/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 3, 2015, 368-374 ISSN 1307-5543 – www.ejpam.com A Generalization of the Calderón Admissibility Condition Ali Akbar Arefijamaal1,∗, Mehdi Mohammadzadeh Karizaki2 1 Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran 2 Kashmar Higher Education Institute, Kashmar, Iran Abstract. Many authors have been considered several conditions equivalent to the Calderón admis- sibility condition. In this paper, we review these results and give a characterization of generalized Calderón admissibility condition. 2010 Mathematics Subject Classifications: 46C50; 42C99 Key Words and Phrases: Calderón admissibility condition, continuous wavelet transform, semidirect product 1. Introduction For every ψ ∈ L2(R) the continuous wavelet transform (CWT) of f ∈ L2(R) is given by (Wψ f )(a, b) = |a| −1 2 ∫ R f (x)ψ( x − b a )d x , (a ∈ R \ {0}, b ∈ R). The mapping Wψ is well-defined if ψ satisfies the Calderón admissibility condition ∫ R\{0} |Òψ(ξ)|2 |ξ| dξ= 1. (1) The generalization of this construction, in particular to higher-dimensional Euclidean space, has been studied early on, see e.g. [4]. One class of groups and representations attracting particular attention are the semidirect products of the type G = H ×τ R n. Here H is a closed matrix group, the so-called dilation group. To construct continuous wavelet transforms from quasi-regular representations of a semidirect product topological group we require a square integrable function whose Plancherel transform satisfies Calderón admissibility condition. The ∗Corresponding author. Email addresses: arefijamaal@hsu.ac.ir;arefijamaal@gmail.com (A. Arefijamaal), mohammadzadehkarizaki@gmail.com (M. Karizaki) http://www.ejpam.com 368 c© 2015 EJPAM All rights reserved. A. Arefijamaal, M. Karizaki / Eur. J. Pure Appl. Math, 8 (2015), 368-374 369 question then arises under what conditions such square-integrable functions exist. The CWT on these groups was discussed in [2, 7, 8, 11]. Many authors have been considered several conditions equivalent to the Calderón admis- sibility condition, [1, 2, 6, 8]. In this paper, we first review these results and then introduce a more general setting for admissible groups. Moreover, some necessary conditions are pro- vided for a class of admissible groups. Let us shortly sketch the group-theoretic framework for the construction of continuous wavelet transforms on locally compact abelian groups. It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. Let G be a locally compact topological group with the left Haar measure µG and modular function ∆G . If π is a unitary representation of G on a Hilbert spaceH , then a vector ψ ∈H where Cψ := 1 ‖ψ‖2 ∫ G |<ψ,π(x)ψ>|2 dµG(x)<∞ (2) is called an admissible vector. The existence of an admissible vector is not generally guaranteed [10]. Now for a fixed admissible vector ψ in H the linear isometry Wψ : H → L2(G) given by (Wψη)(x) = C −1 2 ψ < η,π(x)ψ>, (η ∈H , x ∈ G) is called the CWT on G. Also we refer to the inequality (2) as the admissibility condition. Among the many useful aspects of wavelets, probably the most fundamental one is the wavelet inversion formula, usually given by ∫ G (Wψη)(x)π(x)ψ dµG(x) = η, (η ∈H ). For locally compact groups H and K where K is also abelian, let h 7−→ τh be a homomor- phism of H into the group of automorphisms of K denoted by Aut(K). Also, assume that the mapping (h, x) 7−→ τh(x) from H ×K onto K is continuous. Then the set H ×K endowed with the product topology and the operations: (h, x).(h′, x ′) = (hh′, x .