On a class of severely ill-posed problems

The problem of calculating the value of the pseudo-differential operator a(D, y)φ, y ∈ [0, 1] with unbounded symbol a(iξ, y) is ill-posed. If the symbol a(iξ, y) behaves like an exponential function of ξ, then the problem is severely ill-posed. This note is devoted to the last case. The mollification method [2] is used to regularize the problem in the general Lp space setting. Error estimates of Hölder type for the regularized values and the exact values are derived. Applications of the general scheme to concrete problems from practice are presented.