A spectral Paley-Wiener theorem

The Fourier inversion formula in polar form is \(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ? ⁿ , whereP _λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP _λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

A Paley-Wiener theorem and Wiener-Hopf-type integral equations in Clifford analysis

Using the properties of the monogenic extension of the Fourier transform, we state a Paley-Wiener-type theorem for monogenic functions. Based on an multiplier algebra related to boundary values of monogenic functions we consider integral equations of Wiener-Hopf-typeK±u _±=f on ℝ ⁿ , whereK ∈S′ andu _± are boundary values of monogenic functions in ℝ₊ ⁿ⁺¹ and ℝ_{_} ⁿ⁺¹ respectivly.     

Generalization of the Paley-Wiener theorem in weighted spaces

Translated from Matematicheskie Zametki, Vol. 48, No. 5, pp. 80–87, November, 1990.     

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     

The Paley-Wiener theorem for Schwartz distributions with support on a half-line

LetD₊ be the space of Schwartz distributions with support on the closed positive half-line [0, +). We give a generalization of the Paley-Wiener theorem to the case of the distributions inD₊.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 54–57.     

A Paley-Wiener theorem for the spherical Laplace transform on causal symmetric spaces of rank 1

We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform defined on the rank 1 causal symmetric spaces , for .

     

Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions

We introduce a global wave front set suitable for the analysis of tempered ultradistributions of quasi-analytic Gelfand–Shilov type. We study the transformation properties of the wave front set and use them to give microlocal existence results for pullbacks and products. We further study quasi-analytic microlocality for classes of localization and ultradifferential operators, and prove microellipticity for differential operators with polynomial coefficients.     

The Paley-Wiener theorem in the non-commutative and non-associative octonions

The Paley-Wiener theorem in the non-commutative and non-associative octonion analytic function space is proved. This work was supported by the National Basic Research Program of China (Grant No. 1999075105), the National Natural Science Foundation of China (Grant No. 10471002) and Research Foundation for Doctoral Programm (Grant No. 20050574002)     

Moment sequences for ultradistributions

Supported by DAAD, West Germany     

Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces

We study Fourier transforms of distributions on a symmetric space X. Eguchi et al. [1] characterized the image of $\text{E}$′(X)-distributions of compact support under the Fourier transform. We give a simpler proof of Eguchi's result and characterize the size of the singular support for the K-finite members of E′(X). We apply this Paley-Wiener type theorem to invariant differential equations on X.