Some Questions of Approximation Theory

V. V. Volchkov
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V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 2 in Integral Geometry and Convolution Equations, 2003, pp 359-365 from Springer

Abstract: Abstract The classical Wiener theorem on the closure of shifts asserts that linear combinations of shifts of function ψ m ∊ L 1(ℝ1), m = 1, 2,..., are dense in L 1(ℝ1) if and only if there is no point x ∊ ℝ1 at which the Fourier transforms of all the functions ψ m are simultaneously equal to zero (see, for example, [A19], [E4], [L8]). Wiener has also proved the necessary and sufficient conditions for linear span of shifts for given functions in L 2(ℝ1) to be dense in L 2(ℝ1). To date, a number of analogs of Wiener’s theorem on noncompact groups have been obtained (see [E4]). L p analogues of Wiener’s theorem, even in the one dimensional case, when p ≠ 1 or 2 are quite hard. In this section we investigate approximation of functions on open subset $$ \mathcal{U} \subset \mathbb{R}^1 $$ in $$ L^p \left( \mathcal{U} \right), 1 \leqslant p

Keywords: Linear Combination; Bounded Domain; Linear Subspace; Approximation Theory; Linear Span (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-010-0023-9_28

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DOI: 10.1007/978-94-010-0023-9_28

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