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SAT 6 (2011), 2
Surveys in Approximation Theory, 6 (2011), 1-23.
On the Power of Function Values for the Approximation
Problem in Various Settings
Erich Novak and Henryk Woźniakowski
Abstract. This is an expository paper on
approximating functions from general Hilbert or Banach spaces in the worst
case, average case and randomized settings with error measured in the
Lp
sense. We define the power function as the ratio between the best
rate of convergence of algorithms that use function values over the best
rate of convergence of algorithms that use arbitrary linear functionals for
a worst possible Hilbert or Banach space for which the problem of
approximating functions is well defined. Obviously, the power function takes
values at most one. If these values are one or close to one than the power
of function values is the same or almost the same as the power of arbitrary
linear functionals. We summarize and supply a few new estimates on the power
function. We also indicate eight open problems related to the power function
since this function has not yet been studied in many cases. We believe that
the open problems will be of interest
to a general audience of mathematicians.
E-print: arXiv:1011.3682
Published: 2 June 2011.
Erich Novak
Mathematisches Institut
Universität Jena
Ernst-Abbe-Platz 2, 07740 Jena, Germany
E-mail: novak@mathematik.uni-jena.de
Homepage: http://users.minet.uni-jena/~novak
Henryk Woźniakowski
Department of Computer Science, Columbia
University
New York, NY 10027, USA, and
Institute of Applied Mathematics, University of Warsaw
ul. Banacha 2,
02-097 Warszawa, Poland
E-mail: henryk@cs.columbia.edu
Homepage: http://www.cs.columbia.edu/~henryk