Integral equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:02:57Z http://mathoverflow.net/feeds/question/108704 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108704/integral-equation Integral equation Alex A 2012-10-03T12:46:54Z 2012-10-19T13:06:33Z <p>Assume (for definiteness) $g:\mathbb{R} \to \mathbb{R}$ is continuous and that $f$ is defined by <code>$$f(E) = \int _0^{E-1} \Big ( (E - t)^2 - 1\Big )^{3/2} g(t) \, dt.$$</code> I'm interested in whether $g$ can be recovered assuming we know $f$. </p> <p>Does anyone know if this type of integrals have been studied before? </p> <p>For instance I am familiar to the fact that (Riemann-Liouville) integrals of the form <code>$$(J^\alpha g)(E) = \frac{1}{\Gamma (\alpha )}\int _0 ^E(E - t)^{\alpha -1}g(t) \, dt$$</code> can be inverted when $\alpha$ is a half-integer by using identities of the form $J^\alpha \circ J^\beta = J^{\alpha + \beta }$ and then differentiate. </p> <p>EDIT: I would just like to point out that I'm not necessarily looking for an explicit inversion formula. If the above equation fits into some general theory which concludes that $g$ can be recovered I'm happy. </p> <p>EDIT II: I have narrowed the problem down into finding $g_0$ (only depending on $t$) with <code>$$\int _1 ^{E-1} \Big ( (E - t)^2 - 1\Big )^{3/2} g_0(t) \, dt = 1, \qquad E&gt;1.$$</code> Not sure whether that helps though. </p> <p>EDIT III: If it helps I actually do know the solution in my particular case is <code>$$g(t) = \int _{\{h^{-1}(t)\}} \frac{1}{|\nabla h|}\,dS$$</code> for some $h$ for which the gradient never vanishes on <code>$\{h^{-1}(t)\}$</code>. Here $dS$ is surface measure. (The reason I still want to solve the equation is that I know $f$ is a certain invariant and I need to show $g$ is also invariant.) </p>