INTERPOLATION PROBLEMS FOR FUNCTIONS WITH ZERO BALL MEANS

Abstract

Let 𝑛 >= 2, 𝑉_𝑟(R^𝑛) be the set of functions with zero integrals over all balls in R^𝑛 of radius 𝑟. Various interpolation problems for the class 𝑉_𝑟(R^𝑛) are studied. In the case when the set of interpolation nodes is finite, we solve the interpolation problem under general conditions. For the problems with infinite set of nodes, some sufficient conditions of solvability are founded. Note that an essential condition is that the definition of the class 𝑉_𝑟(R^𝑛) involves integration over balls. For instance, it can be shown that the analogues of our results in which the class of functions is defined using zero integrals over all shifts of a fixed parallelepiped in R^𝑛 do not hold true.

Keywords

• interpolation problems
• spherical means
• mean periodicity