Estimates with constants for some classes of oscillatory integrals

D. A. Popov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 52 (1997) no. 1, pp. 73-145

Bibliographie

[1] Fedoryuk M. V., Metod perevala, Nauka, M., 1977

[2] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, T. 2, Nauka, M., 1984 | Zbl

[3] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., “Trigonometricheskie integraly”, Izv. AN SSSR. Ser. matem., 43:5 (1979), 971–1003 | MR | Zbl

[4] Teoriya kratnykh trigonometricheskikh summ, Nauka, M., 1987

[5] Khermander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi. T. 1. Teoriya raspredelenii i analiz Fure, Mir, M., 1986

[6] Giiemin V., Sternberg S., Geometricheskie asimptotiki, Mir, M., 1981

[7] Arnold V. I., “Zamechaniya o metode statsionarnoi fazy i chislakh Kokstera”, UMN, 28:5 (1973), 17–44 | MR | Zbl

[8] Duistermat J. J., “Oscillatory integrals, Lagrange immersions and unfoldings of Singularities”, Comm. Pure and Appl. Math., 27 (1974), 207–281 | DOI | MR

[9] Colin de Verdier, “Nombre de pointe entere dans une famille homothetique de Domaines in $\mathbb R^n$”, Ann. Sci. Ecoll. Norm. Sup. Ser. 4, 40:4 (1977), 559–575

[10] Varchenko A. N., “Mnogogrannik Nyutona i otsenki ostsilliruyuschikh integralov”, Funkts. analiz i ego prilozh., 10:3 (1976), 13–38 | MR | Zbl

[11] Karpushkin V. N., “Ravnomernye otsenki ostsilliruyuschikh integralov s parabolicheskoi i giperbolicheskoi fazoi”, Trudy seminara im. I. G. Petrovskogo, 9, 1983, 3–39 | MR | Zbl

[12] Karpushkin V. N., “Teorema o ravnomernykh otsenkakh ostsilliruyuschikh integralov s fazoi, zavisyaschei ot dvukh peremennykh”, Trudy seminara im. I. G. Petrovskogo, 10, 1984, 250–269 | MR

[13] Bokhner S., Lektsii ob integralakh Fure, Fizmatgiz, M., 1962

[14] Titchmarsh E., Vvedenie v teoriyu integralov Fure, OGIZ, M., 1948

[15] Venkov A. B., “Spektralnaya teoriya avtomorfnykh funktsii, dzeta-funktsiya Selberga i nekotorye problemy analiticheskoi teorii chisel i matematicheskoi fiziki”, UMN, 34:3 (1979), 69–135 | MR | Zbl

[16] Khua Lo-gen, Metod trigonometricheskikh summ i ego prilozheniya v teorii chisel, Mir, M., 1964 | Zbl

[17] Varchenko A. N., “O chisle tselykh tochek v semeistve gomoteticheskikh oblastei v $\mathbb R^n$”, Funkts. analiz i ego prilozh., 17:2 (1983), 1–6 | MR | Zbl

[18] Pinsky M. A., Stanton N. K., Trapa P. F., “Fourier Series of Radial Functions in Several Variables”, J. Funct. Anal., 116 (1993), 111–132 | DOI | MR | Zbl

[19] Alimov Sh. A., Ilin V. A., Nikishin E. E., “Voprosy skhodimosti kratnykh trigonometricheskikh ryadov i spektralnykh razlozhenii, I”, UMN, 31:6 (1976), 28–83 | MR | Zbl

[20] Alimov Sh. A., Ashurov R. R., Pulatov A. K., “Kratnye ryady i integraly Fure”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 42, VINITI, M., 1989, 7–104 | MR

[21] Natterer F., Matematicheskie aspekty kompyuternoi tomografii, Mir, M., 1990 | Zbl

[22] Khermen G., Vosstanovlenie izobrazhenii po proektsiyam: Osnovy rekonstruktivnoi tomografii, Mir, M., 1983

[23] “Computed Tomography”, Proc. Symp. in Appl. Math., 27, ed. Shepp L. A., Amer. Math. Soc., 1983 | Zbl

[24] Pikalov V. V., Preobrazhenskii N. G., Rekonstruktivnaya tomografiya v gazodinamike i fizike plazmy, Nauka, Novosibirsk, 1987

