Inversion of fractional integrals related to the spherical Radon transform
The author considers the spherical convolution operator \[ T^\alpha f= \gamma_{n,\alpha}\int_{S^n}k(x\cdot\sigma)f(\sigma)d\sigma,\quad x\in S^n \] with the power type kernel of the inner product: \[ k(t)=| t| ^{\alpha -1}, -1<t<1,\quad x\cdot\sigma =x_1\sigma_1+\cdots+ x_{n+1}\sigma_{n+1}, \] \(S_n\) being the unit sphere in \(R^{n+1}\), \(\text{Re }\alpha>0\). The author constructs the analytical continuation of this operator, with respect to the parameter \(\alpha\), to the whole complex plane. The case \(\alpha=0\) corresponds to the Radon transforms. The normalizing constant \(\gamma_{n,\alpha}\) is chosen in such a way that \[ (T^\alpha)^{-1}=T^{1-n-\alpha} \] on nice functions, the latter being a consequence of the known formula for the spherical Fourier multiplier of the operator \(T^\alpha\). The author develops two methods to construct the operator inverse to \(T^\alpha\), that is, to realize the operator \(T^{1-n-\alpha}\). A special attention is paid to the case \(\alpha=0\), corresponding to the Radon transforms.
scientific article; zbMATH DE number 1211641