Wavelets and Linear Algebra

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1 Wavelets and Linear Algebra 5(1) (18) 11-6 Wavelets and Linear Algebra Vali-e-Asr University of afsanjan A Necessary Condition for a Shearlet System to be a Frame via Admissibility M. Amin khah a, A. Askari Hemmat b,. aisi Tousi c, a Department of Applied Mathematics, Faculty of Sciences and new Technologies, Graduate University of Advanced Technology, Kerman, Islamic epublic of Iran. b Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar Uninersity of Kerman, Kerman, Islamic epublic of Iran. c Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box , Mashhad, Islamic epublic of Iran. Article Info Article history: eceived 7 February 17 Accepted 11 June 17 Available online 4 June 17 Communicated by Farshid Abdollahi Abstract Necessary conditions for shearlet and cone-adapted shearlet systems to be frames are presented with respect to the admissibility condition of generators. c (18) Wavelets and Linear Algebra Keywords: Shearlet system, Cone-adapted shearlet system, Shearlet frame, Admissibility condition. MSC: 4C15, 4C4. Corresponding author addresses: m.aminkhah@student.kgut.ac.ir (M. Amin khah), askari@uk.ac.ir (A. Askari Hemmat), raisi@um.ac.ir (. aisi Tousi) c (18) Wavelets and Linear Algebra

2 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) Introduction and Preliminaries Shearlets were introduced by Guo, Kutyniok, Labate, Lim and Weiss in [8, 14] and developed by some others in e.g. [1, 13] as the first directional representation system which allows a unified treatment of the continuum and digital world similar to wavelets. Shealets were derived within a larger class of affine-like systems, composite wavelets, using shearing to control directional selectivity. In contrast to other x-lets which mostly utilize the geometry of the data, shearlet systems form an affine system, generated by dilations and translations of a generator, where the dilation matrix is the product of a parabolic scaling matrix and a shear matrix. This makes the shearlet approach more remunerative for obtaining the anisotropic and directional features of multidimensional data [13]. This property provides additional simplicity of construction and a connection with the theory of square integrable group representations of the affine group [1,, 4, 5, 1]. Of particular importance for the shearlet transform is the situations under which any vector in L ( ) can be reconstructed from shearlet atoms. Admissibility condition is a sufficient condition for this facility. Discrete and cone-adapted discrete shearlet systems are studied by Kutyniok and Labate in [11, 13]. They have derived sufficient conditions in [11] for a discrete shearlet system to form a frame for L ( ), whereas in this paper, we establish a necessary condition for both discrete and cone-adapted discrete shearlet systems to be frames via admissibility. In fact, we provide a relation between shearlet frames and admissibility condition of the generators. We propose here some preliminaries and notation about shearlets. We define the shearlet group S, as the semi-direct product ( + ) equipped with group multiplication given by (a, s, t).(a, s, t ) (aa, s + s a, t + S s A a t ), where the parabolic scaling matrices A a and the shearing matrix S s are given by [ ] [ ] a 1 s A a, S a 1 s. 1 The left-invariant Haar measure of this group is da a 3 dsdt. Let ψ L ( ). The continuous shearlet system associated with ψ is defined by { ψa,s,t T t D Aa D S s ψ : a >, s, t }, (1.1) where T and D are translation and dilation operators, respectively defined as T t f (x) f (x t), D B f (x) detb 1 f (B 1 x), where t and B is an invertible matrix. The continuous shearlet transform of f L ( ) is the mapping f SH ψ f (a, s, t) f, ψ a,s,t, (a, s, t) S. One of our concerns in shearlet theory is the reconstruction formula which is associated with the admissibility condition on ψ.

