Analytical solution of a new class of integral equations

Let (1)Rh = f , 0 ≤ x ≤ L, Rh = ∫ L 0 R(x, y)h(y) dy, where the kernel R(x, y) satisfies the equation QR = Pδ(x− y). Here Q and P are formal differential operators of order n and m < n, respectively, n and m are nonnegative even integers, n > 0, m ≥ 0, Qu := qn(x)u + ∑n−1 j=0 qj(x)u (j), Ph := h(m) + ∑m−1 j=0 pj(x)h (j), qn(x) ≥ c > 0, the coefficients qj(x) and pj(x) are smooth functions defined on R, δ(x) is the delta-function, f ∈ Hα(0, L), α := n−m 2 , H α is the Sobolev space. An algorithm for finding analytically the unique solution h ∈ Ḣ−α(0, L) to (1) of minimal order of singularity is given. Here Ḣ−α(0, L) is the dual space to Hα(0, L) with respect to the inner product of L2(0, L). Under suitable assumptions it is proved that R : Ḣ−α(0, L) → Hα(0, L) is an isomorphism. Equation (1) is the basic equation of random processes estimation theory. Some of the results are generalized to the case of multidimensional equation (1), in which case this is the basic equation of random fields estimation theory.