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In the present paper, using a development of the technique of transmutation mappings, we obtain the first weighted analog of the well-known Brown–Schreiber–Taylor theorem on the eigenvalues of the Laplacian for corresponding spaces of continuous functio

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.

We obtain new sufficient conditions under which a set on the plane has the Pompeiu property. This result allows us to construct first examples of domains with the Pompeiu property with non-Lipschitz (and even fractal) boundary. In addition, we study the problem of determining the least radius of the ball on the sphere in which a given set is a Pompeiu set. We obtain the solution of this problem in the case of a biangle and a spherical half-disk. We also consider some applications to questions of complex analysis.

The property of being holomorphic is studied for a continuous function whose integrals over the boundaries of the triangles from a specified set are zero. The results substantially strengthen Morera's and Dzyadyk's well-known theorems.

The Pompeiu problem is studied for functions defined on a ballB ⊂ ℝ n and having zero integrals over all sets congruent to a given compact setK ⊂B. The problem of finding the least radiusr=r(K) ofB for whichK is a Pompeiu set is considered. The solution is obtained for the cases in whichK is a cube or a hemisphere.

We construct examples of nonharmonic functions satisfying the mean value equation for some set of spheres. These results permit us to obtain the two-circle theorem in its definitive form.

For some classes of analytic functions we estimate the rate of approximation by polynomials with integer coefficients.