This article by Pierre Eymard describes a method to extend harmonic analysis results from abelian groups to arbitrary locally compact groups. It introduces the Fourier algebra $ B(G) $ and the algebra $ A(G) $, which are Banach algebras associated with a locally compact group $ G $. The Fourier algebra $ B(G) $ is defined as the algebra of complex linear combinations of positive definite continuous functions on $ G $, equipped with the dual norm of the $ C^* $-algebra of $ G $. The algebra $ A(G) $ is a subalgebra of $ B(G) $, consisting of functions with compact support. The paper discusses the properties of these algebras, their duals, and their connections to harmonic analysis on groups. It also presents results on ideals, spectra, and spectral synthesis for these algebras, extending classical theorems from abelian groups to non-abelian ones. The work includes the study of the Gelfand spectrum of $ A(G) $, the characterization of positive functionals, and the relationship between these algebras and von Neumann algebras. The paper concludes with a discussion of the synthesis of spectral functions and the extension of the Wiener-Godement theorem to locally compact groups.This article by Pierre Eymard describes a method to extend harmonic analysis results from abelian groups to arbitrary locally compact groups. It introduces the Fourier algebra $ B(G) $ and the algebra $ A(G) $, which are Banach algebras associated with a locally compact group $ G $. The Fourier algebra $ B(G) $ is defined as the algebra of complex linear combinations of positive definite continuous functions on $ G $, equipped with the dual norm of the $ C^* $-algebra of $ G $. The algebra $ A(G) $ is a subalgebra of $ B(G) $, consisting of functions with compact support. The paper discusses the properties of these algebras, their duals, and their connections to harmonic analysis on groups. It also presents results on ideals, spectra, and spectral synthesis for these algebras, extending classical theorems from abelian groups to non-abelian ones. The work includes the study of the Gelfand spectrum of $ A(G) $, the characterization of positive functionals, and the relationship between these algebras and von Neumann algebras. The paper concludes with a discussion of the synthesis of spectral functions and the extension of the Wiener-Godement theorem to locally compact groups.