Abstract— <p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(| \cdot |\)</EquationSource>  </InlineEquation> be the Euclidean norm in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathbb{R}}^{n}}\)</EquationSource>  </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n \geqslant 2\)</EquationSource>  </InlineEquation>. For <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(r > 0\)</EquationSource>  </InlineEquation>, we denote by <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\({{V}_{r}}({{\mathbb{R}}^{n}})\)</EquationSource>  </InlineEquation> the set of functions <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\(f \in {{L}_{{{\text{loc}}}}}({{\mathbb{R}}^{n}})\)</EquationSource>  </InlineEquation> satisfying the condition<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq7.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="286" /> </InlineMediaObject> <EquationSource Format="TEX">\(\int_{|x| \leqslant r} f(x + y)dx = 0\quad {\text{for}}\,\,{\text{any}}\quad y \in {{\mathbb{R}}^{n}}.\)</EquationSource>  </InlineEquation> The paper investigates the interpolation of tempered growth functions of class <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">\(({{V}_{r}} \cap {{C}^{\infty }})({{\mathbb{R}}^{n}})\)</EquationSource>  </InlineEquation> together with the derivatives of bounded order in a given direction. Let <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(d \in {{\mathbb{R}}^{n}}\)</EquationSource>  </InlineEquation>, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \in {{\mathbb{R}}^{n}}{{\backslash }}\{ 0\} \)</EquationSource>  </InlineEquation> be fixed, <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{ {{a}_{k}}\} _{{k = 1}}^{\infty }\)</EquationSource>  </InlineEquation> be a sequence of points lying on the line <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="278" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{ x \in {{\mathbb{R}}^{n}}:{\kern 1pt} x = d + t\sigma ,{\kern 1pt} t \in ( - \infty , + \infty )\} \)</EquationSource>  </InlineEquation> and satisfying the conditions<InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq13.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="346" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathop {\inf }\limits_{i \ne j} {\kern 1pt} {\text{|}}{{a}_{i}} - {{a}_{j}}{\text{|}} > 0,\quad {\text{|}}{{a}_{k}}{\text{|}} \leqslant {\text{|}}{{a}_{{k + 1}}}{\text{|}}\quad {\text{for}}\,\,{\text{all}}\quad k \in \mathbb{N}.\)</EquationSource>  </InlineEquation>Let also <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq14.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(m \in {{\mathbb{Z}}_{ + }}\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\({{b}_{{k,j}}} \in \mathbb{C}\)</EquationSource>  </InlineEquation> (<InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq16.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(k \in \mathbb{N}\)</EquationSource>  </InlineEquation>, <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq17.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">\(j \in \{ 0, \ldots ,m\} \)</EquationSource>  </InlineEquation>) be a set of numbers satisfying the condition<InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq18.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="159" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathop {\max }\limits_{0 \leqslant j \leqslant m} {\kern 1pt} {\text{|}}{{b}_{{k,j}}}{\text{|}} \leqslant {{(k + 1)}^{\alpha }}\)</EquationSource>  </InlineEquation>for all <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq16.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(k \in \mathbb{N}\)</EquationSource>  </InlineEquation> and some <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq20.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \geqslant 0\)</EquationSource>  </InlineEquation> independent of <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq21.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\)</EquationSource>  </InlineEquation>. It is shown (Theorem) that, under the indicated conditions, the interpolation problem<InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq22.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="449" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\left( {{{\sigma }_{1}}\frac{\partial }{{\partial {{x}_{1}}}} + \ldots + {{\sigma }_{n}}\frac{\partial }{{\partial {{x}_{n}}}}} \right)}^{j}}f({{a}_{k}}) = {{b}_{{k,j}}},\quad k \in \mathbb{N},\quad j \in \{ 0, \ldots ,m\} ,\)</EquationSource>  </InlineEquation>is solvable in a class of functions belonging to <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">\(({{V}_{r}} \cap {{C}^{\infty }})({{\mathbb{R}}^{n}})\)</EquationSource>  </InlineEquation>, 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 <InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{ {{a}_{k}}\} _{{k = 1}}^{\infty }\)</EquationSource>  </InlineEquation> 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 <InlineEquation ID="IEq25"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\({{V}_{r}}({{\mathbb{R}}^{n}})\)</EquationSource>  </InlineEquation> at <InlineEquation ID="IEq26"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq26.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n = 1\)</EquationSource>  </InlineEquation> is <InlineEquation ID="IEq27"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10493_Article_IEq27.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(2r\)</EquationSource>  </InlineEquation>-periodic.</p>

Multiple Interpolation Problem for Functions with Zero Spherical Mean

期刊论文
发表日期：

2025年5月1日

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V. V. Volchkov,
Vit. V. Volchkov

摘要

Abstract—

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.