Criterion of the continuation of harmonic functions in the ball of n­dimensional space and representation of the generalized orders of the entire harmonic functions in ℝn in terms of approximation error

Abstract

A growth of harmonic functions in the whole space ℝn is examined. We found the estimate for a uniform norm of spherical harmonics in terms of the best approximation of harmonic function in the ball by harmonic polynomials. An approximation error of harmonic function in the ball is estimated by the maximum modulus of an entire harmonic function in space, as well as the maximum modulus of an entire harmonic function in space in terms of the maximum modulus of some entire function of one complex variable or the maximal term of its power series. These results allowed us to obtain the necessary and sufficient conditions under which a harmonic function in the ball of an n-dimensional space, n≥3, can be continued to the entire harmonic one. This result is formulated in terms of the best approximation of the given function by harmonic polynomials. In order to characterize growth of an entire harmonic function, we used the generalized and the lower generalized orders. Formulae for the generalized and the lower generalized orders of an entire harmonic function in space are expressed in terms of the approximation error by harmonic polynomials of the function that continues. We also investigated the growth of functions of slow increase. The obtained results are analogues to classical results, which are known for the entire functions of one complex variable.The conducted research is important due to the fact that the harmonic functions occupy a special place not only in many mathematical studies, but also when applying mathematical analysis to physics and mechanics, where these functions are often employed to describe various stationary processes