Bochner's theorem

In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.) Bochner's theorem for a locally compact abelian group G, with dual group , says the following: Theorem For any normalized continuous positive-definite function f on G (normalization here means that f is 1 at the unit of G), there exists a unique probability measure μ on such that i.e. f is the Fourier transform of a unique probability measure μ on . Conversely, the Fourier transform of a probability measure on is necessarily a normalized continuous positive-definite function f on G. This is in fact a one-to-one correspondence. The Gelfand–Fourier transform is an isomorphism between the group C*-algebra C*(G) and C0(Ĝ). The theorem is essentially the dual statement for states of the two abelian C*-algebras. The proof of the theorem passes through vector states on strongly continuous unitary representations of G (the proof in fact shows that every normalized continuous positive-definite function must be of this form). Given a normalized continuous positive-definite function f on G, one can construct a strongly continuous unitary representation of G in a natural way: Let F0(G) be the family of complex-valued functions on G with finite support, i.e. h(g) = 0 for all but finitely many g. The positive-definite kernel K(g1, g2) = f(g1 − g2) induces a (possibly degenerate) inner product on F0(G). Quotiening out degeneracy and taking the completion gives a Hilbert space whose typical element is an equivalence class [h]. For a fixed g in G, the "shift operator" Ug defined by (Ug)(h) (g') = h(g − g), for a representative of [h], is unitary.