Extremal functional interpolation and splines

Yu. N. Subbotin   ; S. I. Novikov   ; V. T. Shevaldin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 3, pp. 200-225

Résumé

The paper is a survey of the results obtained in the problems of extremal function interpolation over the past 50 years. Various statements of problems in this direction are analyzed both for the case of one variable and for the case of several variables. A special focus is put on the role of interpolation splines of different types (polynomial, interpolating in the mean, $\mathcal{L}$-splines, $m$-harmonic, etc.) in solving the problems of extremal function interpolation. Important applications of the results and methods of extremal interpolation to other problems in approximation theory and the theory of splines are specified.

Détail

Mots-clés : interpolation
Keywords: splines, approximation, differential operators, difference operators.

Bibliographie

[1] Gelfond A.O., Ischislenie konechnykh raznostei, Nauka, M., 1967, 376 pp.

[2] Krein M.G., “Integralnye uravneniya na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, Uspekhi mat. nauk, 13:5 (1958), 3–120

[3] Markushevich A.I., Kratkii kurs teorii analiticheskikh funktsii, Nauka, M., 1966, 388 pp.

[4] Novikov S.I., Shevaldin V.T., “Ob odnoi zadache ekstremalnoi interpolyatsii dlya funktsii mnogikh peremennykh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 7:1 (2001), 144–159

[5] Novikov S.I., “Periodicheskaya interpolyatsiya s minimalnym znacheniem normy m-i proizvodnoi”, Sib. zhurn. vychisl. matematiki, 9:2 (2006), 165–172

[6] Novikov S.I., “Interpolyatsiya s minimalnym znacheniem normy operatora Laplasa v share”, Zbirnik prats Institutu matematiki NAN Ukraïni, 5:1 (2008), 248–262

[7] Novikov S.I., “Zadachi interpolyatsii s minimalnym znacheniem normy operatora Laplasa na klasse interpoliruemykh dannykh”, Tr. Mezhdunar. letnei mat. shk.-konf. S. B. Stechkina po teorii funktsii (Tadzhikistan), Izd.-vo OOO "Ofset", Dushanbe, 2016, 182–185

[8] Patsko N.L., “Priblizhenie splainami na otrezke”, Mat. zametki, 16:3 (1974), 491–500

[9] Patsko N.L., “Priblizhenie splainami na otrezke v prostranstve $L_p$”, Mat. zametki, 58:2 (1995), 281–294

[10] Ryabenkii V.S., Filippov A.F., Ob ustoichivosti raznostnykh uravnenii, GITTL, M., 1956, 171 pp.

[11] Sobolev S.L., Lektsii po teorii kubaturnykh formul, Ch. 2, Izd-vo Novosibir. gos. un-ta, Novosibirsk, 1965, 263 pp.

[12] Stechkin S.B., Subbotin Yu.N., Splainy v vychislitelnoi matematike, Nauka, M., 1976, 248 pp.

[13] Subbotin Yu.N., “O svyazi mezhdu konechnymi raznostyami i sootvetstvuyuschimi proizvodnymi”, Tr. MIAN SSSR, 78 (1965), 24–42

[14] Subbotin Yu.N., “Funktsionalnaya interpolyatsiya v srednem s naimenshei n-i proizvodnoi”, Tr. MIAN SSSR, 88 (1967), 30–60

[15] Subbotin Yu.N., “Priblizhenie splain-funktsiyami i otsenki poperechnikov”, Tr. MIAN SSSR, 109 (1971), 35–60

[16] Subbotin Yu.N., “Priblizhenie splainami i gladkie bazisy v C(0,2$\pi$)”, Mat. zametki, 12:1 (1972), 43–51

[17] Subbotin Yu.N., “Ekstremalnaya funktsionalnaya interpolyatsiya i splainy”, Dokl. AN SSSR, 214:1 (1974), 56–58

[18] Subbotin Yu.N., “Ekstremalnye zadachi funktsionalnoi interpolyatsii i interpolyatsionnye v srednem splainy”, Tr. MIAN SSSR, 138 (1975), 118–173

[19] Subbotin Yu.N., “Ekstremalnye i approksimativnye svoistva splainov”, Teoriya priblizheniya funktsii, Tr. Mezhdunar. konf. po teorii priblizheniya funktsii (Kaluga, 24-28 iyulya 1975 g), Nauka, M., 1977, 341–345

[20] Subbotin Yu.N., “Ekstremalnaya funktsionalnaya interpolyatsiya v srednem s naimenshim znacheniem n-i proizvodnoi pri bolshikh intervalakh usredneniya”, Mat. zametki, 59:1 (1996), 114–132

[21] Subbotin Yu.N., “Ekstremalnaya v $L_p$ interpolyatsiya v srednem pri peresekayuschikhsya intervalakh usredneniya”, Izv. RAN. Ser. matematicheskaya, 61:1 (1997), 177–198

[22] Timofeev V.G., “Neravenstva tipa Kolmogorova s operatorom Laplasa”, Teoriya funktsii i approksimatsii, sb. st., Izd-vo SGU, Saratov, 1983, 84–92

[23] Tikhomirov V.M., Boyanov B.D., “O nekotorykh vypuklykh zadachakh teorii priblizhenii”, Serdica, 5:1 (1979), 83–96

[24] Chui Ch., Vvedenie v veivlety, Mir, M., 2001, 412 pp.

