Nonstandard Methods in Commutative Harmonic Analysis

This monograph discusses nonstandard analysis (NSA) and its applications to harmonic analysis on locally compact Abelian (LCA) groups. A new notion of approximation of topological groups by finite groups is introduced and investigated. Based on this notion, new results are obtained on convergence of finite Fourier transformations (FT) to the FT on an LCA group. These results, formulated in standard terms in the Introduction, are proved by means of NSA. The book also includes new results on the theory of relatively standard elements and extensions of results of $S$-integrable liftings in Loeb measure spaces to the case of $\sigma$-finite Loeb measures. Basic concepts of NSA are included.