BochnersTheorem

Bochner's theorem is a significant result in the field of harmonic analysis and probability theory, and it can be viewed as a consequence of Stone's theorem due to the deep connections between these two theorems in functional analysis and the theory of topological groups.

1. Stone's Theorem: This theorem, formulated by Marshall H. Stone, concerns the representation of one-parameter unitary groups. In its simplest form, it states that every strongly continuous one-parameter unitary group on a Hilbert space is the exponential of a self-adjoint operator. Mathematically, for a strongly continuous group $U(t)$, there exists a self-adjoint operator $A$ such that $U(t) = e^{itA}$. This theorem is a cornerstone in the study of quantum mechanics and functional analysis.

2. Bochner's Theorem: Salomon Bochner formulated this theorem, which characterizes the Fourier transform of a positive definite function on a locally compact abelian group. In essence, Bochner's theorem states that a function $f$ on a group is positive definite if and only if it is the Fourier transform of a positive measure. This theorem is fundamental in the study of stochastic processes and harmonic analysis.

The connection between these two theorems lies in the interplay between operators on Hilbert spaces and the Fourier transform on groups:

• From Stone to Bochner: Stone's theorem deals with unitary operators and their generators in Hilbert spaces. When you apply this theorem to the specific case of L2 spaces (spaces of square-integrable functions), you deal with unitary representations of groups and their infinitesimal generators. Bochner's theorem, which characterizes positive definite functions in terms of measures (and hence in terms of their Fourier transforms), can be derived by considering the implications of Stone's theorem on these unitary representations and their spectral properties.

In summary, Stone's theorem provides a foundational understanding of the behavior of unitary groups in Hilbert spaces, which, when applied to specific cases like L2 spaces and unitary representations, leads to results that encompass the essence of Bochner's theorem, thus establishing it as a consequence.