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

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Veröffentlicht/Copyright: 11. September 2020
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 23 Heft 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.

References

[1] A. Erdélyi, On some functional transformations. Univ. e Politec. Torino Rend. Sem. Mat. 10 (1951), 217–234.Suche in Google Scholar

[2] P.G. Funk, Über Flächen mit lauter geschlossenen geodätschen Linen. Math. Ann. 74 (1913), 278–300.10.1007/BF01456044Suche in Google Scholar

[3] I.M. Gelfand, S.G. Gindikin, and M.I. Graev, Selected Topics in Integral Geometry. Translations of Mathematical Monographs, AMS, Providence, Rhode Island (2003).10.1090/mmono/220Suche in Google Scholar

[4] F.B. Gonzalez, Radon Transform on Grassmann Manifolds. Thesis, MIT, Cambridge, MA (1984).Suche in Google Scholar

[5] F.B. Gonzalez, Radon transform on Grassmann manifolds. Journal of Func. Anal. 71 (1987), 339–362.10.1016/0022-1236(87)90008-5Suche in Google Scholar

[6] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products. Academic Press, New York (1980).Suche in Google Scholar

[7] L. A-M. Hanna, M. Al-Kandari, and Yu. Luchko, Operational method for solving fractional differential equations with the left- and right-hand sided Erdélyi-Kober fractional derivatives. Fract. Calc. Appl. Anal. 23, No 1 (2020), 103–125; 10.1515/fca-2020-0004; https://www.degruyter.com/view/journals/fca/23/1/fca.23.issue-1.xml.Suche in Google Scholar

[8] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113 (1965), 153–180.10.1007/BF02391776Suche in Google Scholar

[9] S. Helgason, Integral Geometry and Radon Transform. Springer, New York-Dordrecht-Heidelberg-London (2011).10.1007/978-1-4419-6055-9Suche in Google Scholar

[10] R. Herrmann, Towards a geometric interpretation of generalized fractional integrals–Erdélyi-Kober type integrals on ℝn, as an example. Fract. Calc. Appl. Anal. 17, No 2 (2014), 361–370; 10.2478/s13540-014-0174-4; https://www.degruyter.com/view/journals/fca/17/2/fca.17.issue-2.xml.Suche in Google Scholar

[11] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Res. Notes in Math. Ser. 301, Longman Scientific & Technical, Harlow; copubl. John Wiley & Sons, Inc., New York (1994).Suche in Google Scholar

[12] H. Kober, On fractional integrals and derivatives. Quart. J. Math. Oxford Ser. 11 (1940), 193–211.10.1093/qmath/os-11.1.193Suche in Google Scholar

[13] G. Pagnini, Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, No 1 (2012), 117–127; 10.2478/s13540-012-0008-1; https://www.degruyter.com/view/journals/fca/15/1/fca.15.issue-1.xml.Suche in Google Scholar

[14] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. - Nat. Kl. 69 (1917), 262–277.10.1090/psapm/027/692055Suche in Google Scholar

[15] B. Rubin, Radon transforms on affine Grassmannians. Trans. Amer. Math. Soc. 356 (2004), 5045–5070.10.1090/S0002-9947-04-03508-1Suche in Google Scholar

[16] B. Rubin, Reconstruction of functions from their integrals over k-planes. Israel J. of Math. 141 (2004), 93–117.10.1007/BF02772213Suche in Google Scholar

[17] B. Rubin, Combined Radon transforms, Preprint (2006).Suche in Google Scholar

[18] B. Rubin, Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis. Cambridge University Press, New York (2015).Suche in Google Scholar

[19] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Sc. Publ., New York etc. (1993).Suche in Google Scholar

[20] I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory. North-Holland Publ. Co., Amsterdam (1966).Suche in Google Scholar

[21] S. Yakubovich, and Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Kluwer Acad. Publ., Dordrecht (1994).10.1007/978-94-011-1196-6Suche in Google Scholar

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

© 2020 Diogenes Co., Sofia

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Erdélyi–Kober fractional integrals; Radon transforms; Grassmann manifolds

Artikel in diesem Heft

  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
Heruntergeladen am 22.4.2026 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2020-0050/html
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