3 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) A discrete shearlet system associated with ψ is defined by { ψ j,k,m a 3 4 j ψ(s k A j a m) : j, k Z, m Z }, a >. (1.) The discrete shearlet transform of f L ( ) is the mapping defined by Definition 1.1. If ψ L ( ) satisfies f SH ψ f ( j, k, m) f, ψ j,k,m, ( j, k, m) Z Z Z. ψ(, ξ ) c ψ : d dξ <, (1.3) it is called an admissible shearlet. We denote by c + ψ, c ψ the following formulas c + ψ ξ 1 ψ(, ξ ) dξ d, c ψ ξ 1 ψ(, ξ ) dξ d. (1.4) Here, we recall the definitions of a cone-adapted discrete shearlet system and transform from [13]. For φ, ψ, ψ L ( ) and c (c 1, c ) ( + ), the cone-adapted discrete shearlet system is defined by Φ(φ; c 1 ) Ψ(ψ; c) Ψ( ψ; c), (1.5) ξ 1 where Φ(φ; c 1 ) { φ m φ( c 1 m) : m Z }, Ψ(ψ; c) { ψ j,k,m a 3 4 j ψ(s ka j a M c m) : j, k [a j ], m Z}, Ψ( ψ; c) { ψ j,k,m a 3 4 j ψ(s T k Ãa j M c m) : j, k [a j ], m Z}, with M c [ ] [ ] c1 c, M c c. c 1 The system Φ(φ; c 1 ) is associated with and the systems Ψ(ψ; c) and Ψ( ψ; c) are associated with E 1 E 3 and E E 4, respectively, where { (, ξ ) :, ξ 1 }, E 1 E 3 { (, ξ ) : ξ 1, > 1 }, E E 4 { (, ξ ) : ξ > 1, ξ > 1 }, ψ(, ξ ) ψ(ξ, ). The cone-adapted discrete shearlet transform of f L ( ) is the mapping defined by f SH φ,ψ, ψ f ( m, ( j, k, m), ( j, k, m ) ) ( f, φ m, f, ψ j,k,m, f, ψ j,k,m ),

4 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) with ( m, ( j, k, m), ( j, k, m ) ) Z Λ Λ, where Λ N { [a j ],..., [a j ]} Z. We define for C, L (C) { f : f L ( ) : supp f C}. In a discrete shearlet system {ψ j,k,m } j,k,m, in order to have a numerically stable reconstruction algorithm for f from the coefficients f, ψ j,k,m, we require that {ψ j,k,m } j,k,m constitutes a frame. In this paper, using several ideas in [7] we establish a relation between shearlet frames and admissibility condition. The manuscript is organized as follows. In Section, we give a necessary condition via admissibility, for a discrete shearlet system to be a frame. In fact, we show that if a discrete shearlet system {ψ j,k,m } j,k,m is a frame, then ψ is admissible. In Section 3, we establish such a condition for cone-adapted discrete shearlet systems. Finally, we give a similar result for higher dimensions.. The necessary condition for discrete shearlet systems In this section, we will consider a discrete shearlet system {ψ j,k,m } j,k,m as defined in (1.) and we establish a necessary condition for this system to be a frame. The system {ψ j,k,m } j,k,m is called a shearlet frame for L ( ), if there exist constants < A B < such that for all f L ( ), A f f, ψ j,k,m B f. (.1) j,k,m ecall that an operator E is called of trace-class if n Ee n, e n is finite for all orthonormal bases {e n }. The trace of E is defined to be TrE Ee n, e n. n Theorem.1. If the discrete shearlet system {ψ j,k,m } j,k,m constitutes a frame for L ( ) with frame bounds A, B, then ψ(, ξ ) αa dξ d αb, (.) for some constant α >, i.e. ψ is an admissible shearlet. ξ 1 Proof. Let {ψ j,k,m } j,k,m constitute a frame with bounds A, B and {e l } l be an orthonormal basis for L ( ). Put f e l in (.1). Then for coefficients c l with l c l e l <, we obtain A c l e l c l e l, ψ j,k,m B c l e l. (.3) l l j,k,m If C is any positive trace-class operator, then C l c l., e l e l and l c l TrC >. We have therefore, by (.3) A TrC Cψ j,k,m, ψ j,k,m B TrC. (.4) j,k,m l

