Stechkin's problem on the best approximation of an unbounded operator by bounded ones and related problems

V. V. Arestov ; R. R. Akopyan

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 7-31

Résumé

This paper discusses Stechkin's problem on the best approximation of a linear unbounded operator by bounded linear operators and related extremal problems. The main attention is paid to the approximation of differentiation operators in Lebesgue spaces on the axis and to the operator of the continuation of an analytic function to a domain from a part of the boundary of the domain. This is a review paper based on the materials of the authors' lecture on September 14, 2020, at the X Internet video-conference “Day of Mathematics and Mechanics” of four institutes of the Russian Academy of Sciences: Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of RAS (Yekaterinburg), Sobolev Institute of the Siberian Branch of RAS (Novosibirsk), Steklov Mathematical Institute (Moscow), and the St. Petersburg Department of the Steklov Mathematical Institute. The lecture of the authors was dedicated to the 100th anniversary of the birth of Sergei Borisovich Stechkin. The problem of the best approximation of a linear unbounded operator by bounded ones is one of his legacies. We tried to at least partially reflect the new results, methods, and statements that appeared in this topic after the publication of the review papers (Arestov, Gabushin, 1995–1996). The material on this topic is wide; the selection of the material for the lecture and paper is the responsibility of the authors.

Détail

Keywords: Stechkin's problem, recovery, unbounded linear operator, differentiation operator, Kolmogorov inequality, analytic functions, boundary values.

Bibliographie

[1] Stechkin S.B., “Neravenstva mezhdu normami proizvodnykh proizvolnoi funktsii”, Acta Sci. Math., 26:3–4 (1965), 225–230

[2] Stechkin S.B., “Nailuchshee priblizhenie lineinykh operatorov”, Mat. zametki, 1:2 (1967), 137–148

[3] Stechkin S.B., Izbrannye trudy. Matematika, Nauka. Fizmatlit, M., 1998, 384 pp.

[4] Ivanov V.K., Vasin V.V., Tanana V.P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978, 206 pp.

[5] Arestov V.V., Gabushin V.N., “Nailuchshee priblizhenie neogranichennykh operatorov ogranichennymi”, Izv. vuzov. Mat., 1995, no. 11, 42–68

[6] Arestov V.V., “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, Uspekhi mat. nauk, 51:6 (1996), 89–124 | DOI

[7] Tikhomirov V.M., Magaril-Ilyaev G.G., “Neravenstva dlya proizvodnykh”: Kolmogorov A.N., Izbrannye trudy. Matematika i mekhanika, Nauka, M., 1985, 387–390

[8] Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A., Neravenstva dlya proizvodnykh i ikh prilozheniya, Nauk. dumka, Kiev, 2003, 591 pp.

[9] Gabushin V.N., “Nailuchshee priblizhenie funktsionalov na nekotorykh mnozhestvakh”, Mat. zametki, 8:5 (1970), 551–562

[10] Hardy G.H., Littlewood J.E., “Contribution to the arithmetic theory of series”, Proc. London Math. Soc. (2), 11 (1912), 411–478

[11] Landau E., “Einige Ungleichungen fur zweimal differentierbare Funktionen”, Proc. London Math. Soc. (2), 13 (1913), 43–49

[12] Hadamard J., “Sur le module maximum d'une fonction et de ses derivees”, Soc. math. France, Comptes rendus des Seances, 41 (1914), 68–72

[13] Bosse Yu.G. (Shilov G.E.), “O neravenstvakh mezhdu proizvodnymi”, Sb. rabot stud. nauch. kruzhkov MGU, 1 (1937), 17–27

[14] Kolmogorov A.N., “O neravenstvakh mezhdu verkhnimi granyami posledovatelnykh proizvodnykh proizvolnoi funktsii na beskonechnom intervale”, Izbrannye trudy. Matematika, mekhanika, Nauka, M., 1985, 252–263; Уч. зап. Моск. ун-та. Математика, 30, No 3, 1939, 3–16

[15] Gabushin V.N., “Neravenstva dlya norm funktsii i ee proizvodnykh v metrikakh $L_p$”, Mat. zametki, 1:3 (1967), 291–298

[16] Buslaev A.P., Magaril-Ilyaev G.G., Tikhomirov V.M., “O suschestvovanii ekstremalnoi funktsii v neravenstve dlya proizvodnykh”, Mat. zametki, 32:6 (1982), 823–834

[17] Arestov V.V., “O nailuchshem priblizhenii operatorov differentsirovaniya”, Mat. zametki, 1:2 (1967), 149–154

