Premium
Polynomial approximation on the sphere using scattered data
Author(s) -
Filbir Frank,
Themistoclakis W.
Publication year - 2008
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710633
Subject(s) - mathematics , bounded function , unit sphere , polynomial , degree of a polynomial , norm (philosophy) , operator (biology) , degree (music) , euclidean space , approximation error , mathematical analysis , minimax approximation algorithm , quadrature (astronomy) , discretization , euclidean geometry , combinatorics , geometry , biochemistry , chemistry , physics , electrical engineering , repressor , political science , transcription factor , acoustics , law , gene , engineering
We consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space ℝ q +1 by using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi‐projection in the de la Vallée Poussin style, i.e. it reproduces spherical polynomials up to a certain degree and has uniformly bounded L p operator norm for 1 ≤ p ≤ ∞. Using certain positive quadrature rules for scattered sites due to Mhaskar, Narcowich and Ward, we discretize this operator obtaining a polynomial approximation of the target function which can be computed from scattered data and provides the same approximation degree of the best polynomial approximation. To establish the error estimates we use Marcinkiewicz–Zygmund inequalities, which we derive from our continuous approximating operator. We give concrete bounds for all constants in the Marcinkiewicz–Zygmund inequalities as well as in the error estimates. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)