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

The paper discusses integral equations of the type k($\xi _{1}\,\ldots \,\xi _{n}$) = $\int_{-\infty}^{\infty}\cdots \int_{-\infty}^{\infty}\psi $(x$_{1}\,\ldots $ x$_{n}$) L($\xi _{1}$-x$_{1}\,\ldots \,\xi _{n}$-x$_{n}$; $\alpha $) dx$_{1}\,\ldots $ dx$_{n}$, where L is L-integrable, and $\psi $ and k are bounded. Since rapidly oscillating $\psi $ have a small k, and since measurements of k are necessarily uncertain within non-zero limits of experimental error, very different $\psi $ are consistent with any given set of measurements of k. Thus $\psi $ is not determined by measurements of k. Instead of $\psi $, partial information about $\psi $ that is not sensitive to rapid oscillations of $\psi $, can be obtained from k. In the present paper we consider smoothed versions of $\psi $, and their applications to gravity survey and the theory of surface waves. (1) For given L we construct normalized smoothing functions $\mu $(x$_{1}\,\ldots $ x$_{n}$), so that $\overline{\psi}$(x$_{1}\,\ldots $ x$_{n}$) = $\int_{-\infty}^{\infty}\cdots \int_{-\infty}^{\infty}\psi $(u$_{1}\,\ldots $ u$_{n}$) $\mu $(x$_{1}$ - u$_{1}\,\ldots $ x$_{n}$ - u$_{n}$) du$_{1}\,\ldots $ du$_{n}$ can be calculated from measurements of k($\xi _{1}\,\ldots \,\xi _{n}$). The method is applied to gravity survey, where the distribution $\psi $ of masses on a plane $\Sigma ^{\prime}$ is to be calculated from the normal force k on another (parallel) plane $\Sigma $. (2) By studying suitable smoothing functions we get a lower bound for the maximum modulus of the functions $\psi $ which are consistent with given experimental values of k. The bound is large if k is known to vary rapidly. The bound is also large if $\alpha $ is large and if the Fourier transform of L tends to 0 when $\alpha $\rightarrow $\infty $. The results are applied to gravity survey where now we consider the normal force on $\Sigma $ due to masses of bounded density distributed in the space below $\Sigma ^{\prime}$, where $\Sigma ^{\prime}$ is itself below $\Sigma $. If the normal force is not uniform the distance between $\Sigma $ and $\Sigma ^{\prime}$ must not be too large, the estimate depending on the bound for the density. Also fairly general conditions are imposed on $\psi $ so that $\psi $ is approximately determined by measurements of k, 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.