Solution of a finite convolution equation with a Hankel kernel by matrix factorization

The following problem is considered. The convolution f of a function g supported in the interval [-1, 1] with the function H-0(k\.\) is known on [-1, 1]. An expression for g is searched for. It is shown that the problem of continuing the convolution f from the interval [-1, 1] to the whole real axis in a consistent way is equivalent to solving a certain Hilbert boundary-value problem for two unknown functions. This Hilbert boundary-value problem differs essentially from the corresponding one from the modern theory of finite convolution equations in Sobolev spaces (cf. [B. V. Pal'cev, Math. USSR Sb., 41 (1982), pp. 289-328]), which has not yet been factored, in that it is set up in the original space, not in the range of the Fourier transform operator, and it does not contain the full convolution kernel but only its asymptotics at infinity. It is shown that a factorization for this problem can be given in terms of solutions of a certain singular algebraic ordinary differential equation. This factorization leads to an integral representation of the unknown function g. Finally, the singular differential equation, which remains to be solved, is discussed. At this point, work should be continued.