Related papers: Numerical reconstruction from the Fourier transfor…
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.…
Recasting the $N$-point one loop scalar integral from Feynman to Schwinger parameters gives an integrand with a Gaussian form. By application of a Fourier transform, it is easy to derive explicit expressions for the two, three and…
We consider holomorphic self-maps $\v$ of the unit ball $\B^N$ in $\C^N$ ($N=1,2,3,...$). In the one-dimensional case, when $\v$ has no fixed points in $\D\defeq \B^1$ and is of hyperbolic type, there is a classical renormalization…
In the present work, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of new monogenic polynomials are provided based on 2-parameters weight functions. Such classes extend the well…
A recurring task in image processing, approximation theory, and the numerical solution of partial differential equations is to reconstruct a piecewise-smooth real-valued function f(x) in multiple dimensions from its truncated Fourier…
We consider Boolean functions f:{-1,1}^n->{-1,1} that are close to a sum of independent functions on mutually exclusive subsets of the variables. We prove that any such function is close to just a single function on a single subset. We also…
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo-…
We introduce a general framework for the reconstruction of periodic multivariate functions from finitely many and possibly noisy linear measurements. The reconstruction task is formulated as a penalized convex optimization problem, taking…
Spatial resolution of most imaging devices is fundamentally restricted by diffraction. This limitation is manifested in the loss of high spatial frequency information contained in evanescent waves. As a result, conventional far-field optics…
Diffraction tomography aims to recover an object's scattering potential from measured wave fields. In the classical setting, the object is illuminated by plane waves from many directions, and the Fourier diffraction theorem provides a…
The spheroidal wave functions, which are the solutions to the Helmholtz equation in spheroidal coordinates, are notoriously difficult to compute. Because of this, practically no programming language comes equipped with the means to compute…
An algorithm for computing eigenvalues and eigenfunctions of the angular spheroidal wave equation, based on a known but scarcely used method, is developed. By requiring the regularity of the wave function, represented by its series…
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the…
Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called…
This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples…
For precise QCD prediction of observables, the ambiguity due to renormalons in perturbative calculations should be appropriately separated from Wilson coefficients in the framework of the operator-product-expansion. Recently, a new method…
The famous Fourier theorem states that, under some restrictions, any periodic function (or real world signal) can be obtained as a sum of sinusoids, and hence, a technique exists for decomposing a signal into its sinusoidal components. From…
Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. Solution of this problem is needed (both in 2-D and…
We study the Beurling-Selberg problem of finding band-limited $L^1$-functions that lie below the indicator function of an Euclidean ball. We compute the critical radius of the support of the Fourier transform for which such construction can…
We propose a scheme for imaging periodic surfaces using a superlens. By employing an inverse scattering model and the transformed field expansion method, we derive an approximate reconstruction formula for the surface profile, assuming…