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Numerical Algorithms

, Volume 80, Issue 2, pp 661–685 | Cite as

Truncation dimension for linear problems on multivariate function spaces

  • Aicke Hinrichs
  • Peter KritzerEmail author
  • Friedrich Pillichshammer
  • G. W. Wasilkowski
Original Paper
  • 38 Downloads

Abstract

The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are the following: when is it possible to approximate the solution for the original function of very many variables by the solution for the same function, however with all but the first k variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number k = k(ε) such that the corresponding error is bounded by a given error demand ε? Surprisingly, k(ε) could be very small even for weights with a modest speed of convergence to zero.

Keywords

Multivariate problems Weighted function spaces Truncation algorithms Truncation dimension 

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Notes

Acknowledgments

The authors would like to thank two anonymous referees for their suggestions for improving the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for AnalysisJohannes Kepler University LinzLinzAustria
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria
  3. 3.Department of Financial Mathematics and Applied Number TheoryJohannes Kepler University LinzLinzAustria
  4. 4.Computer Science DepartmentUniversity of KentuckyLexingtonUSA

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