A mollification framework for improperly posed problems
Let \(A\) represent a linear, translation-invariant operator between given Hilbert spaces for which the range is not dense. A general theory for the mollification of the improperly posed equation \(Au=f\) is developed utilizing the classical theory of semi-groups. The choice of the mollifier is fairly general and includes B-splines as well as the classical Gaussians. A translation-operator representation for the error associated with the application of mollification is derived and an error estimate for the mollification method as applied to the numerical solution of translation-invariant operator equation is given. Then the connection is analyzed between the degree of the mollification and the size of the discretization necessary to guarantee the boundedness of the amplification error. The results are applied to the construction of finite difference formulas for the inversion of translation-invariant operator equations and their use in connection with the numerical differentiation and half-differentiation of observational data.
translation-invariant operator equations
scientific article; zbMATH DE number 1111589