## Abstract

The paper discusses integral equations of the type *k *(ξ_{1}... ξ_{n}) = ∫ _{-∞}^{∞}... ∫_{-∞}^{∞}ψ (*x *_{1}... *x*_{n})*L*(ξ_{1}*-x*_{1}... ξ_{n}*-x*_{n};α) *d**x *_{1}... *d*x_{n}, where*L*is* L*-integrable, and ψ and *k *are bounded. Since rapidly oscillating ψ have a small *k*, and since measurements of *k * are necessarily uncertain within non-zero limits of experimental error, very different ψ are consistent with any given set of measurements of*k*. Thus ψ is not determined by measurements of* k*. Instead of ψ, partial information about ψ that is not sensitive to rapid oscillations of ψ, can be obtained from *k*. In the present paper we consider *smoothed versions* of ψ, and their applications to gravity survey and the theory of surface waves. (1) For given*L* 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 of *k(*(ξ_{1}... ξ_{n}). The method is applied to gravity survey, where the distribution ψ of masses on a plane ∑' is to be calculated from the normal force * k*on another (parallel) plane∑. (2) By studying suitable smoothing functions we get *a lower bound for the maximum modulus of the functions* ψ 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 α is large and if the Fourier transform of* L* 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 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.

## Footnotes

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- Received January 6, 1948.

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