5 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) Suppose supp(ĥ) [, ) and [13]). We consider C where h a,s,t is defined as in (1.1) and ĥ(ξ) dξ ξ1 d < (e.g. h may be a classical shearlet, see., h a,s,t h a,s,t c(a, s, t) dtdsda a 3, (.5) w( s a, t a ), 1 a a c(a, s, t) (.6), otherwise with t (t 1, t ) and w positive and integrable i.e. w( s, t )dtds <. We then have a C., h a,s,t h a,s,t w( s 1 a, t a )dtdsda a. 3 So We calculate Cψ j,k,m, ψ j,k,m j,k,m j,k,m a ψ j,k,m, h a,s,t a 3 4 j.a 3 4 a 3 4 j.a 4 3 ψ, h aa j,s a j 1 w( s a, t a ) ψ j,k,m, h a,s,t dtds da a 3. (.7) ψ ( S k A j a (x m) ).h ( A 1 ψ(y).h ( A 1 aa j +k,s ka j(t m) a a S 1 s (x t) ) dx S 1 (y S k A s a j j a (t m)) ) dy +k where in the second equality above, we have chosen the change of variable y S k A j a (x m). After the change of variables, the sum in (.7) becomes j,k,m a j+1 a j Now consider w as a aa j, s s w ( j a s a j k, a j a a j A 1 a j k,m, + k, t S k A j a (t m), S 1 k t + m ψ, h a,s,t w ( a j (s k), a ) ψ, h a j a,s,t 3 a j a dt a j a j ds da a 3 a 3 j A 1 S 1 a j k t + m a j a w(s, t) λ 3 e λ πs e λ πt 1 e λ πt, s, t (t1, t ). ) dtds da a 3. (.8) (.9)

6 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) By a similar argument as in the proof of [6, Lemma.], we get Hence w(αs + β, γt + η)dtds w max w(αm + β, γn + η) m Z n Z w(αs + β, γt + η)dtds + w max. j w(s, t)dtds (α detγ )w max (α detγ ) w(αm + β, γn + η) m n w(s, t)dtds + (α detγ )w max, where α a, detγ 1 S 1 a a j a deta 1 a j k a. a Then, we have w(αm + β, γn + η) a + ρ(a, s, t), m n such that ρ(a, s, t) w(, ) λ 3. Therefore continuing from (.9), (.7) will be Cψ j,k,m, ψ j,k,m ψ, h a,s,t ( a + ρ(a, s, t) ) dtds da a 3 j,k,m where j ψ, h a,s,t dtds da a +, ψ, h a,s,t ρ(a, s, t)dtds da. Note that is bounded. Indeed, a 3 ψ, h a,s,t ρ(a, s, t) dtds da a 3 λ 3 ψ, h a,s,t dtds da a 3 λ 3 ψ h a,s, (t) da dtds a 3 λ 3 λ 3 ψ(ξ). ĥ a,s,(ξ) dξds da a 3 ψ(ξ).a 3. ĥ ( a, a(ξ + s ) ) dξdsda, (.1) (.11)

7 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) in which h (x) h( x). Moreover, the first term in (.1), using the Plancherel theorem,is computed as follows ψ, h a,s,t dtds da ψ(ξ).a 1. ĥ ( a, a(ξ + s ) ) dξdsda a ψ(ξ). 1 ξ ĥ(w 1, w ) dξdw dw c + ψ ĥ, ψ(, ξ ) d dξ ξ 1 ψ(, ξ ) d dξ ξ 1 ĥ(w 1, w ) dw dw 1 ĥ(w 1, w ) dw dw 1 where ξ (, ξ ), w 1 a, w a(ξ + s ). Furthermore, a TrC h w ( s 1 a, t ) da dtds a a 3 h ln a w( s, t )dtds. Since w( s, t )dsdt 1, then by (.1), TrC h ln a. Hence by (.4) (.1) A ( h ln a ) c + ψ h + B ( h ln a ), (.13) where λ 3 ψ (c + h + c h ). If we divide (.13) by h and let λ tend to zero, then the result follows by considering α : ln a. In the following example, we give a Parseval shearlet frame which is admissible by Theorem.1. Example.. Let ψ 1 L () be a Lemarie -Meyer wavelet that satisfies the discrete Caldero n condition ψ 1 ( j w) 1, j Z with ψ 1 C () and supp ψ 1 [ 1, 1 16 ] [ 1 16, 1 ], Consider ψ L () is a bump function such that ψ 1 and for all w [ 1, 1], 1 ψ (w + k) 1, k 1

