Related papers: $L^2-$interpolation with error and size of spectra

Given $iid$ observations from an unknown absolute continuous distribution defined on some domain $\Omega$, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function.…

In 2002 A.\ Hartmann and X.\ Massaneda obtained necessary and sufficient conditions for interpolation sequences for classes of analytic functions in the unit disc such that $\log M(r,f)=O((1-r)^{-\rho})$, $0<r<1$, $\rho \in (0 , +\infty)$,…

This paper deals with probabilistic upper bounds for the error in functional estimation defined on some interpolation and extrapolation designs, when the function to estimate is supposed to be analytic. The error pertaining to the estimate…

In statistical learning theory, interpolation spaces of the form $[\mathrm{L}^2,H]_{\theta,r}$, where $H$ is a reproducing kernel Hilbert space, are in widespread use. So far, however, they are only well understood for fine index $r=2$. We…

Ingrid Carbone introduced the notion of so-called LS-sequences of points, which are obtained by a generalization of Kakutani's interval splitting procedure. Under an appropriate choice of the parameters $L$ and $S$, such sequences have low…

We introduce and analyze a framework for function interpolation using compressed sensing. This framework - which is based on weighted $\ell^1$ minimization - does not require a priori bounds on the expansion tail in either its…

We prove sharp interpolatory estimates between Riesz Transforms and directional Haar projections. We obtain applications to the theory of compensated compactness and prove a conjecture of L. Tartar on semi-continuity of separately convex…

We provide matching upper and lower bounds of order $\sigma^2/\log(d/n)$ for the prediction error of the minimum $\ell_1$-norm interpolator, a.k.a. basis pursuit. Our result is tight up to negligible terms when $d \gg n$, and is the first…

Standard interpolation techniques are implicitly based on the assumption that the signal lies on a homogeneous domain. In this letter, the proposed interpolation method instead exploits prior information about domain inhomogeneity,…

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions…

In this paper we explore the effect of $\delta$-ray emission, fluctuations in th e signal deposition on the detection of charged particles in silicon-based detec tors. We show that these two effects ultimately limit the resolution that can…

The prospects for a determination of the strong coupling constant $\alpha_s$ via scaling violations of fragmentation functions in deeply inelastic scattering are studied. The statistical error in the case of an integrated luminosity of $250…

If two random variables X and A are functionally related via f(X)=A for some strictly monotone continuously differentiable function f:R->R, the distribution of X may easily be computed from the distribution of A.

This expository article explores the vital role of interpolation theory and Lorentz spaces in the rigorous analysis of partial differential equations (PDEs). While classical Lebesgue spaces ($L_{p}$) successfully measure the magnitude of…

We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha\in(1,\frac{3}{2}]$, we prove that for any $\sim|A|^{-1}$-separated set $A\subset[1,2]$ and $\delta=|A|^{-\alpha}$, we…

We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an…

Numerical interpolation techniques are widely employed for calculating large rational functions in scattering amplitude computations. It has been observed in recent years that these rational functions greatly simplify upon partial…

Let $S$ be a sequence of points in ${\mathbb{D}}^{n}.$ Suppose that $S$ is $H^{p}$ interpolating. Then we prove that the sequence $S$ is Carleson, provided that $p>2.$ We also give a sufficient condition, in terms of dual boundedness and…

We present a review of the discrete dipole approximation (DDA), which is a general method to simulate light scattering by arbitrarily shaped particles. We put the method in historical context and discuss recent developments, taking the…

The main result in this paper is an error estimate for interpolation biharmonic polysplines in an annulus $A\left( r_{1},r_{N}\right) $, with respect to a partition by concentric annular domains $A\left( r_{1} ,r_{2}\right) ,$ ....,…