Abstract
Polynomials are not as stiff as they are sometimes thought to be. We report on a positive summation method for Appell series with favourable localisation properties, which we gain from the Newman—Shapiro operators which belong to a properly chosen hypersphere. Moreover, the Appell approximants can be calculated from the image under the Radon transform of the approximated function. We discuss the complexity and the stability problem of the resulting reconstruction method for tomography.
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References
P. Appell, J. Kampé de Fériet: Fonctions Hypergéometriques et Hypersphériques. Polynomes d’Hermite, Gauthier—Villars, Paris, 1926.
M. E. Davison, F. A. Grünbaum: Tomographic reconstruction with arbitrary directions, Comm. Pure Appl. Math. 34 (1983), 428–448.
E. Kogbetliantz: Recherches sur la sommabilité de séries ultra-sphériques par la méthode de moyennes arithmétiques, J. Math. Pure Appl. Ser. 9, 3 (1924), 107–187.
U. Maier: Tomographic reconstruction using Cesàro means and Newman—Shapiro operators. In: Algorithms for Approximation IV, J. Levesley, I. J. Anderson, J. C. Mason (eds.), 478–485, University of Huddersfield 2002.
D. J. Newman, H. S. Shapiro: Jackson’s theorem in higher dimensions. In: Über Approximationstheorie, 208–219, Birkhäuser, Basel 1964.
M. Reimer: Discretized Newman—Shapiro operators and Jackson’s inequality on the sphere, Result. Math. 36 (1999), 331–341.
M. Reimer: Hyperinterpolation on the sphere at the minimal projection order, J. Approx. Theory 104 (2000), 272–286.
M. Reimer: Generalized hyperinterpolation on the sphere and the Newman—Shapiro operators, Constr. Approx. 18 (2002), 183–204.
M. Reimer Multivariate Polynomials in Approximation, to appear in ISNM, Birkhäuser, Basel.
I. H. Sloan, R. S. Womersley: Constructive polynomial approximation on the sphere, J. Approx. Theory 103 (2000), 91–98.
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© 2003 Springer Basel AG
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Reimer, M. (2003). Approximation of Density Functions and Reconstruction of the Approximant. In: Haussmann, W., Jetter, K., Reimer, M., Stöckler, J. (eds) Modern Developments in Multivariate Approximation. International Series of Numerical Mathematics, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8067-1_14
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DOI: https://doi.org/10.1007/978-3-0348-8067-1_14
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9427-2
Online ISBN: 978-3-0348-8067-1
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