In-orbit Vignetting Calibrations of XMM-Newton Telescopes

We describe measurements of the mirror vignetting in the XMM-Newton Observatory made in-orbit, using observations of SNR G21.5-09 and SNR 3C58 with the EPIC imaging cameras. The instrument features that complicate these measurements are briefly described. We show the spatial and energy dependences of measured vignetting, outlining assumptions made in deriving the eventual agreement between simulation and measurement. Alternate methods to confirm these are described, including an assessment of source elongation with off-axis angle, the surface brightness distribution of the diffuse X-ray background, and the consistency of Coma cluster emission at different position angles. A synthesis of these measurements leads to a change in the XMM calibration data base, for the optical axis of two of the three telescopes, by in excess of 1 arcminute. This has a small but measureable effect on the assumed spectral responses of the cameras for on-axis targets.Comment: Accepted by Experimental Astronomy. 26 pages, 18 figure

Pointwise Estimates Formultivariate Interpolation Using Conditionally Positive Definite Functions

. We seek pointwise error estimates for interpolants, on scattered data, constructed using a basis of conditionally positive definite functions of order m, and polynomials of degree not exceeding m-1. Two different approaches to the analysis of such interpolation are considered. The former uses distributions and reproducing kernel ideas, whilst the latter is based on a Lagrange function approach. Error estimates in terms of a point density measure are given for both methods of analysis. 1. Introduction Over recent years it has become clear that radial functions are very useful tools for multivariate approximation. Two radial functions in particular have found favour with practitioners, the multiquadric, h(x) = p 1 + kxk 2 , and the thin plate spline, h(x) = kxk 2 log kxk. In this paper we shall be considering the pointwise convergence of interpolation schemes employing conditionally positive definite (CPD) functions, a class which includes multiquadrics and thin plate splines, a..