Admissible wavelets on the Siegel domain of type one

LetSp(n, R) be the sympletic group, and letK _n * be its maximal compact subgroup. ThenG=Sp(n,R)/K _n * can be realized as the Siegel domain of type one. The square-integrable representation ofG gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent state. With the Laguerre polynomials, Hermite polynomials and Jacobi polynomials, two kinds of orthogonal bases for AW are given, and they then give orthogonal decompositions ofL ²-space on the Siegel domain of type one ℒ(ℋ _n , |y| *dxdy). Project supported in part by the National Natural Science Foundation of China (Grant No. 19631080).     

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

On some class of nonseparable continuous wavelet transforms

It is well-known that only a single condition (called the admissibility condition) is sufficient for L ₂-convergence of multiple continuous wavelet transforms (MCWT). However known results suggest that to guarantee the pointwise convergence of MCWT for L _p -functions wavelets should vanish quite rapidly at infinity. In this article, we consider the class of nonseparable multiple wavelets with square-symmetric Fourier transforms. For these wavelets ψ we prove that the admissibility condition and the condition ψ∈L ₁(R ⁿ ) are sufficient for the convergence of corresponding MCWT almost everywhere.     

Criteria for monogenicity of Clifford algebra-valued functions on fractal domains

Suppose that Ω is a bounded domain with fractal boundary Γ in ${\mathbb R^{n+1}}Suppose that Ω is a bounded domain with fractal boundary Γ in \mathbb Rⁿ⁺¹{\mathbb R^{n+1}} and let \mathbb R_0,n{\mathbb R_{0,n}} be the real Clifford algebra constructed over the quadratic space \mathbb Rⁿ{\mathbb R^{n}}. Furthermore, let U be a \mathbb R_0,n{\mathbb R_{0,n}}-valued function harmonic in Ω and H?lder-continuous up to Γ. By using a new Clifford Cauchy transform for Jordan domains in \mathbb Rⁿ⁺¹{\mathbb R^{n+1}} with fractal boundaries, we give necessary and sufficient conditions for the monogenicity of U in terms of its boundary value u = U|_Γ. As a consequence, the results of Abreu Blaya et al. (Proceedings of the 6th International ISAAC Congress Ankara, 167–174, World Scientific) are extended, which require Γ to be Ahlfors-David regular.     

Wavelets on the 2-Sphere: A Group-Theoretical Approach

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S², based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and ⁺_* for dilations on S², which are embedded into the Lorentz group SO₀(3, 1) via the Iwasawa decomposition, so that X SO₀(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO₀(3, 1) acting in the space L²(S², d μ) of finite energy signals on S². This representation is square integrable over X; thus it yields immediately the wavelets on S² and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S² is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R → ∞. Then the parameter space goes into the similitude group of ² and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.     

A Clifford Algebra Realization of Supersymmetry and its Polyvector Extension in Clifford Spaces

It is shown explicitly how to construct a novel (to our knowledge) realization of the Poincaré superalgebra in 2D. These results can be extended to other dimensions and to (extended) superconformal and (anti) de Sitter superalgebras. There is a fundamental difference between the findings of this work with the other approaches to Supersymmetry (over the past four decades) using Grassmannian calculus and which is based on anti-commuting numbers. We provide an algebraic realization of the anticommutators and commutators of the 2D super-Poincaré algebra in terms of the generators of the tensor product Cl_1,1(R) ?A{Cl_{1,1}(R) \otimes \mathcal{A}} of a two-dim Clifford algebra and an internal algebra A whose generators can be represented in terms of powers of a 3 × 3 matrix Q{\mathcal{Q}} , such that Q³ = 0{\mathcal{Q}^3 = 0} . Our realization differs from the standard realization of superalgebras in terms of differential operators in Superspace involving Grassmannian (anti-commuting) coordinates θ ^α and bosonic coordinates x ^μ . We conclude in the final section with an analysis of how to construct Polyvector-valued extensions of supersymmetry in Clifford Spaces involving spinor-tensorial supercharge generators Q_a^m₁m₂?m_n{{{\mathcal {Q}}_{{\alpha}}^{\mu_1\mu_2\ldots\mu_n}}} and momentum polyvectors P_m1m2?mn{P_{\mu_1\mu_2\ldots\mu_n}} . Clifford-Superspace is an extension of Clifford-space and whose symmetry transformations are generalized polyvector-valued supersymmetries.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

Inversion of the Weyl integral transform and the radon transform on Rn using generalized wavelets

We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL , >–1/2. Next, we use the connection between radial classical wavelets onR ⁿ and generalized wavelets associated with the Bessel operatorL( _n–2)/2 to derive new inversion formulas for the Radon transform onR ⁿ ,n2.     

