Abstract:
Let $n\geq 2$, $V_r(\mathbb{R}^n)$ be the set of functions with zero integrals over all balls in $\mathbb{R}^n$ of radius $r$. Various interpolation problems for the class $V_r(\mathbb{R}^n)$ 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 $V_r(\mathbb{R}^n)$ 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 $\mathbb{R}^n$ do not hold true.
Keywords:interpolation problems, spherical means, mean periodicity.
Citation:
V. V. Volchkov, Vit. V. Volchkov, “Interpolation problems for functions with zero ball means”, Probl. Anal. Issues Anal., 10(28):3 (2021), 129–140
\Bibitem{VolVol21}
\by V.~V.~Volchkov, Vit.~V.~Volchkov
\paper Interpolation problems for functions with zero ball means
\jour Probl. Anal. Issues Anal.
\yr 2021
\vol 10(28)
\issue 3
\pages 129--140
\mathnet{http://mi.mathnet.ru/pa336}
\crossref{https://doi.org/10.15393/j3.art.2021.10751}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000701564800001}
Linking options:
https://www.mathnet.ru/eng/pa336
https://www.mathnet.ru/eng/pa/v28/i3/p129
This publication is cited in the following 3 articles:
V. V. Volchkov, Vit. V. Volchkov, “Interpolation of Functions with Zero Spherical Averages Obeying Growth Constraints”, Sib Math J, 65:5 (2024), 1043
V. V. Volchkov, Vit. V. Volchkov, “Interpolyatsiya funktsii s nulevymi sharovymi srednimi s ogranicheniem rosta”, Sib. matem. zhurn., 65:5 (2024), 841–851
V. V. Volchkov, Vit. V. Volchkov, “Interpolation Problem with Knots on a Line for Solutions of a Multidimensional Convolution Equation”, Lobachevskii J Math, 44:8 (2023), 3630