BJC Baxter

EM Stein

EW Cheney

J Levesley

English

Levesley, Jeremy

Hubbert, Simon

Name not available

An open problem concerning Fourier transforms of radial functions in Euclidean space and on spheres

Birkbeck Institutional Research Online

  Birkbeck ePrints: an open access repository of the research output of Birkbeck College  http://eprints.bbk.ac.uk  Hubbert, S. and Levesley, J., (2001). An open problem concerning Fourier transforms of radial functions in Euclidean space and on spheres. In W. Haussmann; K. Jetter and M. Reimer, eds. Recent Progress in Multivariate Approximation. 4th International Conference, September 2000, Witten-Bommerholz. International Series of Numerical Mathematics, 137. Basel: Birkhäuser, pp. 225-226.      This is an author-produced version of a paper published in Recent Progress in Multivariate Approximation (ISBN 3764365056). This version has been peer-reviewed but does not include the final publisher proof corrections, published layout or pagination. All articles available through Birkbeck ePrints are protected by intellectual property law, including copyright law. Any use made of the contents should comply with the relevant law.   Citation for this version: Hubbert, S. and Levesley, J., (2001). An open problem concerning Fourier transforms of radial functions in Euclidean space and on spheres. London: Birkbeck ePrints. Available at: http://eprints.bbk.ac.uk/archive/00000397  Citation for the publisher’s version:  Hubbert, S. and Levesley, J., (2001). An open problem concerning Fourier transforms of radial functions in Euclidean space and on spheres. In W. Haussmann; K. Jetter and M. Reimer, eds. Recent Progress in Multivariate Approximation. 4th International Conference, September 2000, Witten-Bommerholz. International Series of Numerical Mathematics, 137. Basel: Birkhäuser, pp. 225-226.    http://eprints.bbk.ac.ukContact Birkbeck ePrints at lib-eprints@bbk.ac.uk Open ProblemJ Levesley and S HubbertLet P n be the Gegenbauer polynomials orthogoanl on    with respectto the weight   t  normalised byP n 	n  		n Let  be continuous and  mbe completely monotone on  for somem  IN Let x  kxk have a generalised Fourier transform withpolynomial decay that is  Okk d for some    and d    Then the restriction of  to the spherex y   q  xTyhas a representation as a spherical Fourier seriesx y XkdkXlckYklxYklywhose spherical Fourier coecients fckg have the analagous decay rateck Ok d Here fYklg l       dk k        form an orthonormal basis for thespherical harmonics of degree k which has dimension dk

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Open Problem Concerning Fourier Transforms of Radial Functions in Euclidean Space and on Spheres

An open problem concerning Fourier transforms of radial functions in Euclidean space and on spheres

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