8 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) where ψ C () and supp ψ [ 1, 1]. Suppose ψ L ( ) is given by ψ(, ξ ) ψ 1 ( ) ψ ( ξ ). By [13, Proposition ], the shearlet system {ψ j,k,m } j,k,m as defined in (1.) with a is a Parseval frame for L ( ). So by Theorem.1, we have C ψ + C ψ ln. In [3] the continuous shearlet transform is generalized to higher dimensions. Here we give the discrete version and state our main result in this setting. In fact, for ψ L ( d ), we define the discrete shearlet system as { ( ψ j,k,m (a j 1 ) d ψ S k A j a (x m) ) : j Z, k Z d 1, m Z d}, a >, where A a j [ a j ] T d 1 d 1 sgn(a j ) a j d 1.Id 1, S k [ ] 1 k T. d 1 I d 1 Proposition.3. If the system {ψ j,k,m } j,k,m constitutes a frame for L ( d ) with frame bounds A, B, then ψ is admissible, in the sense that αa d 1 ψ(, ξ ) dξ d αb, (.14) for some constant α >, ((.14) is the admissibility condition appeared in [3, Theorem.4]). The proof of Proposition.3 is straightforward and therefore is omitted. The sufficient condition for the shearlet system {ψ j,k,m } j,k,m to be a frame for L ( ) is proposed in [11, Theorem 3.1]. 3. The necessary condition for cone-adapted discrete shearlet systems Similar to Theorem.1 a necessary condition can be given for a cone-adapted discrete shearlet system to be a frame. For convenience we denote the cone-adapted discrete shearlet system (1.5) by {g α } α. We define a cone-adapted discrete shearlet system {g α } α to be a frame for L ( ) if there exists < A, B < such that A f f, φ m + f, ψ j,k,m + f, ψ j,k,m B f, (3.1) m j,k,m j,k,m for all f L ( ). The following theorem is our main result of this section which is a necessary condition via admissibility for a cone-adapted discrete shearlet system to be a frame. Theorem 3.1. If the cone-adapted discrete shearlet system {g α } α is a frame for L ( ), then there exists such that the following admissibility condition holds ξ d 1 Aζ φ(ξ) + c + ψ + c Bζ, ξ. (3.) + ψ

9 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) Proof. Let the system {g α } α constitute a frame with bounds A, B. Consider {e l } l an orthonormal basis for L ( ). Put f e l in (3.1). Then for coefficients c l with l c l e l <, we obtain A c l e l c l e l, g α B c l e l. (3.3) l l α If C is any positive trace-class operator, then as in the proof of Theorem.1 A TrC Cg α, g α B TrC. (3.4) Suppose that h L ( ), with supp(ĥ) [, ) and α l ĥ(ξ) dξ ξ1 d <. Also assume that for a + {1}, s {} and t, we have h a,s,t L ((E 1 E 3 ) (E E 4 )), and for a 1, s, t, we have h a,s,t L (). Consider C., h a,s,t h a,s,t c(a, s, t)dtds da a 3 +., h 1,,t h 1,,t c(1,, t)dt, (3.5) in which c(a, s, t) is defined as (.6) for 1 < a a, s {}, t (t 1, t ) and c(1,, t) λ e πλ t. Then we have Cg α, g α Cφ m, φ m + Cψ j,k,m, ψ j,k,m + C ψ j,k,m, ψ j,k,m. (3.6) α m j,k,m j,k,m By definition of C as in (3.5), we obtain m Cφ m, φ m m m φ m, h 1,,t h 1,,t c(1,, t)dt, φ m φ m, h 1,,t c(1,, t)dt φ, h 1,,t λ e πλ t+m dt, m where m λ e πλ t+m 1 + ρ(t), such that ρ(t) λ. Hence we have Cφ m, φ m φ, h 1,,t dt + φ, h 1,,t ρ(t)dt m φ, h 1,,t dt + 1,

