Real Paley-Wiener theorems for the Clifford Fourier transform

Associated with the Dirac operator and partial derivatives, this paper establishes some real Paley-Wiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform (CFT) has compact support. Based on the Riemann-Lebesgue theorem for the CFT, the Boas theorem is provided to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin.     

Real Paley-Wiener Theorems

The aim of this paper is to exhibit a real Paley–Wienerspace sitting inside the Schwartz space, and to give a quickand simple proof of a Paley–Wiener-type theorem. A simpleand elementary proof of a theorem postulated by H. H. Bang isalso given. 2000 Mathematics Subject Classification 42A38.     

Paley-Wiener type theorems by transmutations

The classical Paley-Wiener theorem for functions in L _dx ² relates the growth of the Fourier transform over the complex plane to the support of the function. In this work we obtain Paley-Wiener type theorems where the Fourier transform is replaced by transforms associated with self-adjoint operators on L _dμ ² , with simple spectrum, where dμ is a Lebesgue-Stieltjes measure. This is achieved via the use of support preserving transmutations. Communicated by Paul L. Butzer     

Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator

In the present paper, we investigate some subordination- and superordination-preserving properties for certain classes of analytic and multivalent functions in the open unit disk, which are associated with such multiplier transformations as the Srivastava-Attiya operator. Various sandwich-type theorems for functions belonging to these classes are also obtained.     

Growth and covering theorems associated with the Roper-Suffridge extension operator

The Roper-Suffridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univalent function to a (locally) univalent map . If belongs to a class of univalent functions which satisfy a growth theorem and a distortion theorem, we show that satisfies a growth theorem and consequently a covering theorem. We also obtain covering theorems of Bloch type: If is convex, then the image of (which, as shown by Roper and Suffridge, is convex) contains a ball of radius . If , the image of contains a ball of radius .

     

Real Paley-Wiener Theorems for the Hankel Transform

We first characterize the image of the compactly supported smooth even functions under the Hankel transform as a subspace of the Schwartz space. We then describe the space of smooth L^p-functions whose Hankel transform has compact support as a subspace of the space of L^p-functions.     

Paley-Wiener theorems on groups of split rank one

Let G be a linear semisimple Lie group of split rank one with K a maximal compact subgroup. In this paper, we consider the space C_c^∞(G:F) of all functions in C_c^∞(G) whose left and right K-translates span a finite-dimensional space. Using the analytic continuation of the principal series to define the Fourier transform, we give a characterization of the Fourier transform of the space C_c^∞(G:F). This gives an analog of the classical Paley-Wiener theorem which gives a characterization of the Fourier transform of the space C_c^∞(Rⁿ).     

On real Paley-Wiener theorems for certain integral transforms

We prove real Paley-Wiener theorems for the (inverse) Jacobi transform, characterising the space of L²-functions whose image under the Jacobi transform are (smooth) functions with compact support.     

New type Paley-Wiener theorems for the modified multidimensional Mellin transform

New type Paley-Wiener theorems for the modified multidimensional Mellin and inverse Mellin transforms are established. The supports of functions are described in terms of their modified Mellin (or inverse Mellin) transform without passing to the complexification. Acknowledgments and Notes. The work is supported by the Kuwait University research grant SM 112.     

Paley-Wiener type theorems on harmonic extensions ofH-type groups

Letn=vz be anH-type group, and letna be the harmonic semidirect product ofn withaR. LetNA be the corresponding simply connected Lie group. If dimv=m and dimz=k, denoteQ=m/2+k. We prove that the spherical Fourier transform is a topological isomorphism between thep-Schwartz spacel ^p (N,A),(0<p2), (0<p2), and the space of holomorphic rapidly decreasing functions on the strip {sC:|Re(s)|<Q/2} with =2/p–1.     

Sobolev type theorems for an operator with singularity

Spaces of Sobolev type are discussed, which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. Smoothness of and continuity of () are studied where is in each space of Sobolev type, and results similar to Sobolev's lemma are obtained. The proofs are carried out based on a generalization of the Fourier transform. The results are applied to the Schrödinger equation.

     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.