[25] “Rekonstruktivnaya vychislitelnaya tomografiya”, TIIER, 71:3 (1983)

[26] “Mathematical Aspects of Computerized Tomography”, Lect. Notes in Med. Inform., 8, eds. Herman G. H., Nattere F., Springer-Verlag, Berlin, 1981

[27] Popov D. A., “Issledovanie algoritma svertki i obratnoi proektsii”, Voprosy rekonstruktivnoi tomografii, VTs SO AN SSSR, Novosibirsk, 1985, 125–136

[28] Popov D. A., “On convergence of a class of algorithms for the inversion of numerical Radon Transform”, Mathematical Problems of Tomography, Transl. Math. Monographs, 81, eds. Gelfand J. M., Gindikin S. G., Amer. Math. Soc., 1990, 7–65

[29] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, T. 1, Nauka, M., 1982

[30] Randol B., “On the Fourier Transform of the indicator function of a planar set”, Trans. Amer. Math. Soc., 139 (1969), 271–278 | DOI | MR | Zbl

[31] Randol B., “On the Asymptotic behavior of the Fourier Transform of the indicator function of a convex set”, Trans. Amer. Math. Soc., 139 (1969), 279–285 | DOI | MR | Zbl

[32] Ikromov I. A., “Ob otsenke preobrazovaniya Fure indikatora nevypuklykh oblastei”, Funkts. analiz i ego prilozh., 29:3 (1995), 16–24 | MR | Zbl

[33] Popov D. A., “O primenenii gladkikh regulyarizatorov dlya vychisleniya svertki”, DAN SSSR, 276:1 (1984), 38–42 | MR | Zbl

[34] Popov D. A., Sushko D. V., “O skhodimosti algoritmov chislennogo resheniya uravneniya svertki”, DAN SSSR, 315:2 (1990), 309–313

[35] Popov D. A., Suchko D. V., “Computation of Singular Convolutions”, Applied Problems of Radon Transform, Amer. Math. Soc. Transl. Ser. 2, 162, ed. Gindikin S., Amer. Math. Soc., 1994, 43–127 | Zbl

[36] Van der Corput, “Zahlentheoretische Abschatzungen”, Math. Ann., 84:1 (1921), 33–79 | MR

[37] Vinogradov I. M., Metod trigonometricheskikh summ, Nauka, M., 1971

[38] Chubarikov V. N., “Ob odnom kratnom trigonometricheskom integrale”, DAN SSSR, 227:6 (1976), 1308–1310 | MR | Zbl

[39] Chubarikov V. N., “O kratnykh ratsionalnykh trigonometricheskikh summakh i kratnykh integralakh”, Matem. zametki, 20:1 (1976), 61–68 | MR | Zbl

[40] Arkhipov G. I., Karatsuba A. A., Chubarikov V. N., Kratnye trigonometricheskie summy, Trudy MIAN, 151, Nauka, M., 1980 | MR | Zbl

[41] Yurinskii V. V., “Otsenki kharakteristicheskikh funktsii nekotorykh vyrozhdennykh raspredelenii”, Teoriya veroyatn. i ee primen., 18:1 (1972), 99–110 | MR

[42] Ikromov I. A., “Invariantnye otsenki dvumernykh trigonometricheskikh integralov”, Matem. sb., 180:8 (1989), 1017–1032

[43] Phong D. H., Stein E. M., “Oscillatory integrals with polinomial phases”, Invent. Math., 110 (1992), 39–62 | DOI | MR | Zbl

[44] Phong D. H., “Regularity of Fourier Integral Operators”, Proc. Intern. Congr. Math., V. 2 (Zurich, 1994), Basel, 1995, 862–874 | MR

[45] Stein E. M., Harmonic Analysis: Real Variable Mathods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, New York, 1993 | Zbl

[46] Narasimkhan R., Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, Mir, M., 1971 | Zbl

[47] Spravochnik po spetsialnym funktsiyam, eds. Abramovits M., Stigan I. M., Nauka, 1979

[48] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, T. 3, Fizmatgiz, M., 1960 | Zbl

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

[50] Vatson G. N., Teoriya besselevykh funktsii, T. 1, IL, M., 1949