[25] Sharma A., Tsimbalario I., “Nekotorye lineinye differentsialnye operatory i obobschennye raznosti”, Mat. zametki, 21:2 (1977), 161–173

[26] Shevaldin V.T., “Ekstremalnaya interpolyatsiya s naimenshim znacheniem normy lineinogo differentsialnogo operatora”, Mat. zametki, 27:5 (1980), 721–740

[27] Shevaldin V.T., “Ob odnoi zadache ekstremalnoi interpolyatsii”, Mat. zametki, 29:4 (1981), 603–622

[28] Shevaldin V.T., “Nekotorye zadachi ekstremalnoi interpolyatsii v srednem”, Dokl. AN SSSR, 267:4 (1982), 803–805

[29] Shevaldin V.T., “Nekotorye zadachi ekstremalnoi interpolyatsii v srednem dlya lineinykh differentsialnykh operatorov”, Tr. MIAN SSSR, 164 (1983), 203–240

[30] Shevaldin V.T., “$\cal L$-splainy i poperechniki”, Mat. zametki, 33:5 (1983), 735–744

[31] Shevaldin V.T., “Otsenki snizu poperechnikov nekotorykh klassov periodicheskikh funktsii”, Tr. MIAN SSSR, 198 (1992), 242–267

[32] Shevaldin V.T., “Ekstremalnaya interpolyatsiya v srednem pri perekryvayuschikhsya intervalakh usredneniya i L-splainy”, Izv. RAN. Ser. matematicheskaya, 62:4 (1998), 201–224

[33] Atteia M., “Functions “spline”, definies sur un ensemble convexe”, Numer. Math., 12 (1968), 192–210

[34] Behforooz H., “Approximation by integro cubic splines”, Appl. Math. Comput., 175 (2006), 8–15 | DOI

[35] de Boor C., “How small can one make the derivatives of an interpolating function?”, J. Approx. Theory, 13:2 (1990), 105–116 | DOI

[36] de Boor C., H$\ddot{\mathrm{o}}$llig K., Riemenschneider S., Box splines, Springer, New York etc., 1993, 200 pp.

[37] Burenkov V.I., Sobolev spaces on domains, Teubner Texts in Math., 137, B. G. Teubner Verlag GmbH, Stuttgart, 1998, 312 pp.

[38] de Concini C., Procesi C., Topics in hyperplane arrangements, polytopes and box-splines, Springer, N Y etc., 2010, 384 pp.

[39] Favard J., “Sur l'interpolation”, J. Math. Pures Appl., 19:9 (1940), 281–306

[40] Fisher S., Jerome J., “Minimum norm extremals in function spaces”, Lecture Notes in Math., 479 (1975), 1–209

[41] Holmes R., Geometric functional analysis and its applications, Springer Verlag, N.Y. etc., 1975, 246 pp.

[42] Kunkle T., “Favard's interpolation problem in one or more variables”, Constr. Approx., 18:4 (2002), 467–478 | DOI

[43] Madych W.R., Nelson S.A., “Polyharmonic cardinal splines”, J. Approx. Theory, 60:2 (1990), 141–156 | DOI

[44] Madych W.R., Nelson S.A., “Polyharmonic cardinal splines: a minimization property”, J. Approx. Theory, 63:3 (1990), 303–320 | DOI

[45] Micchelli C.A., “Cardinal $\cal L$-splines”, Studies in spline functions and approximation theory, Acad. Press, N Y, 1976, 203–250

[46] Novikov S.I., “Generalization of the Rolle theorem”, East J. Approx, 1:4 (1995), 571–575

[47] Schoenberg I.J., “Cardinal interpolation and spline functions”, J. Approx. Theory, 2:2 (1969), 167–206 | DOI

[48] Schoenberg I.J., “On Micchelli's theory of cardinal L-splines”, Studies in spline functions and approximation theory, Acad. Press, N Y, 1976, 251–276

[49] Sharma A., Tzimbalario J., “A generalization of a result of Subbotin”, Approximation theory - ll, Proc. Internat. Sympos., Univ. Texas, 1976, Acad. Press, N Y, 1976, 557–562

[50] Subbotin Yu.N., “Some extremal problems of interpolation and interpolation in the mean”, East J. Approx., 2:2 (1996), 155–167

[51] Zhanlav T., Mijiddorj R., “Integro quintic splines and their approximation properties”, Appl. Math. Comput., 231 (2014), 536–543 | DOI

[52] Zhanlav T., Mijiddorj R., Behforooz H., “Construction of local integro quintic splines”, Commun. Numer. Anal., 2016, no. 2, 167–179 | DOI

[53] Zhanlav T., Mijiddorj R., “Convexity and monotonicity properties of the local integro cubic spline”, Appl. Math. Comput., 293 (2017), 131–137 | DOI