The Generalized Radon Transform on the Plane, the Inverse Transform, and the Cavalieri Conditions

D. A. Popov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 4, pp. 38-53.

Résumé

In the two-dimensional case, the generalized Radon transform takes each function supported in a disk to the values of the integrals of that function over a family of curves. We assume that the curves differ only slightly from straight lines and the network formed by these curves has the same topological structure as the network of straight lines. Thus, the generalized Radon transform specifies a function on the set of straight lines. Under these conditions, we obtain a solution of the inversion problem for the generalized Radon transform and indicate a Cavalieri condition describing the range of this transform in the space of functions on the set of straight lines.

Bibliographie

[1] Gelfand I. M., Graev M. I., Vilenkin N. Ya., Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii. Obobschennye funktsii, vyp. 5, Fizmatgiz, M., 1967 | MR

[2] Gelfand I. M., Gindikin S. G., Graev M. I., Izbrannye zadachi integralnoi geometrii, Dobrosvet, M., 2000 | MR

[3] Khelgason S., Preobrazovanie Radona, Mir, M., 1983 | MR | Zbl

[4] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987 | MR

[5] Gelfand I. M., Graev M. I., Shapiro Z. Ya., “Differentsialnye formy i integralnaya geometriya”, Funkts. analiz i ego pril., 3:2 (1969), 24–40 | MR

[6] Guillemin V., “On some results of Gelfand in integral geometry”, Pseudodifferential operators and Applications (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., 43, Providence, RI, 1985, 149–155 | DOI | MR | Zbl

[7] Giiemin V., Sternberg S., Geometricheskie asimptotiki, Mir, M., 1981 | MR

[8] Quinto E. T., “The dependence of the generalized Radon transform on defining measures”, Trans. Amer. Math. Soc., 257:2 (1980), 331–346 | DOI | MR | Zbl

[9] Greenleaf A., Uhlmann G., “Nonlocal inversion formulas for the $X$-ray transformation”, Duke Math. J., 58:1 (1989), 205–240 | DOI | MR | Zbl

[10] Stein E. M., Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993 | MR | Zbl

[11] Phong D. H., “Regularity of Fourier integral operators”, Proc. Intern. Congress of Math., Vol. 2 (Zurich, 1994), Birkhauser, Basel, 1995, 862–874 | DOI | MR | Zbl

[12] Phong D. H., Stein E. M., “Singular Radon transforms and oscillatory integrals”, Duke Math. J., 58:2 (1989), 347–369 | DOI | MR | Zbl

[13] Quinto E. T., “Radon transforms on curves in the plane”, Thomography, Impedance Imaging, and Integral Geometry, Lectures in Appl. Math., 30, Amer. Math. Soc., Providence, RI, 1994, 231–244. | MR | Zbl

[14] Boman J., Quinto E. T., “Support theorems for real-analytical Radon transforms”, Duke Math. J., 55:4 (1987), 943–948 | DOI | MR | Zbl

[15] Agranovsky M. L., Quinto E. T., “Injectivity sets for the Radon transform over circles and complete systems of radial functions”, J. Funct. Anal., 139 (1996), 383–414 | DOI | MR | Zbl

[16] Greenleaf A., Uhlmann G., “Estimates for singular Radon transforms and pseudodifferential operators with singular symbols”, J. Funct. Anal., 89 (1990), 202–239 | DOI | MR

[17] Thomson A. A., “Sobolev estimates for singular Radon transforms”, J. Funct. Anal., 112 (1993), 61–96 | DOI | MR

[18] Ion F., Ploskie volny i sfericheskie srednie v prilozhenii k differentsialnym uravneniyam s chastnymi proizvodnymi, Mir, M., 1958

[19] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR

[20] Lavrentev M. M., Vasilev V. G., Romanov V. G., Mnogomernye obratnye zadachi dlya differentsialnykh uravnenii, Nauka, Novosibirsk, 1969 | MR

[21] Romanov V. G., “Vosstanovlenie funktsii cherez integraly po semeistvu krivykh”, Sib. matem. zh., 7:5 (1967), 1206–1208 | MR

[22] Cormack A. M., “Radon transform on a family curves in the plane, I”, Proc. Amer. Math. Soc., 83 (1981), 325–330 ; II 86 (1982), 293–298 | DOI | MR | Zbl | DOI | MR | Zbl

[23] Lax P. D., Phillips P. S., “The Paley–Wiener theorem for the Radon transform”, Comm. Pure Appl. Math., 23:3 (1970), 409–424 ; Matematika, 16:3 (1972), 97–112 | DOI | MR | Zbl | MR | Zbl

[24] Mukhometov R. G., “O zadache integralnoi geometrii”, Matematicheskie problemy geofiziki, vyp. 6, Novosibirsk, 1975, 212–242 | Zbl

[25] Anikonov Yu. E., “O razreshimosti odnoi zadachi integralnoi geometrii”, Matem. sb., 101(143):2 (1976), 271–279 | MR

[26] Mukhometov R. G., “Zadacha vosstanovleniya dvumernoi rimanovoi metriki i integralnaya geometriya”, DAN SSSR, 232:1 (1977), 32–35 | MR | Zbl

[27] Bukhgeim A. L., Vvedenie v teoriyu obratnykh zadach, Nauka, Novosibirsk, 1988 | MR

[28] Marr R. B., “On the reconstruction of a function on a circular domain from a sampling of its line integrals”, J. Math. Anal. Appl., 45 (1974), 357–374 | DOI | MR | Zbl

[29] Suetin P. K., Ortogonalnye polinomy, Nauka, M., 1979 | MR | Zbl

[30] Trev F., Vvedenie v teoriyu psevdodifferentsialnykh operatorov i integralnykh operatorov Fure, T. 1, 2, Mir, M., 1984 | Zbl

[31] Popov D. A., “Otsenki s konstantami dlya nekotorogo klassa ostsilliruyuschikh integralov”, UMN, 52:1 (1997), 77–148 | DOI | MR | Zbl

[32] Erdeii A., Asimptoticheskie razlozheniya, Fizmatgiz, M., 1962

[33] Hörmander L., “The spectral function of elliptic operator”, Acta Math., 121:3–4 (1968), 193–218 ; Matematika, 13:6 (1969), 114–137 | DOI | MR | Zbl

[34] Deans R. S., The Radon Transform and Some Its Applications, John Wiley and Sons, New York, 1983 | MR | Zbl

[35] Born M., Volf E., Osnovy optiki, Nauka, M., 1973

[36] Beitmen G., Erdeii A., Tablitsy integralnykh preobrazovanii, T. 2, Nauka, M., 1970

[37] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady spetsialnykh funktsii, Nauka, M., 1983 | MR | Zbl