Open Access

Inequalities for differentiable reproducing kernels and an application to positive integral operators

Journal of Inequalities and Applications20062006:53743

DOI: 10.1155/JIA/2006/53743

Received: 18 October 2005

Accepted: 13 November 2005

Published: 3 May 2006

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq1_HTML.gif be an interval and let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq2_HTML.gif be a reproducing kernel on https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq3_HTML.gif . We show that if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq4_HTML.gif is in the appropriate differentiability class, it satisfies a 2-parameter family of inequalities of which the diagonal dominance inequality for reproducing kernels is the 0th order case. We provide an application to integral operators: if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq5_HTML.gif is a positive definite kernel on https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq6_HTML.gif (possibly unbounded) with differentiability class https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq7_HTML.gif and satisfies an extra integrability condition, we show that eigenfunctions are https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq8_HTML.gif and provide a bound for its Sobolev https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F53743/MediaObjects/13660_2005_Article_1608_IEq9_HTML.gif norm. This bound is shown to be optimal.

[12345678]

Authors’ Affiliations

(1)
Departamento de Matemática, Instituto Superior Técnico
(2)
Departamento de Engenharia Mecânica, ISEL

References

  1. Aronszajn N: Theory of reproducing kernels. Transactions of the American Mathematical Society 1950,68(3):337–404. 10.1090/S0002-9947-1950-0051437-7MathSciNetView ArticleMATHGoogle Scholar
  2. Buescu J: Positive integral operators in unbounded domains. Journal of Mathematical Analysis and Applications 2004,296(1):244–255. 10.1016/j.jmaa.2004.04.007MathSciNetView ArticleMATHGoogle Scholar
  3. Buescu J, Garcia F, Lourtie I, Paixão AC: Positive-definiteness, integral equations and Fourier transforms. Journal of Integral Equations and Applications 2004,16(1):33–52. 10.1216/jiea/1181075257MathSciNetView ArticleMATHGoogle Scholar
  4. Buescu J, Paixão AC: Positive definite matrices and integral equations on unbounded domains. Differential and Integral Equations 2006,19(2):189–210.MathSciNetMATHGoogle Scholar
  5. Moore EH: General Analysis. Pt. I, Memoirs of Amer. Philos. Soc.. American Philosophical Society, Pennsylvania; 1935.Google Scholar
  6. Moore EH: General Analysis. Pt. II, Memoirs of Amer. Philos. Soc.. American Philosophical Society, Pennsylvania; 1939.Google Scholar
  7. Riesz F, Nagy B: Functional Analysis. Ungar, New York; 1952.MATHGoogle Scholar
  8. Saitoh S: Theory of Reproducing Kernels and Its Applications, Pitman Research Notes in Mathematics Series. Volume 189. Longman Scientific & Technical, Harlow; 1988.Google Scholar

Copyright

© J. Buescu and A. C. Paix˜ao. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.