Related papers: Elementary Inversion of Riesz Potentials and Radon…
The windowed ray transform is a natural generalization of the "Analytic-Signal Transform" which is developed to extend arbitrary functions from $\RR^n$ to $\CC^n$. We present several inversion formulas here.
In 1927 Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $\bbr^n$. These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader's formulas to totally geodesic…
The accuracy of reconstruction of a response function from its Lorentz integral transform is studied in an exactly solvable model. An inversion procedure is elaborated in detail and features of the procedure are studied. Unlike results in…
The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from…
In this paper, we study the general orthogonal Radon transform $R_{p,q}^k$ first studied by R.S Strichartz in \cite{Stri}. An sharp existence condition of $R_{p,q}^k f$ on $L^p$-spaces will be given. Then we devote to the relation formulas…
We employ the Riesz transform as a means for describing geometric properties of sets in ${\mathbb{R}}^n$, and study the extent to which they can be used to characterize function spaces defined on said sets. In particular, characterizations…
As a generalization of Riemann-Liouville integral, we introduce integral transformations of convergent power series which can be applied to hypergeometric functions with several variables.
We show how the integral formula of Poisson for holomorphic functions on the right half plane can be used to quickly evaluate certain integrals from the Table of Gradshteyn and Ryzhik. In addition, we prove a version of this formula for…
Riesz potentials are well known objects of study in the theory of singular integrals that have been the subject of recent, increased interest from the numerical analysis community due to their connections with fractional Laplace problems…
The inversion theorem for the k-plane Radon transform in R^n is often stated for Schwartz functions, and lately for smooth functions on R^n fulfilling that f(x)=O(|x|^{-N}) for some N>n. In this paper it will be shown, that it suffices to…
We establish sharp pointwise inequalities for the Riesz potential and its gradient in $\mathbb{R}^{n}$ and indicate their usefulness for potential analysis, moment theory and other applications.
The aim of this paper is to present inversion methods for the classical Radon transform which is defined on a family of $k$ dimensional planes in $\Bbb R^{n}$ where $1\leq k\leq n - 2$. For these values of $k$ the dimension of the set…
We introduce bi-parametric fractional integrals of the Erdelyi-Kober type that generalize known Garding-Gindikin constructions associated to the cone of positive definite matrices. It is proved that the Radon transform, which maps a zonal…
The Lorentz Integral Transform approach allows microscopic calculations of electromagnetic reaction cross sections without explicit knowledge of final state wave functions. The necessary inversion of the transform has to be treated with…
Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT),…
Let $G\subset \C P^n$ be a linearly convex compact with smooth boundary, $D={\C}P^n\setminus G$, and let $D^* \subset (\C P^n)^*$ be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety $V$ of…
We study X-ray and Divergent beam transforms of Trkalian fields and their relation with Radon transform. We make use of four basic mathematical methods of tomography due to Grangeat, Smith, Tuy and Gelfand-Goncharov for an integral…
A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.
For the reconstruction problem, the universal representation of inverse Radon transforms implies the needed complexity of the direct Radon transforms which leads to the additional contributions. In the standard theory of generalized…
The light field reconstruction from the focal stack can be mathematically formulated as an ill-posed integral equation inversion problem. Although the previous research about this problem has made progress both in practice and theory, its…