[18] Buslaev A.P., “O priblizhenii operatora differentsirovaniya”, Mat. zametki, 29:5 (1981), 731–742

[19] Timofeev V.G., “Metod N. P. Kuptsova postroeniya ekstremalnoi funktsii v neravenstve mezhdu ravnomernymi normami proizvodnykh funktsii na poluosi”, Tr. In-ta matematiki i mekhaniki UrO RAN, 25:2 (2019), 220–239 | DOI

[20] Subbotin Yu.N., Taikov L.V., “Nailuchshee priblizhenie operatora differentsirovaniya v prostranstve $L_2$”, Mat. zametki, 3:2 (1968), 157–164

[21] Taikov L.V., “Neravenstva tipa Kolmogorova i nailuchshie formuly chislennogo differentsirovaniya”, Mat. zametki, 4:2 (1968), 233–238

[22] Arestov V.V., “O tochnykh neravenstvakh mezhdu normami funktsii i ikh proizvodnykh”, Acta Sci. Math., 33:3–4 (1972), 243–267

[23] Arestov V.V., “Priblizhenie operatorov, invariantnykh otnositelno sdviga”, Tr. MIAN SSSR, 138 (1975), 43–70

[24] Arestov V.V., O nailuchshem priblizhenii operatorov differentsirovaniya v ravnomernoi metrike, Dis. ... kand. fiz.-mat. nauk, Sverdlovsk, 1969, 89 pp.

[25] Berdyshev V.I., “Nailuchshee priblizhenie v $L[0,\infty)$ operatora differentsirovaniya”, Mat. zametki, 9:5 (1971), 477–481

[26] Gabushin V.N., “O nailuchshem priblizhenii operatora differentsirovaniya na polupryamoi”, Mat. zametki, 6:5 (1969), 573–582

[27] Arestov V.V., Filatova M.A., “Best approximation of the differentiation operator in the space $L_2$ on the semiaxis”, J. Approx. Theory, 187 (2014), 65–81 | DOI

[28] Khardi G.G., Littlvud Dzh.E., Polia G., Neravenstva, IL, M., 1948, 456 pp.

[29] Gabushin V.N., “O nailuchshem priblizhenii operatora differentsirovaniya v metrike $L_p$”, Mat. zametki, 12:5 (1972), 531–538

[30] Arestov V.V., “Nailuchshee priblizhenie neogranichennykh operatorov, invariantnykh otnositelno sdviga, lineinymi ogranichennymi operatorami”, Tr. MIAN, 198 (1992), 3–20

[31] Arestov V.V., “O ravnomernoi regulyarizatsii zadachi vychisleniya znachenii operatora”, Mat. zametki, 22:2 (1977), 231–244

[32] Arestov V.V., “Nailuchshee vosstanovlenie operatorov i rodstvennye zadachi”, Tr. MIAN SSSR, 189 (1989), 3–20

[33] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974, 333 pp.

[34] Lorentz G.G., Approximation of functions, Holt, Rinehalt Winston, N Y, 1966, 188 pp.

[35] Tikhomirov V.M., Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976, 304 pp.

[36] Arestov V.V., “Priblizhenie invariantnykh operatorov”, Mat. zametki, 34:1 (1983), 9–29

[37] Arestov V.V., “Priblizhenie operatorov tipa sverki lineinymi ogranichennymi operatorami”, Tr. MIAN, 145 (1980), 3–19

[38] Arestov V.V., “O nailuchshem priblizhenii operatora differentsirovaniya”, Priblizhenie funktsii polinomami i splainami, cb. st., Sverdlovsk, 1985, 3–14

[39] Arestov V.V., “On the best approximation of the differentiation operator”, Ural Math. J., 1:1 (2015), 20–29 | DOI

[40] Khermander L., Otsenki dlya operatorov, invariantnykh otnositelno sdviga, IL, M., 1962, 71 pp.

[41] Larsen R., An introduction to the theory of multipliers, Springer, Berlin etc., 1971, 282 pp.

