Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation

In the present paper, we consider the approximation of the solution of an ill-posed spherical pseudo-differential equation at a given point. While the methods for approximating the whole solution are well-studied in Hilbert spaces, such as the space of square-summable functions, the computation of values of the solution at given points is much less studied. This can be explained, in particular, by the fact that for square-summable functions the functional of pointwise evaluation is, in general, not well defined. To overcome this limitation we adjust the regularized least-squares method of An, Chen, Sloan and Womersley [Siam J. Numer. Anal. 50 (2012), no. 3, 1513–1534] by using a special a posteriori parameter choice rule. We also illustrate our theoretical findings by numerical results for the reconstruction of the solution at a given point.