Some remarks on integral equations with kernels: L (ξ 1 - x 1 ,..., ξ n - x n ; α)

The paper discusses integral equations of the typek (ξ1 ... ξn ) = ∫-∞ ∞ ... ∫-∞ ∞ ψ (x 1 ...x n )L (ξ1 -x 1 ... ξn -x n ;α)d x 1 ...d xn , whereL isL -integrable, and ψ andk are bounded. Since rapidly oscillating ψ have a smallk , and since measurements ofk are necessarily uncertain within non-zero limits of experimental error, very different ψ are consistent with any given set of measurements ofk . Thus ψ is not determined by measurements ofk . Instead of ψ, partial information about ψ that is not sensitive to rapid oscillations of ψ, can be obtained fromk . In the present paper we considersmoothed versions of ψ, and their applications to gravity survey and the theory of surface waves. (1) For givenL we construct normalized smoothing functions μ(x 1 ...x n ), so that ψ(x 1 ...x n ) = ∫-∞ ∞ ... ∫-∞ ∞ ψ(u 1 ...u n )μ(x 1 -u 1 ...x n -u n )d u 1 ....d u n can be calculated from measurements ofk( (ξ1 ... ξn ). The method is applied to gravity survey, where the distribution ψ of masses on a plane ∑' is to be calculated from the normal forcek on another (parallel) plane∑. (2) By studying suitable smoothing functions we geta lower bound for the maximum modulus of the functions ψ which are consistent with given experimental values ofk . The bound is large ifk is known to vary rapidly. The bound is also large if α is large and if the Fourier transform ofL tends to 0 when α→∞. The results are applied to gravity survey where now we consider the normal force on ∑ due to masses of bounded density distributed in the space below ∑', where ∑' is itself below ∑. If the normal force is not uniform the distance between ∑ and ∑' must not be too large, the estimate depending on the bound for the density. Also fairly general conditions are imposed on ψ so that ψ that is approximately determined by measurements ofk , and an example from the theory of propagation in dispersive media is given where such conditions may be justified. The gist of the paper is contained in theorems A and B.