Geodesic
Optimal quadrature formulae
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1035-1055 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this survey we consider results and open problems related to two major ideas in the theory of optimal quadrature formulae: the ideas of Gauss and Kolmogorov. 
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B. D. Boyanov. Optimal quadrature formulae. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1035-1055. http://geodesic.mathdoc.fr/item/RM_2005_60_6_a3/

[1] J.-E. Andersson, B. D. Bojanov, “A note on the optimal quadrature in $H^p$”, Numer. Math., 44:2 (1984), 301–308 | DOI | MR | Zbl

[2] R. B. Barrar, H. L. Loeb, “On a nonlinear characterization problem for monosplines”, J. Approx. Theory, 18:3 (1976), 220–240 | DOI | MR | Zbl

[3] D. L. Barrow, “On multiple node Gaussian quadrature formulae”, Math. Comp., 32:142 (1978), 431–439 | DOI | MR | Zbl

[4] B. D. Boianov, G. R. Grozev, A. A. Zhensykbaev, “Generalized Gaussian quadrature formulas for weak Tchebyshev systems”, Optimal Recovery, Proceedings of the 2nd International Symposium on Optimal Control (Varna, 1989), eds. B. Bojanov and H. Wozniakowski, Nova Science, Commack, 1992, 115–140 | MR

[5] B. D. Boyanov, “Optimalnaya skorost integrirovaniya i $\varepsilon$-entropii odnogo klassa analiticheskikh funktsii”, Matem. zametki, 14:1 (1973), 3–10 | MR

[6] B. D. Bojanov, “On the existence of optimal quadrature formulae for smooth functions”, Calcolo, 16:1 (1979), 61–70 | DOI | MR | Zbl

[7] B. D. Bojanov, “Uniqueness of the optimal nodes of quadrature formulae”, Math. Comp., 36:154 (1981), 525–546 | DOI | MR | Zbl

[8] B. Bojanov, “Optimal recovery of functions and integrals”, First European Congress of Mathematics, V. I. Invited Lectures (Part 1) (Paris, 1992), Progr. Math., 119, Birkhäuser, Basel, 1994, 371–390 | MR | Zbl

[9] B. Bojanov, “An extension of the Pizzetti formula for polyharmonic functions”, Acta Math. Hungar., 91:1–2 (2001), 99–113 | DOI | MR | Zbl

[10] B. Bojanov, D. Braess, N. Dyn, “Generalized Gaussian quadrature formulas”, J. Approx. Theory, 48:4 (1986), 335–353 | DOI | MR | Zbl

[11] B. D. Bojanov, D. K. Dimitrov, “Gaussian extended cubature formulae for polyharmonic functions”, Math. Comp., 70:234 (2001), 671–683 | DOI | MR | Zbl

[12] B. Bojanov, V. Gushev, “A quadrature formula of Gaussian type on the sphere”, East J. Approx., 11:2 (2005), 119–130 | MR

[13] B. Bojanov, G. Nikolov, “Comparison of Birkhoff type quadrature formulae”, Math. Comp., 54:190 (1990), 627–648 | DOI | MR

[14] B. D. Bojanov, P. P. Petrov, “Gaussian interval quadrature formula”, Numer. Math., 87:4 (2001), 625–643 | DOI | MR | Zbl

[15] B. Bojanov, P. Petrov, “Uniqueness of the Gaussian interval quadrature formula”, Numer. Math., 95:1 (2003), 53–62 | DOI | MR | Zbl

[16] B. Bojanov, P. Petrov, “Gaussian interval quadrature formulae for Tchebycheff systems”, SIAM J. Numer. Anal., 43:2 (2005), 787–795 | DOI | MR | Zbl

[17] B. Bojanov, G. Petrova, “Numerical integration over a disc. A new Gaussian quadrature formula”, Numer. Math., 80:1 (1998), 39–59 | DOI | MR | Zbl

[18] B. Bojanov, G. Petrova, “Uniqueness of the Gaussian quadrature for a ball”, J. Approx. Theory, 104:1 (2000), 21–44 | DOI | MR | Zbl

[19] D. K. Dimitrov, “Integration of polyharmonic functions”, Math. Comp., 65:215 (1996), 1269–1281 | DOI | MR | Zbl

[20] A. Ghizzetti, A. Ossicini, “Sull'esistenza e unicitá delle formule di quadratura gaussiane”, Rend. Mat. (6), 8 (1975), 1–15 | MR | Zbl

[21] H. A. Hakopian, “Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type”, J. Approx. Theory, 34:3 (1982), 286–305 | DOI | MR | Zbl

[22] H. A. Hakopian, “Multivariate spline functions, $B$-splines bases and polynomial interpolation. II”, Studia Math., 79:1 (1984), 91–102 | MR | Zbl

[23] K. Jetter, G. Lange, “Die Eindeutigkeit $L_2$-optimaler polinomialer Monosplines”, Math. Z., 158:1 (1978), 23–34 | DOI | MR

[24] F. John, “Abhängigkeiten zwischen den Flächenintegralen einer stetigen Funktion”, Math. Ann., 111:1 (1935), 541–559 | DOI | MR | Zbl

[25] S. Karlin, A. Pinkus, “An extremal property of multiple Gaussian nodes”, Studies in Spline Functions and Approximation Theory, Academic Press, New York, 1976, 143–162 | MR

[26] M. G. Krein, “Idei P. L. Chebysheva i A. A. Markova v teorii predelnykh velichin integralov i ikh dalneishee razvitie”, UMN, 4:4 (1951), 3–120 | MR | Zbl

