Written by a master mathematical expositor, this classic text reflects the results of the intense period of research and development in the area of Fourier analysis in the decade preceding its first publication in 1962. The enduringly relevant treatment is geared toward advanced undergraduate and graduate students and has served as a fundamental resource for more than five decades. The self-contained text opens with an overview of the basic theorems of Fourier analysis and the structure of locally compact Abelian groups. Subsequent chapters explore idempotent measures, homomorphisms of group algebras, measures and Fourier transforms on thin sets, functions of Fourier transforms, closed ideals in L1 (G), Fourier analysis on ordered groups, and closed subalgebras of L1 (G). Helpful Appendixes contain background information on topology and topological groups, Banach spaces and algebras, and measure theory.