Related papers: Wiener-Hopf difference equations and semi-cardinal…

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be a maximal subdiagonal subalgebra of $\mathcal{M}$. We have proved that for $0< p<1$, $H^p(\mathcal{A})$ is…

The invertibility of Wiener-Hopf plus Hankel operators $W(a)+H(b)$ acting on the spaces $L^p(\mathbb{R}^+)$, $1 < p<\infty$ is studied. If $a$ and $b$ belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition…

We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…

We develop a unified H\"older Lebesgue scale \(X^p\) and its weighted, higher order variants \(X^{k,p,a}\) to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two…

This expository thesis contains a study of four interpolation theorems, the requisite background material, and a few applications. The materials introduced in the first three sections of Chapter 1 are used to motivate and prove the…

In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more…

Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the…

A proof will be presented that the existence of a non-trivial $\Sigma_1$-elementary embedding $j: V_{\lambda+3} \prec V_{\lambda+3}$ is inconsistent with $\textsf{ZF}$. Sections 1 and 2 shall review various important contributions from the…

We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincar\'e-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat…

We investigate an interpolation/extrapolation method that, given scattered observations of the Fourier transform, approximates its inverse. The interpolation algorithm takes advantage of modelling the available data via a shape-driven…

In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes $L_p$ Sobolev error estimates and shows that the error is controlled by the $L_p$ multiplier norm of a Fourier…

This paper introduces the fractal interpolation problem defined over domains with a nonlinear partition. This setting generalizes known methodologies regarding fractal functions and provides a new holistic approach to fractal interpolation.…

This article considers Whittaker's function $W_{\kappa ,\mu }$ where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu -1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linear…

Scattering of a time harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so called zero-offset case of scattering due to a…

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem…

We extend the analysis of hep-th/0408069 on a Lorentz invariant interpretation of noncommutative spacetime to field theories on non-anticommutative superspace with half the supersymmetries broken. By defining a Drinfeld-twisted Hopf…

Computing electron-defect (e-d) interactions from first principles has remained impractical due to computational cost. Here we develop an interpolation scheme based on maximally localized Wannier functions (WFs) to efficiently compute e-d…

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel'd element differs from the value that it takes on the Drinfel'd element itself at most by a…

We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf)…

Let $V$ be a symmetric space over a connected reductive Lie algebra $G$, with Lie algebra $\mathfrak{g}$ and discriminant $\delta\in \mathbb{C}[V]$. A fundamental object is the invariant holonomic system $\mathcal{G} =\mathcal{D}(V)\Big/…