  The interval $j=[-1,1]$ turns into an Abelian group $\cA(\cJ)$ under the
group operation $x+_\cJ y:=(x+y)(1+xy)^{-1},\qquad x,y\in\cJ$. This enables
definition of the invariant measure $d_\cJ x=(1-x^2)^{-1}dx$ and the Fourier
transform $\cF_\cJ$ on the interval $\cJ$ and, as a consequence, we can
consider Fourier convolution operators $W^0_{\cJ,\cA}:=\cF_\cJ^{-1}\cA\cF_\cJ$
on $\cJ$. This class of convolutions includes celebrated Prandtl, Tricomi and
Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary
order with the natural weighted derivative $\fD_\cJ u(x)=-(1-x^2)u'(x)$,
$t\in\cJ$. Equations are solved in the scale of Bessel potential
$\bH^s_p(\cJ,d_\cJ x)$, $1\leqslant p\leqslant\infty$, and H\"older-Zygmound
$\bZ^\nu(\cJ,(1-x^2)^\mu)$, $0<\mu,\nu<\infty$ spaces, adapted to the group
$\cA(\cJ)$. Boundedness of convolution operators (the problem of multipliers)
is discussed. The symbol $\cA(\xi)$, $\xi\in\bR$, of a convolution equation
$W^0_{\cJ,\cA}u=f$ defines solvability: the equation is uniquely solvable if
and only if the symbol $\cA$ is elliptic. The solution is written explicitely
with the help of the inverse symbol.
  We touch shortly the multidimensional analogue-the Abelian group
$\cA(\cJ^n)$.
