A characterization of the approximation order of translation invariant spaces of functions

Author:
Rong Qing Jia

Journal:
Proc. Amer. Math. Soc. **111** (1991), 61-70

MSC:
Primary 41A65

MathSciNet review:
1010801

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a space of functions on with the following properties:

(i) is translation invariant, i.e., implies ;

(ii) ;

(iii) is closed under uniform convergence on compact sets.

In this paper we characterize the approximation order of by proving the following:

**Theorem.** *provides approximation of order* *if and only if* *contains a compactly supported function* *such that the Fourier transform* *of* *satisfies* *and* *for* *and* .

This result extends a corresponding result of de Boor and DeVore, who proved the above theorem for the case .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1010801-1

Keywords:
Order of approximation,
translation invariance

Article copyright:
© Copyright 1991
American Mathematical Society