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Abstract <p> For each integer nonnegative <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>, in some Hilbert space, we introduce an integral transform <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak{H}_n\)</EquationSource> </InlineEquation>. It is similar to the well-known Hankel (Fourier–Bessel) transform of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>th order due to the fact that it is related to the Fourier transform and its integral kernel is expressed in terms of the Bessel function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(J_n\)</EquationSource> </InlineEquation>. But, unlike the Hankel transform, intended for functions of one variable, the transform <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathfrak{H}_n\)</EquationSource> </InlineEquation> is intended for functions of three variables. For the constructed transform <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathfrak{H}_n\)</EquationSource> </InlineEquation>, we prove Plancherel’s theorem, the inversion formula, the similarity property and formulas of composition with some differential operators. In particular, we found second-order differential operators which under <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathfrak{H}_n\)</EquationSource> </InlineEquation> turn into operators of multiplication by certain functions, and we give an example of using the constructed transform to solve a differential equation for the Laplace operator with a Coulomb-type potential. In addition, we described a family of convolutions which under <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathfrak{H}_n\)</EquationSource> </InlineEquation> turn into a product. </p>

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Three-Dimensional Analog of the Integer-Order Hankel Transform

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • E. M. Novikova

摘要

Abstract

For each integer nonnegative \(n\) , in some Hilbert space, we introduce an integral transform \(\mathfrak{H}_n\) . It is similar to the well-known Hankel (Fourier–Bessel) transform of \(n\) th order due to the fact that it is related to the Fourier transform and its integral kernel is expressed in terms of the Bessel function \(J_n\) . But, unlike the Hankel transform, intended for functions of one variable, the transform \(\mathfrak{H}_n\) is intended for functions of three variables. For the constructed transform \(\mathfrak{H}_n\) , we prove Plancherel’s theorem, the inversion formula, the similarity property and formulas of composition with some differential operators. In particular, we found second-order differential operators which under \(\mathfrak{H}_n\) turn into operators of multiplication by certain functions, and we give an example of using the constructed transform to solve a differential equation for the Laplace operator with a Coulomb-type potential. In addition, we described a family of convolutions which under \(\mathfrak{H}_n\) turn into a product.