

Fig. 1
figure 1

Covering of \(\mathbb {S}\) (restricted to \(a=\alpha ^{-j}\) and \(s=\sigma \alpha ^{-j/2}k\) with \(j=-3,\ldots ,3\))

Full size image

It is clearly visible that \(\Omega _1\setminus \Omega _2\) is nonempty.

Fig. 2
figure 2

Visualization of \({\text {supp}}(\hat{\pi }(g_i)\hat{\psi })\) on the \((\omega _{1},\omega _{2})\)-plane. The light gray framed patches represent \(\Omega _2\) and the black framed patches represent \(\Omega _1\). The black framed patches filled with light gray indicate the overlap area and the black framed patches filled with dark gray represent \(\Omega _1\setminus \Omega _2\)

Full size image

Combining expansions (11) and (12) we obtain

$$\begin{aligned} f= & {} \sum _{i\in \mathcal {I}}\langle \pi (g_i) \psi , f \rangle e_i = \sum _{i\in \mathcal {I}} \langle \pi (g_i)\psi ,\sum _{l\in \mathcal {I}} \langle e_l,f \rangle \pi (g_l) \psi \rangle e_i \nonumber \\= & {} \sum _{i\in \mathcal {I}}\underbrace{\left( \sum _{l\in \mathcal {I}}\underbrace{\overline{\langle e_l,f \rangle }}_{b_l} \underbrace{\langle \pi (g_i)\psi , \pi (g_l) \psi \rangle }_{A_{il}}\right) }_{a_i} e_i = \sum _{i\in \mathcal {I}}a_i e_i = \sum _{i\in \mathcal {I}}(Ab)_i e_i~~. \end{aligned}$$
(14)

On the other hand we have,

$$\begin{aligned} f= & {} \sum _{i\in \mathcal {I}} \langle e_i,f \rangle \pi (g_i) \psi = \sum _{i\in \mathcal {I}} \langle e_i,\sum _{l\in \mathcal {I}} \langle \pi (g_l)\psi ,f \rangle e_l \rangle \pi (g_i)\psi \nonumber \\= & {} \sum _{i\in \mathcal {I}} \underbrace{\left( \sum _{l\in \mathcal {I}}\underbrace{\overline{\langle \pi (g_l)\psi ,f \rangle }}_{d_l}\underbrace{\langle e_i,e_l \rangle }_{B_{il}} \right) }_{c_i} \pi (g_i)\psi = \sum _{i\in \mathcal {I}} c_i \pi (g_i)\psi = \sum _{i\in \mathcal {I}} (Bd)_i \pi (g_i)\psi ~~.\nonumber \\ \end{aligned}$$
(15)

Then, the following three theorems are direct consequences of the fundamental relations (14) and (15), we therefore omit the details.

Theorem 3.4

Let \(f\in \mathcal {S}\mathcal {C}_{p,m}\), hence it can be represented as in (14), where \(\{\pi (g_i)\psi \}_{i\in \mathcal {I}}\) is the Banach frame for \(\mathcal {S}\mathcal {C}_{p,m}\) and \(\{e_i\}_{i\in \mathcal {I}}\) the dual frame respectively as established in Theorem 3.1. Let the pre-frame operator be defined by

$$\begin{aligned} \mathcal {F}_1:\,\mathcal {S}\mathcal {C}_{p,m}\rightarrow & {} \ell _{p,m} \\ f\mapsto & {} \{\langle e_i, f \rangle \}_{i\in \mathcal {I}} =\{b_i\}_{i\in \mathcal {I}} ~~~~. \end{aligned}$$

Let \(i=(j,k,l)\) and \(i'=(j',k',l')\) and \(A=(\langle \pi (g_i)\psi ,\pi (g_{i'})\psi \rangle )_{i,i'\in \mathcal {I}}\) be partitioned as follows:

$$\begin{aligned} A = \left( \begin{array}{c|c} A^{11} &{} A^{12} \\[1mm] \hline \\ [-2mm] A^{21} &{} A^{22} \end{array}\right) , \end{aligned}$$

