Abstract
In this article a special type of Radon transform (Kipriyanov-Radon transform K γ ) is considered and some properties of this transform are proved. The main results of this work are the inversion formulas of K γ , which were obtained with a help of general B-hypersingular integrals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kipriyanov I.A., Lyakhov L.N., On Fourier, Fourier-Bessel, and Radon transforms, Doklady Akademii Nauk, 360:2 (1998), 157–160 [Dokl. Math. 57, 361–364 (1998)].
Kipriyanov I.A., Singular Elliptic Boundary Value Problems, Nauka, Moscow, 1996.
Lyakhov L.N., On the one class of hypersingular integrals, Russian Acad. Sci. Dokl. Math. 49(1), 83–87 (1994).
Aliev I.A., Rubin B., Wavelet-like trans-forms for admissible semi-groups; inversion formulas for potentials and Radon transforms, J. Fourier Anal. Appl. 11 (2005), no. 3, 333–352.
Rubin B., Reconstruction of functions from their integrals over κ-planes, Israel J. Math. 141 (2004), 93–117.
Rubin B., Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math. 29 (2002), no. 3, 471–497.
Rubin B., Helgason-Marchaud inversion formulas for Radon transforms, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3017–3023.
Levitan B.M., Expansion by Bessel functions in the Fourier series and integrals, Usp. Mat. Nauk, 4:2 (1951), 102–143.
John F., Plane Waves and Spherical Means, Wiley (Interscience), New York, 1955.
Lyakhov L.N., On the one class of hypersingular integrals, Dokl. Akad. Nauk SSSR 315(2), 291–296 (1990).
Samko S.G., On the Riesz-potential Spaces, Izv. Akad. Nauk SSSR, Ser. Mat. 40(5), 1443–1472 (1976).
Samko S.G., Kilbas A.A., and Marichev O.I., Integrals and Derivatives of Fractional Orders and Some of Their Applications Nauka i Tekhnika, Minsk, 1987.
Lyakhov L.N., Kipriyanov-Radon Transform, Trudy Mat. Inst. Steklov, 248 (2005), 153–163.
Gelfand I.M., Graev M.I. and Vilenkin I.Ya., Integral Geometry and Representation Theory, GIFML, Moscow, 1962.
Edwards R.E., Functional Analysis. Theory and Application, Mir, Moscow, 1969.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Gots, E., Lyakhov, L. (2008). On a Radon Transform. In: Bastos, M.A., Lebre, A.B., Speck, FO., Gohberg, I. (eds) Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol 181. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8684-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8684-9_8
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8683-2
Online ISBN: 978-3-7643-8684-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)