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 V˜u and Gorenflo [Kim Tuan V˜u, 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 j th singular value of the fractional integral operator of order α >0 is approximately (πj)−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 j th singular value is (πj)−α[1+O(j−1)]. © 2006 Elsevier Inc. All rights reserved