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Radon–Kipriyanov Transform of Laplace Series by Weight Spherical Functions

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Abstract

In this paper we consider the Radon–Kipriyanov transform and its relationship with the special Radon transform. The representation of plane integrals in the Lebesgue–Kipriyanov measure by the corresponding hemisphere integral in \(\mathbb{R}_{n}^{+}\) is given. The definitions and information necessary for calculating the Radon–Kipriyanov transform from Laplace series for one class of weight spherical functions with one weight variable are given. A generalization of the Funk–Hecke formula is obtained. The formula for the Radon–Kipriyanov transform with one special variable of the Laplace series for weight spherical functions is obtained.

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Notes

  1. Here it is an analogy to the classical harmonic polynomials, which are often called spherical polynomial, see, e.g., [13], p. 479.

  2. This equation is called Laplace’s formula for weight spherical functions. For many weight variables, this formula is proved in the book [11], Theorem 1.4.1.

REFERENCES

  1. I. M. Gelfand and Z. Y. Shapiro, ‘‘Homogeneous functions and their applications,’’ Usp. Mat. Nauk 10 (3), 3–51 (1955).

    Google Scholar 

  2. I. A. Kipriyanov and L. N. Lyakhov, ‘‘On Fourier, Fourier–Bessel and Radon transforms,’’ Dokl. Math. 57, 361 (1998).

    MATH  Google Scholar 

  3. L. N. Lyakhov, ‘‘Appeal of the Kipriyanov–Radon transform,’’ Rep. Acad. Sci. 399, 597–600 (2004).

    Google Scholar 

  4. L. N. Lyakhov, ‘‘Kipriyanov–Radon transform,’’ Tr. Mat. Inst. Steklova 248, 153–163 (2005).

    MathSciNet  MATH  Google Scholar 

  5. D. Ludwig, ‘‘The Radon transform on Euclidean space,’’ Comm. Pure Appl. Math. 19, 49–81 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  6. G. T. Herman and F. Natterer, Mathematical Aspects of Computerized Tomography, Proceedings, Oberwolfach, February 10–16, 1980 (Springer, Berlin, 1981).

  7. L. N. Lyakhov, ‘‘On one class of spherical functions and singular pseudodifferential operators,’’ Rep. Acad. Sci. 272, 781–784 (1983).

    Google Scholar 

  8. L. N. Lyakhov, ‘‘About series by weight spherical functions,’’ in Correct Boundary Value Problems for Nonclassical Equations of Mathematical Physics, Collection of Articles (Sib. Otdel. Akad. Nauk, Novosibirsk, 1984), pp. 102–109 [in Russian].

    Google Scholar 

  9. L. N. Lyakhov, ‘‘Weight spherical functions and singular functions pseudo-differential operators,’’ Differ. Equat. 21, 1020–1032 (1985).

    Google Scholar 

  10. L. N. Lyakhov, Weight Spherical Functions and Riess Potentials Generated by a Generalized Shift (VGTA, Voronezh, 1997) [in Russian].

    Google Scholar 

  11. L. N. Lyakhov, Hypersingular Integrals and their Applications to the Description of Kipriyanov’s Functional Classes and to Integral Equations with B-potential Kernels (LGPU, Lipetsk, 2007) [in Russian].

    Google Scholar 

  12. I. A. Kipriyanov, Singular Elliptic Boundary Value Problems (Nauka, Moscow, 1997) [in Russian].

    MATH  Google Scholar 

  13. S. L. Sobolev, Introduction to the Theory of Cubic Formulas (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Semenov-Tyan-Shansky Lipetsk State Pedagogical University, 398020, Lipetsk, Russia

    V. A. Kalitvin & M. G. Lapshina

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  1. V. A. Kalitvin
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  2. M. G. Lapshina
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Correspondence to V. A. Kalitvin or M. G. Lapshina.

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(Submitted by A. B. Muravnik)

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Kalitvin, V.A., Lapshina, M.G. Radon–Kipriyanov Transform of Laplace Series by Weight Spherical Functions. Lobachevskii J Math 44, 3323–3330 (2023). https://doi.org/10.1134/S1995080223080231

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  • DOI: https://doi.org/10.1134/S1995080223080231

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