We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups G; (ii) In case of Lie groups, representations of
the associated Lie algebras La(G) by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
HF​.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form, the problem becomes an operator extension problem, referring to operators
in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we
have stressed hands-on-examples. Extensions are almost never unique, and so we
deal with both the question of existence, and if there are extensions, how they
relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text
overlap with arXiv:1401.478