Multiple interpolation problem for functions with zero spherical mean

V. V. Volchkov, Vit. V. Volchkov

Donetsk State University, 24 Universitetskaya str., Donetsk, 283001 Russia

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*} \int_{|x|\leq r}f(x+y)dx=0 \text{for any} 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*} \underset{i\ne j}\inf |a_i-a_j|>0, |a_k|\leq|a_{k+1}| \text{for all} 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*} \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*}\left(\sigma_1\frac{\partial}{\partial x_1}+\ldots+\sigma_n\frac{\partial}{\partial x_n}\right)^jf(a_k)=b_{k,j}, k\in\mathbb{N}, 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.

Keywords: interpolation, spherical mean, Fourier transform, Bessel function.

UDC: 517.5

Received: 29.03.2024
Revised: 19.04.2024
Accepted: 26.06.2024

DOI: 10.26907/0021-3446-2025-5-44-57