10 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) 11-6 where 1 φ, h 1,,t ρ(t)dt. Also 1 is bounded, since 1 λ φ, h 1,,t dt λ (φ h )(t) dt λ φ ĥ <. Similarly, φ, h 1,,t dt φ(ξ) ĥ(ξ) dξ. So Cφ m, φ m φ(ξ) ĥ(ξ) dξ + 1. m Also, similar to the proof of Theorem.1 for ψ j,k,m and ψ j,k,m, we have Cψ j,k,m, ψ j,k,m c + ψ ĥ(ξ) dξ +, j,k,m where ψ, h a,s,t ρ(a, s, t)dtds da and a 3 C ψ j,k,m, ψ j,k,m ĥ(ξ) c+ ψ dξ + 3, j,k,m where 3 ψ, h a,s,t ρ(a, s, t)dtds da. Then a 3 ( ) Cg α, g α φ(ξ) + c + ψ + c + ψ ĥ(ξ) dξ Furthermore, α TrC + + Ce n, e n n n a 1 e n, h a,s,t h a,s,t c(a, s, t)dtds da a 3, e n e n, h 1,,t h 1,,t c(1,, t)dt, e n n a 1 a h a,s,t n h.ζ, n e n, h a,s,t c(a, s, t)dtds da a 3 e n, h 1,,t c(1,, t)dt 1 c(a, s, t)dtds da a 3 + h 1,,t c(1,, t)dt

11 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) where ζ a c(a, s, t)dtds da + c(1,, t)dt ln a 1 a Hence by (3.4) A( h.ζ) ( φ(ξ) + c + ψ + c + ψ ) ĥ(ξ) dξ B( h.ζ). Since h ĥ, so Aζ ĥ(ξ) dξ ( φ(ξ) + c + ψ + c + ψ ) ĥ(ξ) dξ Bζ ĥ(ξ) dξ. Let λ tend to zero, then Aζ ĥ(ξ) dξ ( φ(ξ) + c + ψ + c + ψ ) ĥ(ξ) dξ Bζ ĥ(ξ) dξ. (3.7) By (3.7), we have and ( φ(ξ) + c + ψ + c + ψ Bζ) ĥ(ξ) dξ, (3.8) ( φ(ξ) + c + ψ + c + ψ Aζ) ĥ(ξ) dξ. (3.9) Now since ĥ(ξ) >, then by (3.8) there exists 1 such that for all ξ 1, we have φ(ξ) + c + ψ + c Bζ, + ψ and by (3.9) there exists such that for all ξ, we have φ(ξ) + c + ψ + c Aζ. + ψ Consider : 1, then for all ξ we have Aζ φ(ξ) + c + ψ + c Bζ. + ψ Example 3.. Consider C 1 {(, ξ ) : ξ 1}, C {(, ξ ) : ξ > 1}. Define f (x), x < 35x 4 84x 5 + 7x 6 x 7, x < 1 1, x 1 v(u) { f (1 + u), u f (1 u), u >.,

12 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) 11-6 It is obvious that f, v C(), also supp v [ 1, 1] and In addition, we have v() 1 and by (3.1), Let φ be given by v(u 1) + v(u) + v(u + 1) 1, for u 1. (3.1) j m j v( j u m) 1, for u 1. (3.11) ˆφ(ξ) c ξ e 1 (5ξ), ξ, in which we have chosen c so that ˆφ 1. (e.g. c ) and supp ˆφ [ 1, 1], ( ˆφ(1) ). Figure 1: The graph of ˆφ (c33.9) Then We consider Now, define ˆΦ(, ξ ) ˆφ( ) ˆφ(ξ ), (, ξ ). (3.1) ˆΦ(, ξ ) 1. (3.13) ˆΨ(, ξ ) ˆψ( ) ˆψ(ξ ), where (, ξ ). ˆψ(ξ) (ξ 1) e (ξ 1), ξ > 1 (ξ + 1) e (ξ+1), ξ < 1, 1 ξ 1,

