Reconstructing a function from its values on a subset of its domain - a Hilbert space approach (Q1076924)

This paper, which should be regarded as mainly expository, presents the general features of a reconstruction (or extrapolation) procedure, various cases of which have been discussed in the literature. The main idea is that the restrictions of functions in a suitable Hilbert space H of functions on a set E, to a subset \(E_ 0\), can be recognized and in principle extrapolated to E, in terms of the spectral resolution of a certain self-adjoint operator on H that can be introduced (in the presence of some natural hypotheses). Examples are given. It has been pointed out to the author that the general results can be obtained from the extant theory of solving ''ill posed problems'' by the method of Tikhonov regularization.