Journal of Mathematical Sciences, Vol. 165, No. 4, 2010

APPROXIMATION IN Lp Rd , p , BY LINEAR COMBINATIONS OF CHARACTERISTIC FUNCTIONS OF BALLS

    ∗

  

We prove that translates of the characteristic function of a ball span the space Lp (Rd ) provided that 0 < p < 1 and d ≥ 2. Similar approximation problems are considered for some other functions. Bibliography: 5 titles.

1. Introduction

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Department of the Steklov Mathematical Institute, St.Petersburg, Russia, e-mail: [email protected].

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 366, 2009, pp. 5–12. Original article submitted August 10, 2009. c 2010 Springer Science+Business Media, Inc. 431 1072-3374/10/1654-0431 

2. Main results

  

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1 Here we do not give a detailed proof of this inclusion because in the applications of this lemma which we consider in this paper, the inclusion F f ∈ C ∞ (Rd ) is fulfilled. Then it is clear that g ∈ S(Rd ) ⊂ L1 (Rd , (1 + |x|)N ). 2 Here and in what follows, a progression is viewed as a set rather than a sequence.

432

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