Distributions Of Compact Support Research Articles

On the Distributional Fourier Duality and Its Applications

Sampling theorems for bandlimited functions or distributions are obtained by exploiting the topological isomorphism between the space E′(R) of distributions of compact support on R and the Paley–Wiener spacePWof entire functions satisfying an estimate of the form |f(z)|≤A(1+|z|)NeB|Imz|for some constantsA,B,N≥0. We obtain sampling theorems forfinPWby expanding its Fourier transformTin a series converging in the topology of E′(R) and whose coefficients are samples taken fromf. By Fourier duality, we obtain a sampling theorem forfin the spacePW. These sampling expansions converge, in fact, uniformly on compact sets of C, since convergence in the topology ofPWimplies uniform convergence on compact sets of C. This procedure allows us to recover previous sampling theorems in a unified way. We also present further expansions of Paley–Wiener functions obtained by expanding their Fourier transform as a series involving Legendre or Hermite polynomials.

Mehler-Fock transforms of generalized functions via the method of adjoints

In this paper we analyse the Mehler-Fock transform of generalized functions via the method of adjoints. For a distribution of compact support, we prove that its Mehler-Fock transform agrees with its transform via the kernel method. A Paley-Wiener type theorem is established.

Support Theorems for Radon Transforms on Real Analytic Line Complexes in Three-Space

In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in ${\mathbb {R}^3}$. Let $f$ be a distribution of compact support on ${\mathbb {R}^3}$. Assume $Y$ is a real analytic admissible line complex and ${Y_0}$ is an open connected subset of $Y$ with one line in ${Y_0}$ disjoint from $\text {supp}\;f$. Under weak geometric assumptions, if the Radon transform of $f$ is zero for all lines in ${Y_0}$, then $\text {supp}\;f$ intersects no line in ${Y_0}$. These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support.

Support theorems for Radon transforms on real analytic line complexes in three-space

In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in R 3 . Let f be a distribution of compact support on R 3 . Assume Y is a real analytic admissible line complex and Y 0 is an open connected subset of Y with one line in Y 0 disjoint from supp f. Under weak geometric assumptions, if the Radon transform of f is zero for all lines in Y 0 , then supp f intersects no line in Y 0 . These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support

Self-similar measures and their Fourier transforms. II

A self-similar measure on R n {{\mathbf {R}}^n} was defined by Hutchinson to be a probability measure satisfying ( ∗ ) ({\ast }) \[ μ = ∑ j = 1 m a j μ ∘ S j − 1 \mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}} \] , where S j x = ρ j R j x + b j {S_j}x = {\rho _j}{R_j}x + {b_j} is a contractive similarity ( 0 > ρ j > 1 , R j (0 > {\rho _j} > 1,{R_j} orthogonal) and the weights a j {a_j} satisfy 0 > a j > 1 , ∑ j = 1 m a j = 1 0 > {a_j} > 1,\sum \nolimits _{j = 1}^m {{a_j} = 1} . By analogy, we define a self-similar distribution by the same identity ( ∗ ) ( {\ast } ) but allowing the weights a j {a_j} to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to ( ∗ ) ( {\ast } ) among distributions of compact support, and show that the space of such solutions is always finite dimensional. If F F denotes the Fourier transformation of a self-similar distribution of compact support, let \[ H ( R ) = 1 R n − β ∫ | x | ≤ R | F ( x ) | 2 d x , H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,} \] where β \beta is defined by the equation ∑ j = 1 m ρ j − β | a j | 2 = 1 \sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1} . If ρ j ν j = ρ \rho _j^{{\nu _j}} = \rho for some fixed ρ \rho and ν j {\nu _j} positive integers we say the { ρ j } \{ {\rho _j}\} are exponentially commensurable. In this case we prove (under some additional hypotheses) that H ( R ) H(R) is asymptotic (in a suitable sense) to a bounded function H ~ ( R ) \tilde H(R) that is bounded away from zero and periodic in the sense that H ~ ( ρ R ) = H ~ ( R ) \tilde H(\rho R) = \tilde H(R) for all R > 0 R > 0 . If the { ρ j } \{ {\rho _j}\} are exponentially incommensurable then lim R → ∞ H ( R ) {\lim _{R \to \infty }}H(R) exists and is nonzero.

Exact deconvolution for multiple convolution operators-an overview, plus performance characterizations for imaging sensors

An updated review of the subject, including physically realizable examples along with explicit inverses and a computer simulation of the resulting large bandwidth, is given. A discussion of the ill-posedness of a single convolution operator clarifies the necessity of multiple operators. A precisely stated necessary and sufficient condition for inevitability is given. The performance of simultaneous convolution operators when there are sources of additive noise prior to the inverse is addressed. The main point is that for noise typical of electrooptical sensors, invertible multiple operators with their inverses will always outperform any set of single or multiple operators with the inverse omitted. A tutorial on the theory of distributions of compact support, which is used freely throughout the paper, is given in the appendix.

