# Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals

Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of ℝ. If k belongs to class [InlineMediaObject not available: see fulltext.] defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C 1 case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals

, Volume 10 (4) – Jul 11, 2006
20 pages

/lp/springer_journal/eigenvalues-of-positive-definite-integral-operators-on-unbounded-x8rI02eGOn
Publisher
Birkhäuser-Verlag
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-0040-z
Publisher site
See Article on Publisher Site

### Abstract

Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of ℝ. If k belongs to class [InlineMediaObject not available: see fulltext.] defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C 1 case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out.

### Journal

PositivitySpringer Journals

Published: Jul 11, 2006

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