# An extension of Mercer’s theory to L p

An extension of Mercer’s theory to L p Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ν and $${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer’s theory for K and $${\mathcal{K}}$$ under the assumption that the function $${x\in X\to K(x,x)}$$ belongs to some L p/2(X, ν), p ≥ 1. In particular, we obtain series representations for K and some powers of $${\mathcal{K}}$$ , with convergence in the p-mean, and show that the range of certain powers of $${\mathcal{K}}$$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of $${\mathcal{K}}$$ acting on L p (X, ν). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# An extension of Mercer’s theory to L p

, Volume 16 (2) – Mar 16, 2011
16 pages

Loading next page...

/lp/springer_journal/an-extension-of-mercer-s-theory-to-l-p-aYYTLAkQJy
Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Basel AG
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Fourier Analysis; Operator Theory; Econometrics; Potential Theory
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0118-8
Publisher site
See Article on Publisher Site

### Abstract

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ν and $${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer’s theory for K and $${\mathcal{K}}$$ under the assumption that the function $${x\in X\to K(x,x)}$$ belongs to some L p/2(X, ν), p ≥ 1. In particular, we obtain series representations for K and some powers of $${\mathcal{K}}$$ , with convergence in the p-mean, and show that the range of certain powers of $${\mathcal{K}}$$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of $${\mathcal{K}}$$ acting on L p (X, ν).

### Journal

PositivitySpringer Journals

Published: Mar 16, 2011

### References

• On the nuclearity of integral operators
Ferreira, J.C.; Menegatto, V.A.; Oliveira, C.P.

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off