On nonsmooth solutions of Abel's integral equation (Q2641102)

The authors describe a regularizing scheme for the approximate integral equation \(\int^{t}_{0}u(s)(t-s)^{-\alpha}dx\approx f(t),\quad 0<t<1,\quad 0<\alpha <1.\) In Abel's classical solution formula (for \(=\) instead of \(\approx)\) they replace the first order derivative by a one- sided difference quotient (forward off 1, backward near 1) whose step- length h serves as parameter of regularization. Assuming \(f\in L_ 1\), \(\| f-Au\|_ 1<\epsilon\), \(V(u)<E\) (where V(u) is the total variation of u), \(0<\epsilon <E/4\), they obtain (by suitable choice of h) an approximate solution v and the stability estimate \(\| v-u\|_ 1\leq 4\sqrt{E\epsilon}\).