Factoring Fourier Transforms with Zeros in a Strip

$f$ is the Fourier transform of an infinitely differentiable function of compact support on ${\mathbf {R}}$ if, and only if, $f$ is entire and of exponential type with $\left | {f\left ( x \right )} \right | = O\left ( {|x{|^{ - N}}} \right )$ for each $N > 0$ as $|x| \to \infty$ for real $x$. In some sense, such an $f$ has its zeros close to the real axis and has positive density of zeros $F$ with $n\left ( r \right ) = Dr + o\left ( r \right )$. It is shown here that if the zeros of $f$ are in a strip parallel to the real axis and if $n\left ( r \right ) = Dr + O\left ( 1 \right )$, then $f$ is the product of two such transforms with zero densities $D/2$ and indicators one-half of the indicator of $f$. There is a factorable $f$ in $\widehat {\mathcal {D}}\left ( {\mathbf {R}} \right )$ with zeros on a line and not satisfying the stricter density condition. Analogous results hold for transforms of distributions of compact support on ${\mathbf {R}}$. The study was motivated by the outstanding problem of Ehrenpreis that asks if $\mathcal {D}\left ( {\mathbf {R}} \right ) * \mathcal {D}\left ( {\mathbf {R}} \right ) = \mathcal {D}\left ( {\mathbf {R}} \right )$.

Algebras of distributions suitable for phase-space quantum mechanics. I

The twisted product of functions on R2N is extended to a *-algebra of tempered distributions that contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.

Moment sequences for a certain class of distributions

We Present the necessary and sufficient conditions for a sequence to be the moment sequence of a distribution in the space of the distributions of compact support. The analysis is based on a series of lemmas that reduce the problem to that of existence of distributional boundary limits of certain analytic functions defined in the unit disk. These results have many applications. We give an application in the theory of orthogonal polynomials.

The invertibility of rotation invariant Radon transforms

Let R μ denote the Radon transform on R n that integrates a function over hyperplanes in given smooth positive measures μ depending on the hyperplane. We characterize the measures μ for which R μ is rotation invariant. We prove rotation invariant transforms are all one-to-one and hence invertible on the domain of square integrable functions of compact support, L 0 2( R n ). We prove the hole theorem: if f ϵ L 0 2 R n and R μ f = 0 for hyperplanes not intersecting a ball centered at the origin, then f is zero outside of that ball. Using the theory of Fourier integral operators, we extend these results to the domain of distributions of compact support on R n . Our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated Radon transform as well as invertibility for other Radon transforms.

Reconstruction of a three-dimensional object from a limited range of views

We define the Radon transform R and the back projection R ∗ (adjoint of R ) on the space of C ∞ rapidly decreasing functions J and on the space of distributions of compact support E ′. We show that R ∗ R is a convolution operator. The result is true in the case of full views as well as in the case of a limited range of views. An application to the medical image reconstruction is given.

Periodic Gibbs states of ferromagnetic spin systems

We give a complete description of the set of periodic Gibbs states at low temperatures for classical spin systems with arbitrary ferromagnetic, finite-range, interactions and fairly general even single-spin distribution of compact support on R. This extends results of Holsztynski and Slawny for the spin-1/2 case. The extension is based on recent ferromagnetic inequalities and low-temperature expansions.

Low temperature expansion for continuous-spin Ising models

We consider general ferromagnetic spin systems with finite range interactions and an even single-spin distribution of compact support on IR. It is shown under mild assumptions on the single-spin distribution that a low temperature expansion, in powers ofT, for the free energy and the correlation functions is asymptotic. We also prove exponential clustering in the pure phases and analyticity of the free energy and of the correlation functions in the reciprocal temperature β for Re β large.

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 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 .

The Kontorovich–Lebedev transformation on distributions of compact support and its inversion

1. During the past decade a variety of integral transformations have been extended to various classes of distributions. However, one important integral transformation, which has defied such generalization, is the Kontorovich–Lebedev transformation. The objective of this note is to alter that situation.

Measure algebras associated with a second order differential operator

Let L = d 2 dx 2 − q(x) be a Sturm-Liouville operator acting on functions defined on R. The authors have recently shown how to construct commutative associative algebras of distributions of compact support for which L is a centralizer (in the sense that L(f ∗ g) = (Lf) ∗ g for distributions f, g of compact support) when q is locally bounded. Here, it is assumed either that q is bounded and x → (1 + ¦ x ¦) q(x) is integrable, or that q is of bounded variation. A function ψ is then found such that M ψ={μ : μ is a measure on R and | μ |(ψ) < & infin;} becomes a Banach algebra containing the algebra of measures of compact support. The representation theory of M ψ is discussed and conditions for its semisimplicity are obtained.