Inverse problem for potentials of measures in a Hilbert space

Abstract

For a Banach space E and a number p εR, under the assumption of the convergence of the integral, one considers the potential g(a)=∝ E ∥x−a∥pdμ(x), where μ is a finite Borel measure on E. One solves the problem of the determination of the measure μ from known values of the function g(a),a ε E. In the note one gives an explicit solution of the problem for an infinite-dimensional Hilbert space, which exists for p≠0, 2, 4,.... For certain finite-dimensional Banach spaces one solves the inverse problem with the aid of the known Levy representations for the norms.