No abstract

We consider the multi-dimensional Ilausdorif moment problem over the unit cube: to reconstruct an unknown function from the (inaccurately) given values of the integrals of the unknown function multiplied by all power-products of the independent variables. We describe a regularization scheme using orthogonalization by the tensor product of (shifted) Legendre polynomials and "approximation" of the unknown function by a finite sum, the dimension of the space of approximation playing the role of the regularization parameter. For the case of square integrability of the unknown function we present an estimate of the regularization error that implies convergence if the data error tends to zero.

The present paper studies the wave patterns generated in an elastic half-space by a line load moving on its surface with a velocity varying as a step function of time. The solution given in closed form is obtained by means of Fourier integral equations techniques following a Laplace transformation with respect to the time variable. The inversion of the Laplace transforms is based on a trick due to Cagniard and De Hoop.

In this paper the authors prove a uniqueness theorem for the electric detection of cavities in a three dimensional solid from Cauchy data measured on the surface. The cavities, finite in number, are assumed to be insulating. The surfaces of the cavities are assumed to be smooth on the complement of a set that is, in some sense, negligible.

We consider the problem of identifying the domain ci C R2 of a semilinear elliptic equation subject to given Cauchy data on part of the known outer boundary I' and to the zero flux condition on the unknown inner boundary y, where it is assumed that r is a piecewise C' curve and that y is the boundary of a finite disjoint union of simply connected domains, each bounded by a piecewiseC' Jordan curve. It is shown that, under appropriate smoothness conditions, the domain ci is uniquely determined. The problem of existence of solution for given data is not considered since it is usually of lesser importance in view of measurement errors giving data for which no solution exists.