Multiple Interpolation Problem for Functions with Zero Spherical Mean

Let $$| \cdot |$$ be the Euclidean norm in $${{\mathbb{R}}^{n}}$$ , $$n \geqslant 2$$ . For $$r > 0$$ , we denote by $${{V}_{r}}({{\mathbb{R}}^{n}})$$ the set of functions $$f \in {{L}_{{{\text{loc}}}}}({{\mathbb{R}}^{n}})$$ satisfying the condition $$\int_{|x| \leqslant r} f(x + y)dx = 0\quad {\text{for}}\,\,{\text{any}}\quad y \in {{\mathbb{R}}^{n}}.$$ 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 $$d \in {{\mathbb{R}}^{n}}$$ , $$\sigma \in {{\mathbb{R}}^{n}}{{\backslash }}\{ 0\} $$ be fixed, $$\{ {{a}_{k}}\} _{{k = 1}}^{\infty }$$ be a sequence of points lying on the line $$\{ x \in {{\mathbb{R}}^{n}}:{\kern 1pt} x = d + t\sigma ,{\kern 1pt} t \in ( - \infty , + \infty )\} $$ and satisfying the conditions $$\mathop {\inf }\limits_{i \ne j} {\kern 1pt} {\text{|}}{{a}_{i}} - {{a}_{j}}{\text{|}} > 0,\quad {\text{|}}{{a}_{k}}{\text{|}} \leqslant {\text{|}}{{a}_{{k + 1}}}{\text{|}}\quad {\text{for}}\,\,{\text{all}}\quad k \in \mathbb{N}.$$ 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 $$\mathop {\max }\limits_{0 \leqslant j \leqslant m} {\kern 1pt} {\text{|}}{{b}_{{k,j}}}{\text{|}} \leqslant {{(k + 1)}^{\alpha }}$$ for all $$k \in \mathbb{N}$$ and some $$\alpha \geqslant 0$$ independent of $$k$$ . It is shown (Theorem) that, under the indicated conditions, the interpolation problem $${{\left( {{{\sigma }_{1}}\frac{\partial }{{\partial {{x}_{1}}}} + \ldots + {{\sigma }_{n}}\frac{\partial }{{\partial {{x}_{n}}}}} \right)}^{j}}f({{a}_{k}}) = {{b}_{{k,j}}},\quad k \in \mathbb{N},\quad j \in \{ 0, \ldots ,m\} ,$$ 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.