[42] Figa-Talamanca A., “Translation invariant operators in $L^p$”, Duke. Math. J., 32 (1965), 495–502

[43] Arestov V.V., “O sopryazhennosti prostranstva multiplikatorov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 25:4 (2019), 5–14 | DOI

[44] Arestov V., “Uniform approximation of differentiation operators by bounded linear operators in the space $L_r$”, Anal. Math., 46:3 (2020), 425–445 | DOI

[45] Arestov V.V., “Nailuchshee ravnomernoe priblizhenie operatora differentsirovaniya ogranichennymi v prostranstve $L_2$ operatorami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 24:4 (2018), 34–56 | DOI

[46] Arestov V.V., “Best approximation of a differentiation operator on the set of smooth functions with exactly or approximately given Fourier transform”, Mathematical Optimization Theory and Operations Research (MOTOR 2019), eds. M. Khachay, Y. Kochetov, P. Pardalos, Springer, Cham, 2019, 434–448

[47] Timan A.F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960, 624 pp.

[48] Konovalov V.N., “Tochnye neravenstva dlya norm funktsii, tretikh chastnykh, vtorykh smeshannykh ili kosykh proizvodnykh”, Mat. zametki, 23:1 (1978), 67–78

[49] Timoshin O.A., “Nailuchshee priblizhenie operatora vtoroi smeshannoi proizvodnoi v metrikakh $L$ i $C$ na ploskosti”, Mat. zametki, 36:3 (1984), 369–375

[50] Timoshin O.A., “Tochnye neravenstva mezhdu normami chastnykh proizvodnykh vtorogo i tretego poryadka”, Dokl. RAN, 344:1 (1995), 20–22

[51] Timofeev V.G., “Neravenstvo tipa Landau dlya funktsii neskolkikh peremennykh”, Mat. zametki, 37:5 (1985), 676–689

[52] Taikov L.V., “O nailuchshem priblizhenii v srednem nekotorykh klassov analiticheskikh funktsii”, Mat. zametki, 1:2 (1967), 155–162

[53] Akopyan R.R., Saidusainov M.S., “Tri ekstremalnye zadachi v prostranstvakh Khardi i Bergmana analiticheskikh funktsii v kruge”, Tr. In-ta matematiki i mekhaniki UrO RAN, 23:3 (2017), 22–32

[54] Taikov L.V., “Analiticheskoe prodolzhenie funktsii s oshibkoi”, Tr. MIAN SSSR, 109 (1971), 61–64

[55] Akopyan R.R., “Nailuchshee priblizhenie operatora analiticheskogo prodolzheniya na klasse analiticheskikh v koltse funktsii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18:4 (2012), 3–13

[56] Osipenko K.Yu., Stesin M.I., “O nekotorykh zadachakh optimalnogo vosstanovleniya analiticheskikh i garmonicheskikh funktsii po netochnym dannym”, Sib. mat. zhurn., 34:3 (1993), 144–160

[57] Osipenko K.Yu., Stesin M.I., “Optimalnoe vosstanovlenie proizvodnykh ogranichennykh analiticheskikh i garmonicheskikh funktsii po netochnym dannym”, Mat. zametki, 53:5 (1993), 87–97

[58] Osipenko K.Yu., “Ob n-poperechnikakh, optimalnykh kvadraturnykh formulakh i optimalnom vosstanovlenii funktsii, analiticheskikh v polose”, Izv. RAN. Ser. matematicheskaya, 58:4 (1994), 55–79

[59] Osipenko K.Yu., “Neravenstva dlya proizvodnykh analiticheskikh v polose funktsii”, Mat. zametki, 56:4 (1994), 114–122

[60] Osipenko K.Yu., Optimal recovery of analytic functions, NOVA Science Publ.Inc., Huntington, 2000, 229 pp.

[61] Osipenko K.Yu., “Ob optimalnykh metodakh vosstanovleniya v prostranstvakh Khardi - Soboleva”, Mat. sb., 192:2 (2001), 67–86

[62] Osipenko K.Yu., “Neravenstvo Khardi - Littlvuda - Polia dlya analiticheskikh funktsii iz prostranstv Khardi - Soboleva”, Mat. sb., 197:3 (2006), 15–34

[63] Osipenko K.Yu., Stessin M.I., “Hadamard and Schwarz type theorems and optimal recovery in spaces of analytic functions”, Constr. Approx., 31, 37–67 | DOI

[64] Akopyan R.R., “Analog teoremy o dvukh konstantakh i optimalnoe vosstanovlenie analiticheskikh funktsii”, Mat. sb., 210:10 (2019), 3–36

[65] Akopyan R.R., “Optimal recovery of a derivative of an analytic function from values of the function given with an error on a part of the boundary. II”, Anal. Math., 46:3 (2020), 409–424 | DOI

[66] Lavrentev M.M., Romanov V.G., Shishatskii S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980, 286 pp.