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.
Positivity – Springer Journals
Published: Jul 11, 2006
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