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This opens a possibility of using methods of commutative harmonic analysis to study certain convolution algebras of radial functions on G. The fist, who considered the subalgebra of radial functions in the regular C∗-algebra of G was J. M. Cohen [1]. He proved that it is isomorphic to C ( [−2ω, 2ω]), the algebra of continuous functions on the interval [−2ω, 2ω], where ω = √2k − 1 and k is the number of free generators in G. In the paper we study other such algebras. One of our results is (Theorem 2. 1) that the algebra `r(G) of radial functions in ` 1(G) has the ellipse

Radial functions on free groups and a decomposition of the regular representation into irreducible components.