Sharp bounds for singular values of fractional integral operators

From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380–391] and Vũ and Gorenflo [Kim Tuan Vũ, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Ap...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2007-03, Vol.327 (1), p.251-256
Main Author: Burman, Prabir
Format: Article
Language:English
Subjects:
Eigenvalues
Exact sciences and technology
Fractional difference
Fractional integral operator
Fractional summation
Integral equations
Mathematical analysis
Mathematics
Operator theory
Real functions
Sciences and techniques of general use
Singular values
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Summary:From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380–391] and Vũ and Gorenflo [Kim Tuan Vũ, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology, Ho Chi Minh City, 1995, Ho Chi Minh City Math. Soc., Ho Chi Minh City, 1995, pp. 174–185] it is known that the jth singular value of the fractional integral operator of order α > 0 is approximately ( π j ) − α for all large j. In this note we refine this result by obtaining sharp bounds for the singular values and use these bounds to show that the jth singular value is ( π j ) − α [ 1 + O ( j −1 ) ] .
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2006.03.073