n-Widths in Approximation Theory pp 63-137 | Cite as

*n*-Widths in Hilbert Spaces

Chapter

## Abstract

The organization of this chapter is as follows. The most general theorem concerning
as a subset of

*n*-widths in Hilbert spaces is the main content of Section 2. Let*T*be a compact operator mapping*H*_{1}to*H*_{2}, where both*H*_{1}and*H*_{2}are Hilbert spaces. Then the Kolmogorov, linear, Gel’fand, and Bernstein*n*-width of the set$$A = \left\{ {T\Phi :{{\left\| \Phi \right\|}_{{H_1}}}\underline{\underline < } 1} \right\}$$

*H*_{2}, 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

Hilbert Space Orthogonal Projection Compact Operator Positive Operator Dimensional Subspace
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 1985