Quadrature formula for computed tomography

AbstractWe give a bivariate analog of the Micchelli–Rivlin quadrature for computing the integral of a function over the unit disk using its Radon projections

On oscillating polynomials

AbstractExtremal problems of Markov type are studied, concerning maximization of a local extremum of the derivative in the class of real polynomials of bounded uniform norm and with maximal number of zeros in [−1,1]. We prove that if a symmetric polynomial f, with all its zeros in [−1,1], attains its maximal absolute value at the end-points, then |f′| attains maximal value at the end-points too. As an application of the method developed here, we show that the classic Zolotarev polynomials have maximal derivative at one of the end-points

Surface approximation by piece-wise harmonic functions

We describe here a new method for surface approximation on the basis of given values at a regular grid. The resulting approximant is a continuous piece-wise harmonic function

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinite norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples

Reconstruction from Radon projections and orthogonal expansion on a ball

The relation between Radon transform and orthogonal expansions of a function on the unit ball in \RR^d is exploited. A compact formula for the partial sums of the expansion is given in terms of the Radon transform, which leads to algorithms for image reconstruction from Radon data. The relation between orthogonal expansion and the singular value decomposition of the Radon transform is also exploited.Comment: 15 page

The Atacama Cosmology Telescope (ACT): Beam Profiles and First SZ Cluster Maps

The Atacama Cosmology Telescope (ACT) is currently observing the cosmic microwave background with arcminute resolution at 148 GHz, 218 GHz, and 277 GHz. In this paper, we present ACT's first results. Data have been analyzed using a maximum-likelihood map-making method which uses B-splines to model and remove the atmospheric signal. It has been used to make high-precision beam maps from which we determine the experiment's window functions. This beam information directly impacts all subsequent analyses of the data. We also used the method to map a sample of galaxy clusters via the Sunyaev-Zel'dovich (SZ) effect, and show five clusters previously detected with X-ray or SZ observations. We provide integrated Compton-y measurements for each cluster. Of particular interest is our detection of the z = 0.44 component of A3128 and our current non-detection of the low-redshift part, providing strong evidence that the further cluster is more massive as suggested by X-ray measurements. This is a compelling example of the redshift-independent mass selection of the SZ effect.Comment: 16 pages, 10 figures. Accepted for publication in ApJS. See Marriage et al. (arXiv:1010.1065) and Menanteau et al. (arXiv:1006.5126) for additional cluster result