Geodesic
Direct and inverse theorems in approximation theory for periodic functions in S. B. Stechkins papers and the development of these theorems
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 5-17
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S. B. Stechkin made a considerable contribution to an important aspect of approximation theory for periodic functions concerned with direct and inverse theorems. We present the history of development of this research area and pose new problems.
Keywords: best approximation, module of continuity, Jackson–Stechkin inequality, inverse inequality. 
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V. I. Ivanov. Direct and inverse theorems in approximation theory for periodic functions in S. B. Stechkins papers and the development of these theorems. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 5-17. http://geodesic.mathdoc.fr/item/TIMM_2010_16_4_a0/

[1] Akhiezer N.I., Lektsii po teorii approksimatsii, Gostekhizdat, M., 1947, 323 pp.

[2] Babenko A.G., “O neravenstve Dzheksona - Stechkina dlya nailuchshikh $L_2$-priblizhenii funktsii trigonometricheskimi polinomami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 7, no. 1, Ekaterinburg, 2001, 30–46

[3] Bari N.K., Stechkin S.B., “Nailuchshie priblizheniya i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tr. Mosk. mat. o-va, 5, 1956, 483–522 | MR | Zbl

[4] Berdyshev V.I., “O teoreme Dzheksona v $L_p$”, Tr. MIAN SSSR, 88, 1967, 3–16 | MR | Zbl

[5] Bernshtein S.N., “O nailuchshem priblizhenii nepreryvnykh funktsii posredstvom mnogochlenov dannoi stepeni”, Soobsch. Khark. mat. o-va, 13, no. 2, 1912, 49–144

[6] Bernshtein S.N., “O svoistvakh odnorodnykh funktsionalnykh klassov”, Dokl. AN SSSR, 57:2 (1947), 111–114 | MR | Zbl

[7] Vasilev S.N., “Neravenstvo Dzheksona - Stechkina v $L_2[-,]$”, Trudy Instituta matematiki i mekhaniki UrO RAN, 7, no. 1, Ekaterinburg, 2001, 75–84

[8] Geit V.E., “O tochnosti nekotorykh neravenstv v teorii priblizhenii”, Mat. zametki, 10:5 (1971), 571–582 | MR | Zbl

[9] Geit V.E., “Teoremy vlozheniya dlya nekotorykh klassov periodicheskikh nepreryvnykh funktsii”, Izv. vuzov. Matematika, 1972, no. 4, 67–77

[10] Ivanov V.I., “Pryamye i obratnye teoremy teorii priblizhenii v metrike $L_p$ dlya $0 p 1$”, Mat. zametki, 18:5 (1975), 641–658 | MR | Zbl

[11] Ivanov V.I., “O priblizhenii funktsii v prostranstvakh $L_p$”, Mat. zametki, 56:2 (1994), 15–40 | MR | Zbl

[12] Ivanov V.I., Smirnov O.I., Konstanty Dzheksona i konstanty Yunga v prostranstvakh $L_p$, TulGU, Tula, 1995, 192 pp.

[13] Ilyasov N.A., “Teoremy vlozheniya dlya nekotorykh klassov periodicheskikh funktsii v $L_p$, $1

\infty$”, Dokl. AN SSSR, 276:6 (1984), 1301–1304 | MR | Zbl

[14] Kozko A.I., Rozhdestvenskii A.V., “O neravenstve Dzheksona s obobschennym modulem nepreryvnosti”, Mat. sb., 195:8 (2004), 3–46 | MR

[15] Kolomoitsev Yu.S., “O modulyakh gladkosti i multiplikatorakh Fure v $L_p$, $0 p 1$”, Ukr. mat. zhurn., 59:9 (2007), 1221–1238 | MR | Zbl

[16] Korneichuk N.P., “Tochnaya konstanta v neravenstve D.Dzheksona o nailuchshem ravnomernom priblizhenii nepreryvnykh periodicheskikh funktsii”, Dokl. AN SSSR, 145:3 (1962), 514–515 | MR | Zbl

[17] Lozinskii S.M., “Obraschenie teoremy Dzheksona”, Dokl. AN SSSR, 83:5 (1952), 645–647 | MR | Zbl

[18] Natanson G.I., Timan M.F., “Srednie geometricheskie posledovatelnosti nailuchshikh priblizhenii”, Vestn. LGU, 3:19 (1979), 50–52

[19] Nikolskii S.M., “Obobschenie odnogo neravenstva S.N. Bernshteina”, Dokl. AN SSSR, 60:9, 1948

[20] Runovskii K.V., “O semeistvakh lineinykh polinomialnykh operatorov v prostranstvakh $L_p$, $0 p 1$”, Mat. sb., 184:2 (1993), 33–42 | Zbl

[21] Runovskii K.V., “Pryamaya i obratnaya teoremy teorii priblizhenii v prostranstvakh $L_p$ s $0 p 1$”, Dokl. RAN, 331:5 (1993), 684–686 | Zbl

[22] Runovskii K.V., “O priblizhenii semeistvami lineinykh polinomialnykh operatorov v prostranstvakh $L_p$, $0 p 1$”, Mat. sb., 185:8 (1994), 81–102 | Zbl

[23] Stechkin S.B., “Obobschenie nekotorykh neravenstv S.N.Bernshteina”, Dokl. AN SSSR, 60:9 (1948), 1511–1514 | MR | Zbl

[24] Stechkin S.B., “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Dokl. SSSR, 65:2 (1949), 135–137 | MR | Zbl

[25] Stechkin S.B., “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Izv. AN SSSR. Ser. mat., 15:3 (1951), 219–242 | MR | Zbl

