n-Widths in Hilbert Spaces

  • Allan Pinkus
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 7)

Abstract

The organization of this chapter is as follows. The most general theorem concerning n-widths in Hilbert spaces is the main content of Section 2. Let T be a compact operator mapping H1 to H2, where both H1 and H2 are Hilbert spaces. Then the Kolmogorov, linear, Gel’fand, and Bernstein n-width of the set
$$A = \left\{ {T\Phi :{{\left\| \Phi \right\|}_{{H_1}}}\underline{\underline < } 1} \right\}$$
as a subset of H2, is simply the (n + 1)st singular value of T and optimal subspaces are easily constructed in terms of eigenvector subspaces. In Section 3, we consider variations on this problem and many examples. The n-widths and optimal subspaces are easily identified if T is given by convolution against a fixed periodic function. This is discussed in Section 4. Sections 5 and 6 are different in nature. In those sections we consider integral operators whose kernels are either totally positive or cyclic variation diminishing (and variants thereof). In these cases we are able to determine additional optimal subspaces of an elementary form.

Keywords

Hull Convolution Dition Suffix Weinstein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Allan Pinkus
    • 1
  1. 1.Department of MathematicsTechnion Israel Institute of TechnologyHaifaIsrael

Personalised recommendations