τh(x ′)), (h, x)−1 = (h−1,τh−1(x−1)) is a locally compact group. This group is called the semidirect product of H and K , respectively, and is denoted by H ×τ K . Let G = H ×τ K . Then the left Haar measure of G is dµG(h, x) = δ(h)dµH(h)dµK(x) and ∆G(h, x) = δ(h)∆H(h) is its modular function, in which δ is a positive continuous homomorphism on H and is given by µK(E) = δ(h)µK(τh(E)), for all measurable subsets E of K , for more details of these facts see [5]. From the canonical action of G on K arises a natural unitary representation, which is called the quasi regular representation on the semidirect product group G. A. Arefijamaal, M. Karizaki / Eur. J. Pure Appl. Math, 8 (2015), 368-374 370 Definition 1. The quasi regular representation (U , L2(K)) on G = H ×τ K is defined by U(h, x) f (y) = δ(h) 1 2 f (τh−1(y x−1)), ( f ∈ L2(K)). U is not generally irreducible [10]. An element ψ ∈ L2(K) is admissible if satisfies the generalized Calderón admissibility condition ∫ H ∫ K |<ψ, U(h, x)ψ> |2δ(h)dµH(h)dµK(x)<∞, ( f ∈ L2(K)). (3) Consider bK as the dual group of the LCA group K and denote its left Haar measure by dω. Then one can define a continuous action from H on bK by (h,ω) 7−→ ω ◦ τh−1 . Now for each ω ∈ bK , the stabilizer and the orbit of ω, that play a key role in our discussion are defined by Hω := {h ∈ H ; ; ω ◦τh =ω}, Oω := {ω ◦τh; h ∈ H}, respectively. The set Hω is a closed subgroup of H and Oω is an H-invariant subset in bK . 2. Admissible Subgroups of GL(n,R) Let H ≤ GL(n,R) be the group consisting of diagonal matrices, also let H ×τ R n be the semidirect product of H and Rn, with the usual action of H on Rn. In [3] it is shown that ψ ∈ L2(Rn) is admissible if and only if ∫ H |Òψ(ξ)|2 |ξ1ξ2 . . .ξn| dξ <∞. A subgroup H of GL(n,R) is said to be admissible if the quasi-regular representation on the semidirect product group H ×τ R n, with the natural action of H on Rn, has an admissible vector ψ ∈ L2(Rn). In [12] it is shown that a subgroup H of GL(n,R) is admissible if and only if there exists ψ ∈ L2(Rn) such that ∫ H |Òψ(ωh)|2dµH(h) = 1 for a.e. ω ∈ Rn. A straightforward calculation gives that H ≤ GL(n,R) is admissible if and only if there exists a Borel measurable function g ∈ L1(Rn) such that g ≥ 0 and ∫ H g(ht x)dµH(h) = 1 for a.e. x ∈ Rn, (4) in which ht is the transpose of h. For example, with the natural action on R2 the Affine group R \ {0} ×τ R is admissible. The best results are due to Laugesen et al. proved in [11] is a characterization of those admissible groups which admit an inversion formula; A. Arefijamaal, M. Karizaki / Eur. J. Pure Appl. Math, 8 (2015), 368-374 371 Theorem 1 ([11]). Let H be a σ-compact, locally compact group, and h 7−→ τh from H to GL(n,R) be a continuous homomorphism. Then (i) If H is admissible, then ∆H 6≡ δ −1 and Hω is compact for a.e. ω ∈ Rn. (ii) If ∆H 6≡ δ −1 and for a.e. ω ∈ Rn there exists an ε > 0 such that Hωε = {h ∈ H; ‖ω ◦τh −ω‖ ≤ ε}, the ε-stabilizer of ω, is compact, then H is admissible. Proposition 1. For any n> 1 the group GL(n,R) is not admissible. Proof. Assume that H = GL(n,R). It is sufficient to show that the stabilizers Hω, for all ω in a positive Lebesgue measure subset of Rn, are not compact. Let E = {ω ∈ Rn;ωi > 0}. Then the equation htω =ω can be solved with respect to any ω ∈ E. In fact, we may find solutions h ∈ H whose some arrays are arbitrary large. So that Hω is not compact for all ω ∈ E. The structure of stabilizers of a group and its subgroup are almost the same. Let H be an admissible group and L ≤ H. Then Lω is compact for a.e. ω ∈ Rn since it is a closed subgroup of Hω. But the condition ∆H 6≡ |det| about H and L may be different. For example, the group SL(2,R) and its subgroup K = �� 1 y 0 1 � , y ∈ R � are not admissible by Theorem 1. Although the subgroup H = �� x y 0 x−1 � x 6= 0, y ∈ R � of SL(2,R) is admissible [11]. The following theorem is about the admissibility of H and H t . Theorem 2. A closed unimodular subgroup H of GL(n,R) is admissible if and only if H t is admissible. Proof. First we would like to compute ∆H t , the modular function of H t . Clearly ν(E) = µH t (E t) defines a right Haar measure on H, so µH(E −1) = cµH t (E t) for all Borel sets E of H and for some c > 