Asymptotic behavior of the singular values of fractional integral operators (Q1260817)

The author considers the convolution operators \(I^ \alpha f(x)=1/\Gamma(\alpha)\int^ x_ 0(x-y)^{\alpha-1}f(y)dy\) in the space \(L^ 2(0,1)\) for \(0<\alpha<1\). The following main results are proved: (i) If \(0<\alpha\leq 1/2\) then the singular numbers \(s_ n=s_ n(I^ \alpha)\) have the asymptotics \(\lim n^ \alpha s_ n=\pi^{-\alpha}\) as \(n\to\infty\). (ii) If \(1/2<\alpha<1\) then \(\pi^{- \alpha}/2\leq\varliminf n^ \alpha s_ n\leq\varlimsup n^ \alpha s_ n\leq e^ \alpha\pi^{-\alpha}\).