[26] Stechkin S.B., “O teoreme Kolmogorova - Seliverstova”, Izv. AN SSSR. Ser. mat., 17:6 (1953), 499–512 | MR | Zbl

[27] Stechkin S.B., “O nailuchshem priblizhenii sopryazhennykh funktsii trigonometricheskimi polinomami”, Izv. AN SSSR. Ser. mat., 20:2 (1956), 197–206 | MR | Zbl

[28] Stechkin S.B., “O priblizhenii periodicheskikh funktsii summami Feiera”, Tr. MIAN SSSR, 62, 1961, 48–60 | MR | Zbl

[29] Stechkin S.B., “Zamechanie k teoreme Dzheksona”, Tr. MIAN SSSR, 88, 1967, 17–19 | MR | Zbl

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

[31] O.V.Besov [i dr.], “Sergei Borisovich Stechkin”, Uspekhi mat. nauk, 51:6 (1996), 3–10 | MR

[32] V.V.Arestov, V.I.Berdyshev, Yu.N.Subbotin, N.I.Chernykh, “S.B. Stechkin i teoriya priblizhenii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 4, Ekaterinburg, 1996, 3–16

[33] S.V.Konyagin, S.A.Telyakovskii, “Pamyati Sergeya Borisovicha Stechkina”, Fundament. i prikl. matematika, 3:4 (1997), 955–960 | MR | Zbl

[34] Storozhenko E.A., Krotov V.G., Osvald P., “Pryamye i obratnye teoremy tipa Dzheksona v prostranstvakh $L_p$, $0 p 1$”, Mat. sb., 98:3 (1975), 395–415 | MR | Zbl

[35] Storozhenko E.A., Osvald P., “Teoremy Dzheksona v prostranstvakh $L_p(\mathbb R_n)$, $0 p 1$”, Sib. mat. zhurn., 19:4 (1978), 888–901 | MR | Zbl

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

[37] Timan A.F., Timan M.F., “Obobschennyi modul nepreryvnosti i nailuchshee priblizhenie v srednem”, Dokl. AN SSSR, 71 (1950), 17–19

[38] Timan M.F., “Obratnye teoremy konstruktivnoi teorii funktsii v prostranstvakh $L_p$”, Mat. sb., 46:1 (1958), 125–132 | MR

[39] Timan M.F., “Osobennosti osnovnykh teorem konstruktivnoi teorii funktsii v prostranstvakh $L_p$”, Issled. po sovrem. probl. konstruktivnoi teorii funktsii, AN Azerb. SSR, Baku, 1965, 18–25

[40] Timan M.F., “O poryadkovoi tochnosti mazhoranty modulya gladkosti nepreryvnykh funktsii”, Nauch. konf. DSKhI, Promin, Dnepropetrovsk, 1965, 62–65

[41] Timan M.F., “O teoreme Dzheksona v prostranstvakh $L_p$”, Ukr. mat. zhurn., 1966, no. 1, 134–137

[42] Timan M.F., Approksimatsiya i svoistva periodicheskikh funktsii, Nauk. dumka, Kiev, 2009, 376 pp.

[43] Trigub R.M., “Konstruktivnye kharakteristiki nekotorykh klassov funktsii”, Izv. AN SSSR. Ser. mat., 29:3 (1965), 615–630 | MR | Zbl

[44] Chernykh N.I., “O neravenstve Dzheksona v $L_2$”, Tr. MIAN SSSR, 88, 1967, 71–74 | MR | Zbl

[45] Chernykh N.I., “O nailuchshem priblizhenii periodicheskikh funktsii trigonometricheskimi polinomami v $L_2$”, Mat. zametki, 2:5 (1967), 513–522 | MR | Zbl

[46] Chernykh N.I., “Neravenstvo Dzheksona v $L_p(0, 2)$ s tochnoi konstantoi”, Tr. MIAN, 198, 1992, 232–241 | Zbl

[47] Foucart S., Kryakin Yu., Shadrin A., “On the exact constant in the Jackson - Stechkin inequality for the uniform metric”, Constr. Approx., 29 (2009), 157–179 | DOI | MR | Zbl

[48] Jackson D., Über die Genauigkeit der Annaheruny stetiger Funktionen durch ganze rationale Funktionen gegebenen Frades und trigonometrischen Summen gegebener Ordnung, Diss., Göttingen, 1911

[49] Quade E.S., “Trigonometric approximation in the mean”, Duke Math. J., 3 (1937), 529–543 | DOI | MR | Zbl

[50] Radoslavova T.V., “Decrease orders of the $L^p$-moduli of continuity ($0

\le\infty$ )”, Anal. Math., 5 (1979), 219–234 | DOI | MR | Zbl

[51] Rathore R.K.S., “The problem of A. F.Timan on the precise order of decrease of the best approximation”, J. Approx. Theory, 77 (1994), 153–166 | DOI | MR | Zbl

[52] Riesz M., “Eine trigonometrische Interpolationsformel und einige Ungleichungen fr Polynome”, Jahresber. Deutsch. Math.-Verein, 23 (1914), 354–368 | Zbl

[53] Salem R., “Sur certaines fonctions continues et le proprits de leur sries de Fourier”, Comptes Rendus Acad. Sci., 201 (1935), 703–705 | Zbl

[54] Valle Poussin C. de la., Lecons sur l'approximation des fonctions d'une variable relle, Paris, 1919, 151 pp. | Zbl

[55] Zygmund A., “Smooth functions”, Duke Math. J., 12 (1945), 47–76 | DOI | MR | Zbl

[56] Zygmund A., “A remark on the modules of continuity”, Unic. Nac. Tucuman Revista A, 7 (1950), 259–269 | MR | Zbl