0. This implies that ∆H t (ht) = µH t (Eht) µH t (E) = µH((hE t)−1) µH((E t)−1) = ∆H(h −1) i.e. ∆H t =∆−1 H . Now assume that H is admissible, then there exists a non-negative g ∈ L1(Rn) such that (4) holds. This implies that ∫ H t g(hx)dµH t (ht) = ∫ H g(ht x)dµH(h −1) A. Arefijamaal, M. Karizaki / Eur. J. Pure Appl. Math, 8 (2015), 368-374 372 = ∫ H g(ht x)∆H(h −1)dµH(h) = ∫ H g(ht x)dµH(h) = 1, for a.e. x ∈ Rn. Therefore, H t is admissible. The following example shows that the condition H is unimodular can not be removed; Example 1. Let H = (R \ {0})×τ R be the affine group. As we have seen before, dµH(h, x) = h−2dhd x is the left Haar measure of H. This shows that every non-negative nor- malized function g ∈ L1(R2) such that x2 1 g(x1, x2) is integrable satisfies in (4). Hence, H is admissible but ∆H t ≡ |det| and so H t is not admissible by Theorem 1. 3. Admissibility of Arbitrary Topological Groups A more general family of admissible groups was studied by Grochenig, Kaniuth and Taylor [9], who focused on certain one-parameter groups; in particular all of the aforementioned ex- amples fall under the class described in [8, 11]. Consider the semidirect product group G×τR n where G is an arbitrary topological group and τ : G→ GL(n,R);a 7−→ τa is a homomorphism such that (a, x) 7−→ τa(x) is continuous. The topological group G is called admissible if the quasi-regular representation on the semidirect group G ×τ R n has an admissible vector. A further extension, replacingRn by a general locally compact abelian group K . In this case, we can also modify (4) to describe admissible groups. A characterization of such admissible groups which extends (1) can be found in [2, 8]. Theorem 3. [2] Equality (3) is valid for ψ ∈ L2(K) if ∫ H |Òψ(ω ◦τh)| 2dµH(h) = 1 for a.e. ω ∈ bK . (5) Moreover, the converse is also true by more assumptions. Let H be a closed subgroup of G. We consider the left multiplication as the usual action of G on quotient space G/H. A Radon measure µ on G/H is called invariant if µ(aB) = µ(B) for every g ∈ G and Borel set B of G/H. There is an invariant measure on G/H if and only if ∆G |H= ∆H , for more details see 2.49 of [5]. The following theorem shows that the admissibility can be extended from a subgroup to own group. Theorem 4. Let H be a closed subgroup of a σ−compact group G such that G/H is compact. If H is admissible and G/H has an invariant measure, then is G also admissible. Proof. Suppose µ is an invariant measure on G/H, then we have ∫ G f (a)dµG(a) = ∫ G/H ∫ H f (ah)dµH(h)dµ(aH), ( f ∈ L1(G)). (6) A. Arefijamaal, M. Karizaki / Eur. J. Pure Appl. Math, 8 (2015), 368-374 373 This identity is known as Weil’s formula and holds also for any f ≥ 0 that vanishes outside a finite set [5]. Now let H be admissible and g ∈ L1(R) a non-negative measurable function such that ∫ H g((τh) t x)dµH(h) = 1, for a.e. x ∈ Rn. Since a 7−→ g((τa) t x) on G is positive by using (6) with the fact that µ is finite we obtain ∫ G g((τa) t x)dµG(a) = ∫ G/H ∫ H g((τah)t x)dµH(h)dµ(aH) = ∫ G/H ∫ H g((τh) t(τa) t x)dµH(h)dµ(aH) = ∫ G/H dµ(aH)<∞, for a.e. x ∈ R. Therefore, G is admissible. Theorem 5. If G1 and G2 are admissible groups, then so is G1 × G2. Proof. Since Gi is admissible there exists a measurable function gi ∈ L1(Rni ) such that gi ≥ 0 and ∫ G gi((τ i a) t x)dµGi (a) = 1, for a.e. x ∈ Rni where ni ∈ N and τi : Gi → GL(ni ,R) is a continuous homomorphism (i = 1,2). Consider G = G1 × G2 and define the continuous homomorphism τ : G→ GL(n1 + n2,R) by τ(a1,a2) = � τ1 a1 0 0 τ2 a2 � . (7) Then the semidirect product group (G1 × G2) ×τ R n1+n2 is well defined and g : Rn1+n2 → C given by g(x1, x2) = g1(x1)g2(x2) where x1 ∈ R n1 and x2 ∈ R n2 is positive and belongs to L1(Rn1+n2). Moreover ∫ G g((τ(a1,a2) )t(x1, x2))dµG(a1, a2) = 1, for a.e. x ∈ Rn1 , x ∈ Rn2 . Therefore, G is admissible. 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