Related papers: Comparison Problems for Radon Transforms

A correction is brought to the opinion expressed in a previous note published in this journal that the off critical line points indicated by some authors as being non trivial zeros of the Davenport and Heilbronn function are affected of…

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form,…

The sonar transform in geometric tomography maps functions on the Euclidean half-space to integrals of those functions over hemispheres centered on the boundary hyperplane. We obtain sharp $L^p$-$L^q$ estimates for this transform and new…

If $G$ is a finite group, is a function $f:G\to\mathbb C$ determined by its sums over all cosets of cyclic subgroups of $G$? In other words, is the Radon transform on $G$ injective? This inverse problem is a discrete analogue of asking…

The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$…

In an earlier paper, we studied solutions g to convolution equations of the form a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+...+a_1*g+a_0=0, where a_0, ..., a_d are given arithmetic functions associated with Dirichlet series which converge on some right…

Motivated by the testing condition for Radon-Brascamp-Lieb multilinear functionals established in arXiv:2201.12201, this paper is concerned with identifying local conditions on smooth maps $u(t)$ with values in the space of decomposable…

Let f(x) belong to L^p(R^n) and R>0. The transform is considered that integrates the function f over (almost) all spheres of radius R in R^n. This operator is known to be non-injective (as one can see by taking Fourier transform). However,…

Let $G(p,n)$ and $G(q,n)$ be the affine Grassmann manifolds of $p$- and $q$- planes in ${\mathbb R}^n$, respectively, and let $\mathcal{R}^{(p,q)}$ be the Radon transform from smooth functions on $G(p,n)$ to smooth functions on $G(q,n)$…

Let $G_{n,k}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. For $1\le k < k^\prime \le n-1$ we define…

The symmetry breaking is obtained for Neumann problems driven by $p$-Laplacian in certain non-convex cones. These problems are generated by the Hardy--Sobolev inequalities. In the case of the Sobolev inequality for the ordinary Laplacian…

We prove sharp $L^p$ regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated to a canonical relation with fold and blowdown singularities. The proof relies on…

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms.…

The generalized spherical Radon transform associates the mean values over spherical tori to a function $f$ defined on $\mathbb{S}^3 \subset \mathbb{H}$, where the elements of $\mathbb{S}^3$ are considered as quaternions representing…

This paper explores possibilities for new uses of the Radon transform for imaging and analysis of lightfields. We show that the previously reported Dimansionality Gap can be derived from an ultrahyperbolic PDE, first proposed by F. John,…

Let $G_{n,r}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine…

In recent years, Radon type transforms that integrate functions over various sets of ellipses/ellipsoids have been considered in SAR, ultrasound reflection tomography, and radio tomography. In this paper, we consider the transform that…

The "proton radius puzzle" was recently solved by reducing the four-standard deviation discrepancy between the results for electronic hydrogen ($H$) and muonic hydrogen ($\mu H$) atoms to $3.3$ value. The value of the root-mean-square…

The image reconstruction problem consists in finding an approximation of a function f starting from its Radon transform Rf. This problem arises in the ambit of medical imaging when one tries to reconstruct the internal structure of the…

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos(kt-\frac{\beta\pi}{2})\varphi(x-t)dt$,…