13 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) Figure : The graph of ˆψ By definition of ˆΨ, it is clear that for ξ (, ξ ) ˆΨ( j ξ) <, for ξ i supp ˆψ, i 1,. j Infact, there exists a positive constant b such that ˆΨ( j ξ) b. (3.14) j Now, for ξ (, ξ ) consider the following cone-adapted shearlet system for L ( ) : {Φ( k) : k Z } {Ψ (d) j,k,m : j, k < j, m Z, d 1, } { Ψ j,k,m : j, k ± j, m Z }, (3.15) where and and A 1 A Also, for j >, define ( Ψ j,k,m ) (ξ) ˆΨ (1) j,k,m (ξ) 3 j ˆΨ( j ξ)v 1 (ξa j 1 S k j 1 )eπiξa 1 S 1 km, [ ] 4, S 1 [ ] 1 1, V 1 1 (, ξ ) v( ξ ), ξ, ˆΨ () j,k,m (ξ) 3 j ˆΨ( j ξ)v (ξa j S k j )eπiξa S km, [ ], S 4 [ ] 1, V 1 1 (, ξ ) v( ), ξ. ξ 3 j 1 ˆΨ( j, j ξ )v( j ξ k)e πiξ 1 A j 1 S 1 km, if ξ C 1 3 j 1 ˆΨ( j, j ξ )v( j ξ k)e πiξ 1 A j 1 S 1 km, if ξ C,,

14 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) and for j, m Z, k ±1, ( Ψ,k,m ) (ξ) ˆΨ(, ξ )v( ξ k)e πiξm, if ξ C 1 ˆΨ(, ξ )v( ξ k)e πiξm, if ξ C. The following calculations show that the shearlet system (3.15) is a frame for L ( ). For f L ( ), we observe that d1 j k < j m Z f, Ψ (d) j,k,m + j k± j m Z f, Ψ j,k,m j k < j m Z ( f ˆ, ˆΨ (1) j,k,m + f ˆ, ˆΨ () j,k,m ) + j k± j f ˆ, ( Ψ j,k,m ) m Z ˆ f (ξ) ˆΨ( j ξ) v( j ξ k) + v( j k) ξ j 1 ξ dξ k < j k < j + C 1 ˆ f (ξ) j ˆΨ( j ξ) v( j ( ξ 1)) dξ ˆ f (ξ) C 1 j ˆ f (ξ) C j ˆ f (ξ) C j ˆΨ( j ξ) v( j ( ξ + 1)) dξ ˆΨ( j ξ) v( j ( ξ 1)) dξ ˆΨ( j ξ) v( j ( ξ + 1)) dξ ˆ f (ξ) ˆΨ( j ξ) ( v( j ξ k) χ C1 (ξ) + v( j k) χ C (ξ))dξ ξ j 1 ξ k j ˆ f (ξ) ˆΨ( j ξ) dξ, j the last equality results from C 1 C and (3.11).

15 Amin khah, Askari Hemmat, aisi Tousi/ Wavelets and Linear Algebra 5(1) (18) Finally, using (3.1), for any f L ( ) we have n Z f, Φ( n) + d1 j k < j m Z f, Ψ (d) j,k,m + j k± j m Z f, Ψ j,k,m f ˆ(ξ) ˆΦ(ξ) dξ + ˆ f (ξ) ˆΨ( j ξ) dξ j f ˆ(ξ) ˆΦ(ξ) + ˆΨ( j ξ) dξ. j It follows from (3.14) there exists a positive constant a, so that a ˆΦ(ξ) + ˆΨ( j ξ) (b + 1). (Note that a. Indeed, for ξ 1, Hence a f f, Φ( n) + n Z j j ˆΨ( j ξ) but ˆφ(1) ). d1 j k < j m Z f, Ψ (d) j,k,m + f, Ψ j,k,m (b + 1) f. j k± j m Z So by Theorem 3.1 the admissibility condition (3.) holds for the shearlet frame (3.15), i.e. there exist ζ +, so that for ξ supp ˆφ, we have Acknowledgments a ζ φ(ξ) + c + + c + + c (b + 1)ζ. ψ (1) ψ + ψ () The authors are indebted to Professor Hartmut Führ for valuable comments and remarks on an earlier version of this paper. eferences [1] S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G. Stark and G. Teschke, The uncertainty principle associated with the continuous shearlet transform, Int. J. Wavelets Multiresolut. Inf. Process., 6() (8), [] S. Dahlke, G. Kutyniok, G. Steidl and G. Teschke, Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harmon. Anal., 7() (9), [3] S. Dahlke, G. Steidl and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl., 16(3) (1), [4] S. Dahlke, G. Steidl and G. Teschke, Multivariate shearlet transform, shearlet coorbit spaces and their structural properties, Shearlets: Multiscale Analysis for Multivariate Data, Springer, (1),

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