Using the representation theory of positive definite sequences some propositions in additive number theory are obtained and H. Bohr's approximation theorem is deduced. A unified approach to theorems by S. Bochner, S, N, Bernstein and H. Hamburger is discussed and some operator versions of numerical moment problems are studied. The Appendix contains comments to J. von Neumann's spectral theorem for self-adjoint operators in Hilbert space. / Thesis / Doctor of Philosophy (PhD)