Multiple interpolation problem for functions with zero spherical mean

Abstract

Let $|\cdot|$ be the Euclidean norm in $\mathbb{R}^n$, $n\geq 2$. For $r>0$, weLet $|\cdot|$ be the Euclidean norm in $\mathbb{R}^n$, $n\geq 2$. For $r>0$, we denote by $V_r(\mathbb{R}^n)$ the set of functions $f\in L_{\mathrm{loc}}(\mathbb{R}^n)$ satisfying the condition \begin{equation*}\label{eq } \int_{|x|\leq r}f(x+y)dx=0\quad\text{for any}\quad y\in\mathbb{R}^n. \end{equation*} The paper investigates the interpolation of tempered growth functions of class $(V_r\cap C^{\infty})(\mathbb{R}^n)$ together with the derivatives of bounded order in a given direction. Let $d\in\mathbb{R}^n$, $\sigma\in\mathbb{R}^n\setminus\{0\}$ be fixed, $\{a_k\}_{k=1}^{\infty}$ be a sequence of points lying on the line $ \{x\in\mathbb{R}^n:\,x=d+t\sigma,\,t\in(-\infty,+\infty)\}$ and satisfying the conditions \begin{equation*}\label{equation} \underset{i\ne j}\inf\, |a_i-a_j|>0,\quad |a_k|\leq|a_{k+1}|\quad\text{for all}\quad k\in\mathbb{N}. \end{equation*} Let also $m\in\mathbb{Z}_+$ and $b_{k,j}\in\mathbb{C}$ ($k\in\mathbb{N}$, $j\in\{0,\ldots,m\}$) be a set of numbers satisfying the condition \begin{equation*}\label{eq} \underset{0\leq j\leq m}\max\, |b_{k,j}|\leq(k+1)^{\alpha} \end{equation*} for all $k\in\mathbb{N}$ and some $\alpha\geq 0$ independent of $k$. It is shown (Theorem) that, under the indicated conditions, the interpolation problem \begin{equation*}\label{ equation*} \left(\sigma_1\frac{\partial}{\partial x_1}+\ldots+\sigma_n\frac{\partial}{\partial x_n}\right)^jf(a_k)=b_{k,j},\quad k\in\mathbb{N},\quad j\in\{0,\ldots,m\}, \end{equation*} is solvable in a class of functions belonging to $(V_r\cap C^{\infty})(\mathbb{R}^n)$, which, together with all their derivatives, have growth no higher than a power-law at infinity. It is noted that the condition of separability of nodes $\{a_k\}_{k=1}^{\infty}$ in  the Theorem cannot be removed, and also that the solution of the considered interpolation problem is not the only one. In addition, it is stated that the one-dimensional analogue of the Theorem is not valid since every continuous function of class $V_r(\mathbb{R}^n)$ at $n=1$ is $2r$-periodic.