Convergence of Multilevel Stationary Gaussian Convolution

  • Simon Hubbert
  • Jeremy LevesleyEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


In this paper we give a short note showing convergence rates for periodic approximation of smooth functions by multilevel Gaussian convolution. We will use the Gaussian scaling in the convolution at the finest level as a proxy for degrees of freedom d in the model. We will show that, for functions in the native space of the Gaussian, convergence is of the order \(d^{-\frac {\ln (d)}{\ln (2)}}\). This paper provides a baseline for what should be expected in discrete convolution, which will be the subject of a follow up paper.


  1. 1.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1964)zbMATHGoogle Scholar
  2. 2.
    R.A. Adams, Sobolev Spaces (Academic, New York, 1975)zbMATHGoogle Scholar
  3. 3.
    R. Bracewell, The Fourier Transform and Its Applications (2nd edn.) (McGrawHill, New York, 1986)zbMATHGoogle Scholar
  4. 4.
    E.H. Georgoulis, J. Levesley F. Subhan, Multilevel sparse kernel-based interpolation. SIAM J. Sci. Comput. 35, 815–831 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Usta, J. Levesley, Multilevel quasi-interpolation on a sparse grid with the Gaussian. Numer. Algorithms 77, 793–808 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Birkbeck, University of LondonLondonUK
  2. 2.Department of MathematicsUniversity of LeicesterLeicesterUK

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