zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The covering number in learning theory. (English) Zbl 1016.68044
Summary: The covering number of a ball of a reproducing kernel Hilbert space as a subset of the continuous function space plays an important role in learning theory. We give estimates for this covering number by means of the regularity of the Mercer kernel K. For convolution type kernels K(x,t)=k(x-t) on [0,1] n , we provide estimates depending on the decay of k ^, the Fourier transform of k. In particular, when k ^ decays exponentially, our estimate for this covering number is better than all the previous results and covers many important Mercer kernels. A counter example is presented to show that the eigenfunctions of the Hilbert-Schmidt operator L K associated with a Mercer kernel K may not be uniformly bounded. Hence some previous methods used for estimating the covering number in learning theory are not valid. We also provide an example of a Mercer kernel to show that L K 1/2 may not be generated by a Mercer kernel.

68Q32Computational learning theory