NS Condition of Admissibility for the Linear Estimator of Normal Mean with Unknown Variance

Suppose Y - N（β, σ^2 In）, where β ∈ R^n and σ^2 〉 0 are unknown. We study the admissibility of linear estimators of mean vector under a quadratic loss function. A necessary and sufficient condition of the admissible linear estimator is given.     

Clifford analysis versus its quaternionic counterparts

For a positive integer n let Cl_0,n be the universal Clifford algebra with the signature (0,n). The name Clifford analysis is usually referred to the function theories for functions in the kernels of the two operators: the (Cliffordian) Cauchy–Riemann operator and the Dirac operator. For n=2, Cl_0,2 becomes the skew‐field of Hamilton's quaternions for which the two operators are widely known: the Moisil–Théodoresco and the Fueter operators. We establish the precise relations between the Moisil–Théodoresco operator and the Dirac operator for Cl_0,3. It turns out that the case of the Cauchy–Riemann operator for Cl_0,3 and the Fueter operator is more sophisticated, and we describe the peculiarities emerging here. Copyright © 2009 John Wiley & Sons, Ltd.     

ADMISSIBLE WAVELETS ASSOCIATED WITH THE TRANSFORM GROUP ON R

Let p be the transform group on R, then P has a natural unitary representation U onL2 (R^n). Decompose L2(R^n) into the direct sum of irreducible invariant closed subspace,s. The re-striction of U on these suhspaces is square-intagrable. In this paper the characterization of admissi-ble condition in tarrns of the Fourier transform is given. The wavelet transform is defined, and theorthogorml direct sum decomposition of function space L2 (P,du1) is obtained.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

Between quantum Virasoro algebra $$\mathcal{L}_c $$ and generalized Clifford algebras

In this paper we construct the quantum Virasoro algebra generators in terms of operators of the generalized Clifford algebras C_n^k. Precisely, we show that can be embedded into generalized Clifford algebras. Junior Associate at The Abdus Salam ICTP, Trieste, Italy.     

On the Classification of Twisting Maps Between Kn and Km

We define the notion of admissible pair for an algebra A, consisting on a couple (Γ, R), where Γ is a quiver and R a unital, splitted and factorizable representation of Γ, and prove that the set of admissible pairs for A is in one to one correspondence with the points of the variety of twisting maps T_Aⁿ:=T(Kⁿ,A)\mathcal{T}_A^n:=\mathcal{T}(K^n,A). We describe all these representations in the case A = K ^m .     

Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius

In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system \frac?f?x₀ + ?_i=1ⁿ e_i\frac?f?x_i=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives.     

Partly Divisible Probability Measures on Locally Compact Abelian Groups

A notion of admissible probability measures μ on a locally compact Abelian group (LCA ‐ group) G with connected dual group Ĝ = ℝ^d × 𝕋ⁿ is defined. To such a measure μ, a closed semigroup Λ(μ) ⊆ (0, ∞) can be associated, such that, for t ∈ Λ(μ), the Fourier transform to the power t, (μˆ)^t, is a characteristic function. We prove that the existence of roots for non admissible probability measures underlies some restrictions, which do not hold in the admissible case. As we show for the example ℤ₂, in the case of LCA ‐ groups with non connected dual group, there is no canonical definition of the set Λ(μ).     

GENERALIZED WAVELETS AND INVERSION OF THE RADON TRANSFORM ON THE LAGUERRE HYPERGROUP

Let X=R ₊ ⁿ ×R denote the underlying manifold of polyradial functions on the Heisenberg group H_n. We construct a generalized translation on X=R ₊ ⁿ ×R, and establish the Plancherel formula on L²(X, dμ). Using the Gelfand transform we give the condition of generalized wavelets on L²(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.     

The Calderón reproducing formula, windowed X-ray transforms, and radon transforms in LP-spaces

The generalized Calderón reproducing formula involving “wavelet measure” is established for functions f ∈ L^p(ℝⁿ). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ. Acknowledgements and Notes. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Dyadic Bivariate Wavelet Multipliers in L^2（R@2）

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L ²(ℝ²). In this paper, we choose $2I_2 = \left( {{*{20}c} 2 & 0 \\ 0 & 2 \\ } \right)$2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right) as the dilation matrix and consider the 2I ₂-dilation multivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f ₁, f ₂, f ₃} a dyadic bivariate wavelet multiplier if Y₁ = { F ^{- 1} ( f₁ [^(y₁ )] ),F ^{- 1} ( f₂ [^(y₂ )] ),F ^{- 1} ( f₃ [^(y₃ )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}, where [^(f)]\hat f and F ⁻¹ denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.