Jeff Ledford

Longwood University | Longwood · Department of Mathematics and Computer Science

The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by Peyerimhoff (J. London Math. Soc. (2) 66: 753-768, 2002). We show that this symmetrization preserves volume in every dimension, and convexity in th...

This note concerns the finite interpolation problem with two parametrized families of splines related to polynomial spline interpolation. We address the questions of uniqueness and establish basic convergence rates for splines of the form $ s_\alpha = p\cosh(\alpha\cdot)+q\sinh(\alpha \cdot)$ and $t_\alpha = p+q\tanh(\alpha \cdot) $ between the nod...

This article is concerned with the problem of placing seven or eight points on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ so that the surface area of the convex hull of the points is maximized. In each case, the solution is given for convex hulls with congruent isosceles or congruent equilateral triangular facets.

Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A Littlewood type inequality is shown which represents an $L_p$ interpolation of the volume and surface area maximizers in the sphere. A variety of coroll...

This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees of the polynomials.

This note mainly concerns the binomial power function, defined as $(1+x^q)^{r}$. We construct systems of polynomials related to non-local approximation, which allows us to establish the density results on $C[a,b]$, where $a,b\in\mathbb{R}$. As a corollary, we show that scattered translated of power functions and certain related functions are dense...

This article pertains to interpolation of Sobolev functions at shrinking lattices \(h\mathbb {Z}^{d}\) from Lp shift-invariant spaces associated with cardinal functions related to general multiquadrics, ϕα, c(x) := (|x|² + c²)α. The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadr...

In this note, we discuss solutions of differential equation \((D^2-\alpha ^2)^{k}u=0\) on \(\mathbb {R}\setminus \mathbb {Z}\), which we call polyhyperbolic splines. We develop the fundamental function of interpolation and prove various properties related to these splines.

This article discusses the structure of various shift-invariant spaces of cardinal functions generated by a single kernel, and their role in some approximate sampling methods. Particularly, conditions on the generating kernel φ are given which imply that the associated Lp shift-invariant space coincides with the shift-invariant space for the cardin...

The structure of certain types of quasi shift-invariant spaces, which take the form $V(\psi,\mathcal{X}):=\overline{\text{span}}^{L_2}\{\psi(\cdot-x_j):j\in\mathbb{Z}\}$ for a discrete set $\mathcal{X}=(x_j)\subset\mathbb{R}$ is investigated. Additionally, the relation is explored between pairs $(\psi,\mathcal{X})$ and $(\phi,\mathcal{Y})$ such tha...

This paper studies the cardinal interpolation operators associated with the general multiquadrics, $\phi_{\alpha,c}(x) = (\|x\|^2+c^2)^\alpha$, $x\in\mathbb{R}^d$. These operators take the form $$\mathscr{I}_{\alpha,c}\mathbf{y}(x) = \sum_{j\in\mathbb{Z}^d}y_jL_{\alpha,c}(x-j),\quad\mathbf{y}=(y_j)_{j\in\mathbb{Z}^d},\quad x\in\mathbb{R}^d,$$ where...

This article studies sufficient conditions on families of approximating kernels which provide $N$--term approximation errors from an associated nonlinear approximation space which match the best known orders of $N$--term wavelet expansion. These conditions provide a framework which encompasses some notable approximation kernels including splines, s...

In this note we discuss solutions of differential equation $(D^2-\alpha^2)^{k}u=0$ on $\mathbb{R}\setminus\mathbb{Z}$, which we call hyperbolic splines. We develop the fundamental function of interpolation and prove various properties related to these splines.

In this short note we show that functions in the modulation space $\mathscr{F}W=\{ f: \sum_{j\in\mathbb{Z}^n}\| \hat{f}(\cdot+2\pi j)\|_{L_\infty([-\pi,\pi]^n)}<\infty \}$ enjoy similar recovery properties as band-limited functions. If $\{\phi_\alpha\}$ is a regular family of cardinal interpolators, then one can build an approximand of $f$ using th...

In this short note, we investigate the relationship between so-called regular families of cardinal interpolators and multiresolution analyses. We focus our studies on examples of regular families of cardinal interpolators whose Fourier transform is unbounded at the origin. In particular, we show that when this is the case there is a multiresolution...

This paper continues the study of interpolation operators on scattered data. We introduce the Poisson interpolation operator and prove various properties of this operator. The main result concerns functions whose Fourier transforms are concentrated near the origin, specifically functions belonging to the Paley-Wiener space PWB beta. We show that on...

This note concerns density properties of the general multiquadric, (x 2 +c 2 ) k-1/2 , where k is a fixed natural number. We establish that scattered translates of the general multiquadric are dense in C([a,b]), where a and b are finite. As a corollary, we show that scattered translated of the general multiquadric are dense in the function spaces L...

If $f\in \{f\in L^p(\mathbb{R}): f(x)=\int_{-\pi}^{\pi}e^{ix\xi}d\beta(\xi), \beta\in B.V.([-\pi,\pi]) \}$, then $f$ is determined by its samples on the integers by taking an appropriate limit. Specifically, $\| f - L_{\phi_\alpha}f \|_{L^p(\mathbb{R})}\to 0$ as $\alpha\to\infty$ provided that $\{\phi_\alpha: \alpha\in A\}$ is what we call a spline...

In this paper we show that functions from the Paley-Wiener amalgam space $(PW,l^1)=\{f\in L^2(\mathbb{R}): \sum\|\hat{f}(\xi+2\pi m) \|_{L^2([-\pi,\pi])} < \infty\}$ enjoy similar recovery properties as the classical Paley-Wiener space. Specifically, if $\{\phi_\alpha(x): \alpha\in A\}$ is a regular family of interpolators and $\{x_n: n\in \mathbb{...

In this note a general way to develop a cardinal interpolant for $l^2$-data on the integer lattice $Z^n$ is shown. Further, a parameter is introduced which allows one to recover the original Paley-Weiner function from which the data came.

This article shows that on a closed interval $[a,b]$ a continuous function may be approximated to an arbitrary degree of accuracy using scattered translates of the general multiquadric $(x^2+c^2)^{k-1/2}$.

In both Yu. Lyubarskii and W. R. Madych [J. Funct. Anal. 125, No. 1, 201–222 (1994; Zbl 0873.41012)], and Th. Schlumprecht and N. Sivakumar [J. Approx. Theory 159, No. 1, 128–153 (2009; Zbl 1166.42017)], it was shown that Paley-Wiener functions may be recovered from their values on a complete interpolating sequence. This paper explores the same phe...

The goal of this note is to show that continuous functions may be approximated using scattered translates of the Poisson kernel.

The goal of this work is to extend the results of [4] and [7]. Both of these works focus on specific examples which allow on to reconstruct a function from the Paley Wiener class given its samples on a complete interpolating sequence. A theorem is proved which allows us to generalize the work in these two papers. New examples are given after the pr...