Abstract
We discuss two questions. First, we consider the existence of close to optimal quadrature formulas with a “bad” L2-discrepancy of their grids, and the second is the question of how much explicit quadrature formulas are preferable to sorting algorithms. Also, in the model case, we obtain the solution to the question of approximative possibilities of Smolyak’s grid in the problems of recovery of functions.
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References
Nauryzbayev, N. Temirgaliyev, N. “An Exact Order of Discrepancy of the Smolyak’s Grid and Some General Conclusions in the Theory of Numerical Integration”, Found. Comput. Math. 12, 139–172 (2012).
Smolyak, S. A. “Quadrature and Interpolation Formulae on Tensor Products of Certain Function Classes”, SovietMath. Dokl., No. 4, 240–243 (1963).
Temirgaliev, N. “Classes Us(β, θ,α; ψ) and Quadrature Formulas”, Dokl.Math. 68, 414–417 (2003).
Temirgaliev, N. “Tensor Products of Functionals and Their Application”, Dokl.Math. 81, 78–82 (2010).
Temirgaliyev, N., Nauryzbayev, N. Zh., and Shomanova, A. A. “Approximative Possibilities of Computational Aggregates of the ‘Smolyak Type’With Dirichlet, Fejer and Valleé-Poussin Kernels in the Scale of Ul’yanov Classes”, RussianMathematics 59, No. 7, 67–72 (2015).
Voronin, S. M. and Temirgaliev, N. “Quadrature Formulas That are Connected With Divisors of the Field of Gaussian Numbers”, Mathematical notes of the Academy of Sciences of the USSR 46, No. 2, 597–602 (1989).
Temirgaliev, N. “Application of Divisor Theory to the Numerical Integration of Periodic Functions of Several Variables”, Math. USSR-Sb. 69, No. 2, 527–542 (1991).
Zhubanysheva, A.Zh., Temirgaliev, N., Temirgalieva, Zh.N. “Application of Divisor Theory to the Construction of Tables of Optimal Coefficients for Quadrature Formulas”, Computational Mathematics and Mathematical Physics 49, No. 1, 12–22 (2009).
Il’in, A. M. and Danilin, A. R. Asymptotic Methods in Analysis (Fizmatlit, Moscow, 2009) [in Russian].
Korobov, N. M. Number-Theoretic Methods in Approximate Analysis (MTsNMO, Moscow, 2004) [in Russian].
Bakhvalov, N. S. “Approximate Computation of Multiple Integrals”, Vestnik Mosk.Univ. Ser. Mat. Mech. 4, 3–18 (1959) [in Russian].
Hua, L. K., Wang, Y. Applications of Number Theory to Numerical Analysis (Springer-Verlag, Berlin–Heidelberg–New York, 1981).
Hlawka, E. “Näherungsformeln zur Berechnung von mehrfachen Integralen mit Anwendungen auf die Berechnungen von Potentialen, Induktionskoeffizienten und Lösungen von Gleichungssystemen”, in Proceedings of Seminar ‘Number-Theoretic Analysis’, Vienna/Austria 1988–89, Lect. Notes Math. 1452, 65–111 (1990).
Kuipers, L. and Niederreiter, G. Uniform Distribution of Sequences (Wiley-Interscience, John Wiley & Sons, New York–London–Sydney, 1974; Nauka, Moscow, 1985).
Temirgaliev, N., Bailov, E. A. and Zhubanisheva, A. Zh. “General Algorithm for the Numerical Integration of Periodic Functions of Several Variables”, Dokl.Math. 76, No. 2, 681–685 (2007).
Bailov, E. A., Sikhov, M. B., Temirgaliev, N. “General Algorithm for the Numerical Integration of Functions of Several Variables”, Comput.Math. and Math. Phys. 54, No. 7, 1061–1078 (2014).
Korobov, N. M. “On Finite Continued Fractions”, Russ. Math. Surv. 52, No. 6, 1302–1304 (1997).
Kan, I. D., Frolenkov, D. A. “A Strengthening of a Theorem of Bourgain and Kontorovich”, Izv.Math. 78, No. 2, 293–353 (2014).
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Original Russian Text © N.Zh. Nauryzbaev, A.A. Shomanova, N. Temirgaliyev, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 3, pp. 96–102.
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Nauryzbaev, N.Z., Shomanova, A.A. & Temirgaliyev, N. On Some Special Effects in Theory on Numerical Integration and Functions Recovery. Russ Math. 62, 84–88 (2018). https://doi.org/10.3103/S1066369X18030118
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DOI: https://doi.org/10.3103/S1066369X18030118