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Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes

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Published/Copyright: September 11, 2020
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 23 Issue 4

Abstract

We apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

MSC 2010: Primary 44A12; Secondary 28A75
Key Words and Phrases: Erdélyi–Kober fractional integrals; Radon transforms; Grassmann manifolds

Acknowledgements

The study of the operators 𝓡j,kf for all j + k < n in the general Lp setting was suggested by the first-named author [17], who was inspired by the works [4, 5, 8]. He is thankful to Fulton Gonzalez, Todd Quinto and Sigurdur Helgason for useful discussions during his visit to Tufts University in April, 2006. Both authors are thankful to Virginia Kiryakova for the valuable comments in Remark 2.1. The second-named author was supported by National Natural Science Foundation of China, 11671414.

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Received: 2020-03-25
Published Online: 2020-09-11
Published in Print: 2020-08-26

© 2020 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–VOLUME 23–4–2020)
  4. Editorial Survey
  5. Fractional derivatives and the fundamental theorem of fractional calculus
  6. Research Paper
  7. Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes
  8. Nontrivial solutions of non-autonomous dirichlet fractional discrete problems
  9. Applications of Hilfer-Prabhakar operator to option pricing financial model
  10. On a quantitative theory of limits: Estimating the speed of convergence
  11. Global solutions and blowing-up solutions for a nonautonomous and nonlocal in space reaction-diffusion system with Dirichlet boundary conditions
  12. On the harmonic extension approach to fractional powers in Banach spaces
  13. Initial-value / Nonlocal Cauchy problems for fractional differential equations involving ψ-Hilfer multivariable operators
  14. Fractional abstract Cauchy problem on complex interpolation scales
  15. On representation formulas for solutions of linear differential equations with Caputo fractional derivatives
  16. Asymptotics of fundamental solutions for time fractional equations with convolution kernels
  17. Attractivity for differential equations of fractional order and ψ-Hilfer type
  18. Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions
https://doi.org/10.1515/fca-2020-0050
Keywords for this article
Erdélyi–Kober fractional integrals; Radon transforms; Grassmann manifolds

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–VOLUME 23–4–2020)
  4. Editorial Survey
  5. Fractional derivatives and the fundamental theorem of fractional calculus
  6. Research Paper
  7. Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes
  8. Nontrivial solutions of non-autonomous dirichlet fractional discrete problems
  9. Applications of Hilfer-Prabhakar operator to option pricing financial model
  10. On a quantitative theory of limits: Estimating the speed of convergence
  11. Global solutions and blowing-up solutions for a nonautonomous and nonlocal in space reaction-diffusion system with Dirichlet boundary conditions
  12. On the harmonic extension approach to fractional powers in Banach spaces
  13. Initial-value / Nonlocal Cauchy problems for fractional differential equations involving ψ-Hilfer multivariable operators
  14. Fractional abstract Cauchy problem on complex interpolation scales
  15. On representation formulas for solutions of linear differential equations with Caputo fractional derivatives
  16. Asymptotics of fundamental solutions for time fractional equations with convolution kernels
  17. Attractivity for differential equations of fractional order and ψ-Hilfer type
  18. Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions
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