Entropy numbers of Sobolev embeddings of radial Besov spaces

  • Authors:

  • Thomas Kühn;Hans-Gerd Leopold;Winfried Sickel;Leszek Skrzypczak

  • Affiliations:

  • Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany;Mathematisches Institut, Friedrich-Schiller-Universität, Ernst-Abbe-Platz 1-4, D-07743 Jena, Germany;Mathematisches Institut, Friedrich-Schiller-Universität, Ernst-Abbe-Platz 1-4, D-07743 Jena, Germany;Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48-49, 60-769 Poznan, Poland

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2003

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Abstract

Let RBp,qs(Rn) be the radial subspace of the Besov space Bp,qs(Rn). We prove the independence of the asymptotic behavior of the entropy numbers ek(id : RBp0,q0s0(Rn) ↦ RBs1p1,q1(Rn) from the difference s0 - s1 as long as the embedding itself RBs0p0,q0(Rn) → RBs1p1,q1(Rn) is compact. In fact, we shall show that ek(id : RBs0p0,q0(Rn) ↦ RBs1p1,q1 (Rn) ∼ k-n(1/p0-1/p1) This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces Bsp,q(Ω) on bounded domains Ω.