[27] A. A. Ligun, “Tochnye neravenstva dlya splain-funktsii i nailuchshie kvadraturnye formuly dlya nekotorykh klassov funktsii”, Matem. zametki, 19:6 (1976), 913–926 | MR | Zbl

[28] H. L. Loeb, “A note on optimal integration in $H_\infty$”, C. R. Acad. Bulgare Sci., 27 (1974), 615–618 | MR | Zbl

[29] B. F. Logan, L. A. Shepp, “Optimal reconstruction of a function from its projections”, Duke Math. J., 42:4 (1975), 645–659 | DOI | MR | Zbl

[30] G. V. Milovanović, A. S. Cvetković, “Uniqueness and computation of Gaussian interval quadrature formula for Jacobi weight function”, Numer. Math., 99:1 (2004), 141–162 | DOI | MR | Zbl

[31] V. P. Motornyi, “O nailuchshei kvadraturnoi formule vida $\sum_{k=1}^np_kf(x_k)$ dlya nekotorykh klassov periodicheskikh differentsiruemykh funktsii”, Izv. AN SSSR. Ser. matem., 38:3 (1974), 583–614 | MR | Zbl

[32] C. R. Morrow, T. N. L. Patterson, “Construction of algebraic cubature rules using polynomial ideal theory”, SIAM J. Numer. Anal., 15:5 (1978), 953–976 | DOI | MR | Zbl

[33] H. M. Möller, “Kubaturformeln mit minimaler Knotenzahl”, Numer. Math., 25:2 (1975/76), 185–200 | DOI | MR

[34] I. P. Mysovskikh, Interpolyatsionnye kubaturnye formuly, Nauka, M., 1981 | MR | Zbl

[35] G. P. Nikolov, “A comparison theorem in the theory of quadrature formulae”, Math. Balkanica (N.S.), 2:1 (1988), 3–10 | MR | Zbl

[36] S. M. Nikolskii, “Priblizhenie funktsii trigonometricheskimi mnogochlenami v srednem”, Izv. AN SSSR. Ser. matem., 10:3 (1946), 207–256 | MR

[37] S. M. Nikolskii, “K voprosu ob otsenkakh priblizhenii kvadraturnymi formulami”, UMN, 5:2 (1950), 165–177 | MR | Zbl

[38] S. M. Nikolskii, “Kvadraturnye formuly”, Izv. AN SSSR. Ser. matem., 16:2 (1952), 181–196 | MR | Zbl

[39] S. M. Nikolskii, Kvadraturnye formuly (s dobavleniyami N. P. Korneichuka), 4-e izd., dop., Nauka, M., 1988 | MR

[40] M. Omladič, S. Pahor, A. Suhadolc, “On a new type of quadrature formulas”, Numer. Math., 25:4 (1975/76), 421–426 | DOI | MR

[41] K. Yu. Osipenko, Optimal Recovery of Analytic Functions, Nova Science, Huntington, NY, 2000

[42] G. Petrova, “Uniqueness of the Gaussian extended cubature for polyharmonic functions”, East J. Approx. 2003, 9:3, 269–275 | MR | Zbl

[43] P. Pizzetti, “Sulla media dei valore che una funzione dei punti della spazio assume alla superficie di una sfera”, Rom. Accad. Lincei Rend. Ser. (5), XVIII; 1 sem. (1909), 182–185

[44] F. Pittnauer, M. Reimer, “Interpolation mit Intervallfunktionalen”, Math. Z., 146:1 (1976), 7–15 | DOI | MR | Zbl

[45] F. Pittnauer, M. Reimer, “Intervallfunktionale vom Gauss–Legendre–Typ”, Math. Nachr., 87 (1979), 239–248 | DOI | MR | Zbl

[46] T. Popoviciu, “Aspura unei generalizari a formulei de integrare numerica a lui Gauss”, Acad. Republ. Popul. Romine Studii Cerc. Mat., 6 (1955), 29–57 | MR | Zbl

[47] J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten”, Ber. Verch. Sächs. Akad. Leipzig Vol., 69 (1917), 262–277 | Zbl

[48] P. W. Smith, “An improvement theorem for Descartes systems”, Proc. Amer. Math. Soc., 70:1 (1978), 26–30 | DOI | MR | Zbl

[49] S. A. Smolyak, Ob optimalnom vosstanovlenii funktsii i funktsionalov ot nikh, Diss. ... kand. fiz.-matem. nauk, MGU, M., 1965

[50] D. C. Solmon, “The $X$-ray transform”, J. Math. Anal. Appl., 56:1 (1976), 61–83 | DOI | MR | Zbl

[51] J. F. Traub, H. Woźniakowski, A General Theory of Optimal Algorithms, Academic Press, New York, 1980 | MR

[52] L. Chakalov, “Obschi kvadraturni formuli ot gausov tip”, BAN Izv. Matem. Inst., 1:2 (1954), 67–84 ; L. Tschakaloff, East J. Approx., 1:2 (1995), 261–276 | MR | Zbl

[53] P. Turán, “On the theory of the mechanical quadrature”, Acta Sci. Math. (Szeged), 12, Part A (1950), 30–37 | MR

[54] A. A. Zhensykbaev, “O nailuchshei kvadraturnoi formule na klasse $W^rL_p$”, Dokl. AN SSSR, 227:2 (1976), 277–279 | MR | Zbl

[55] A. A. Zhensykbaev, “Nailuchshaya kvadraturnaya formula dlya nekotorykh klassov periodicheskikh differentsiruemykh funktsii”, Izv. AN SSSR. Ser. matem., 41:5 (1977), 1110–1124 | MR | Zbl

[56] A. A. Zhensykbaev, Problemy vosstanovleniya operatorov, Institut kompyuternykh issledovanii, M.–Izhevsk, 2003