where \((A^{11})_{ii'}\) is the submatrix with \(|k|\le \alpha ^{j/2}\) and \(|k'|\le \alpha ^{j/2}\), \((A^{12})_{ii'}\) with \(|k|\le \alpha ^{j/2}\) and \(|k'|> \alpha ^{j/2}\), \((A^{21})_{ii'}\) with \(|k|> \alpha ^{j/2}\) and \(|k'|\le \alpha ^{j/2}\), and \((A^{22})_{ii'}\) with \(|k|> \alpha ^{j/2}\) and \(|k'|> \alpha ^{j/2}\). If now there exists a sequence \(\{b_i\}_{i\in \mathcal {I}}\in {\text {Range}} \,(\mathcal {F}_1)\) with \(\{b_i\}_{i\in \mathcal {I}}\not \in {\text {Ker}} \,(A^{11}\,|\, A^{12})\) and \(\{b_i\}_{i\in \mathcal {I}}\in {\text {Ker}} \,(A^{21}\,|\, A^{22})\), then \(\mathcal {NSCC}_{p,m}\not =\{0\}\).

Theorem 3.5

Let \(f\in \mathcal {S}\mathcal {C}_{p,m}\), hence it can be represented as in (15), where \(\{\pi (g_i)\psi \}_{i\in \mathcal {I}}\) is the Banach frame for \(\mathcal {S}\mathcal {C}_{p,m}\) and \(\{e_i\}_{i\in \mathcal {I}}\) the dual frame respectively as established in Theorem 3.1. Let the pre-frame operator be defined by

$$\begin{aligned} \mathcal {F}_2:\,\mathcal {S}\mathcal {C}_{p,m}\rightarrow & {} \ell _{p,m} \\ f\mapsto & {} \{\langle \pi (g_i)\psi ,f\rangle \}_{i\in \mathcal {I}}=\{d_i\}_{i\in \mathcal {I}}. \end{aligned}$$

Let \(B=(\langle e_i,e_{i'}\rangle )_{i,i'\in \mathcal {I}}\) be partitioned as A in Theorem 3.4. If now there exists a sequence \(\{d_i\}_{i\in \mathcal {I}}\in {\text {Range}} \,(\mathcal {F}_2)\) with \(\{d_i\}_{i\in \mathcal {I}}\not \in {\text {Ker}} \,(B^{11}\,|\, B^{12})\) and \(\{d_i\}_{i\in \mathcal {I}}\in {\text {Ker}} \,(B^{21}\,|\, B^{22})\), then \(\mathcal {SCC}_{p,m}\not =\{0\}\).

The next theorem illuminates a relation between the two different cone-adapted spaces.

Theorem 3.6

Suppose that the space \(\mathcal {NSCC}_{p,m}\not =\{0\}\). Let the matrix B be partitioned as A in Theorem 3.4. If there exists for \(B^{21}\) a partition

$$\begin{aligned} B^{21} = \left( \begin{array}{c} B^{21}_1\\[1mm] \hline \\ [-2mm] B^{21}_2 \end{array}\right) , \end{aligned}$$

and an element \(0\not =f\in \mathcal {NSCC}_{p,m}\), \({\mathcal {F}}_2(f) = \{d_i\}_{i\in \mathcal {I}}\) such that \(\{d_i\}_{i\in \mathcal {I},|k|\le \alpha ^{j/2}}\not \in {\text {Ker}} \,(B^{11})\) and \(\{d_i\}_{i\in \mathcal {I},|k|\le \alpha ^{j/2}}\in {\text {Ker}} \,(B^{21}_2)\), then \(\mathcal {SCC}_{p,m}\not =\{0\}\), possibly with respect to another \(\alpha \) for the definition of \({\mathcal {SCC}}_{p,m}\).

4 Conclusion

In this paper, we have stated and discussed the conditions that imply the non-tiviality of the cone-adapted shearlet spaces generated by the coorbit approach. The condition turned out to be quite natural. Cone-adapted shearlet spaces are very important since, e.g., they allow for the derivation of trace theorems and of embedding theorems into Besov spaces, respectively. Similar to the classical shearlet setting, they can also be used to detect geometric information, just by examining the decay of the shearlet transform as the scaling parameter a tends to zero. However, there is no free lunch. The strong analytic properties of the coorbit approach allows for the derivation of beautiful and important theorems, but as a drawback, the cone-adapted shearlet spaces are generated by just one mother shearlet which might cause trouble for the detection of specific coordinate directions. In this case, of course an ad-hoc-construction by just adding a second, rotated shearlet would do the job. However, if this ad-hoc-construction has an interpretation in the realm of coorbit space theory is still an open problem