A range theorem for the Radon transform

Authors:
W. R. Madych and D. C. Solmon

Journal:
Proc. Amer. Math. Soc. **104** (1988), 79-85

MSC:
Primary 44A15; Secondary 26B40

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958047-7

MathSciNet review:
958047

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Abstract | References | Similar Articles | Additional Information

Abstract: Conditions are prescribed for a function which are sufficient to ensure that it is the Radon transform of a continuous function on such that as . Roughly speaking, these criteria involve smoothness and the classical polynomial consistency conditions up to order on . In particular, the result implies Helgason's Schwartz theorem for the Radon transform [Acta Math. **113** (1965)].

**[1]**A. P. Calderón and A. Zygmund,*Singular integral operators and differential equations*, Amer. J. Math.**79**(1957), 901-921. MR**0100768 (20:7196)****[2]**F. Gonzalez,*Radon transforms on Grassmann manifolds*, Ph.D. Thesis, M.I.T., Cambridge, Mass., 1984.**[3]**S. Helgason,*The Radon transform on Euclidean spaces, compact two point homogeneous spaces and Grassmann manifolds*, Acta Math.**113**(1965), 153-180. MR**0172311 (30:2530)****[4]**-,*The Radon transform*, Birkhäuser, Boston, Mass., 1980.**[5]**A. Hertle,*Habilitationsschrift*, Univ. Mainz (to appear).**[6]**P. D. Lax and R. S. Phillips,*The Paley-Wiener theorem for the Radon transform*, Comm. Pure Appl. Math.**23**(1970), 409-424. MR**0273309 (42:8189)****[7]**W. R. Madych,*On Littlewood-Paley functions*, Studia Math.**50**(1974), 43-63. MR**0344797 (49:9536)****[8]**J. Peters,*A Tauberian theorem for the Radon transform*, Houston J. Math.**8**(1982), 565-574. MR**688256 (84g:40003)****[9]**K. T. Smith, D. C. Solmon and S. L. Wagner,*Practical and mathematical aspects of the problem of reconstructing objects from radiographs*, Bull. Amer. Math. Soc.**83**(1977), 1227-1270. MR**0490032 (58:9394a)****[10]**D. C. Solmon,*Asymptotic formulas for the dual Radon transform and applications*, Math. Z.**195**(1987), 321-343. MR**895305 (88i:44006)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958047-7

Keywords:
Radon transform,
polynomial consistency condition,
asymptotic behavior

Article copyright:
© Copyright 1988
American Mathematical Society