BochnerHerglotz

BochnerHerglotz refers to a theorem in harmonic analysis, named after Salomon Bochner and Gerhard Herglotz. This theorem establishes a fundamental connection between positive-definite functions and measures on a locally compact abelian group. Specifically, it states that a continuous function on a locally compact abelian group is positive-definite if and only if it is the Fourier transform of a non-negative finite Radon measure on the dual group.

The dual group of a locally compact abelian group G, denoted by Ĝ, is itself a locally compact abelian group. The theorem essentially asserts that positive-definite functions on G correspond to measures on Ĝ, and vice-versa, through the Fourier transform. This duality is a cornerstone of harmonic analysis, allowing for the translation of problems from one domain to another.

The concept of positive-definiteness is crucial. A function f on an abelian group G is positive-definite if for any finite set of elements g_1, ..., g_n in G and any complex numbers c_1, ..., c_n, the sum sum_{i,j} c_i bar{c_j} f(g_i - g_j) is non-negative. This property is analogous to the positive-definiteness of kernels in machine learning and other areas.

The Bochner-Herglotz theorem has far-reaching implications and applications in various fields, including probability theory, quantum mechanics, and signal processing. It provides a powerful tool for understanding the structure of functions and measures on groups and is a key result in the spectral theory of operators.