The functions with zero integral averages on the balls and densities of the packages (Q2879981)

The present paper is devoted to the study of the different functions having zero integrals under the balls in \({\mathbb R}^n.\) The above investigation is closely related to the Bessel functions \(J_{\mu}\) of the first type of the order \(\mu.\) We consider the main results of the work. Theorem 3 gives an example of such a function \(f\) by the integral of a special type depending on the given function \(\varphi.\) In particular, if \(\varphi\) is odd, then the corresponding function \(f\) is non-zero. Theorem 4 states the existence of the function \(f\) with zero integral on the balls with some prescribed conditions, whose growth is not more than of exponential type. Finally, for a so-called package \({\mathcal K}\) of the given set \(G,\) where \({\mathcal K}\) consists of some number of the balls, it is obtained the estimate of the density of the \({\mathcal K}\). This is proved on the basis of the Theorem 3, compare the Theorem 5. The obtained results can be applicable to many problems of mathematical analysis.