The interval G=(−1 ,1) turns into a Lie group under the group operation x ◦ y : = ( x + y ) ( 1 + x y ) − 1 , x , y ∈ G . This enables definition of the invariant measure d G ( x ) : = ( 1 − x 2 ) − 1 d x and the Fourier transformation F _G on the interval G and, as a consequence, we can consider Fourier convolution operators W G , a 0 : = F G − 1 a F G on G. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative G u ( x ) = ( 1 − x 2 ) u ′ ( x ) , x∈ G. Equations are solved in the scale of Bessel potential H p s ( G , d G ( x ) ) , 1⩽ p⩽∞, and Hölder-Zygmound Z ν ( G ) , 0 ∞ spaces, adapted to the group G. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a( ξ), ξ∈R, of a convolution equation W G , a 0 u = f defines solvability: the equation is uniquely solvable if and only if the symbol a is elliptic. The solution is written explicitly with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Lie group G n .