Abstract
This is a tutorial paper that gives the complete proof of a result of Frolov (Dokl Akad Nauk SSSR 231:818–821, 1976, [4]) that shows the optimal order of convergence for numerical integration of functions with bounded mixed derivatives. The presentation follows Temlyakov (J Complex 19:352–391, 2003, [13]), see also Temlyakov (Approximation of periodic functions, 1993, [12]).
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Notes
- 1.
A polynomial P is called irreducible over \(\mathbb {Q}\) if \(P=GH\) for two polynomials G, H with rational coefficients implies that one of them has degree zero. This implies that all roots of P must be irrational. In fact, every polynomial of the form \(\prod _{j=1}^d(x-b_j)-1\) with different \(b_j\in {\mathbb {Z}}\) is irreducible, but has not necessarily d different real roots.
- 2.
References
Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy theory and quasi-Monte Carlo integration. Cambridge University Press, Cambridge (2010)
Dung, D., Ullrich, T.: Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square, Math. Nachrichten (2015) (to appear)
Fine, B., Rosenberger, G.: The fundamental theorem of algebra. Springer-Verlag, New York, Undergraduate Texts in Mathematics (1997)
Frolov, K.K.: Upper error bounds for quadrature formulas on function classes. Dokl. Akad. Nauk SSSR 231, 818–821 (1976)
Frolov, K.K.: Upper bound of the discrepancy in metric \(L_{p}\), \(2\le p<\infty \). Dokl. Akad. Nauk SSSR 252, 805–807 (1980)
Hinrichs, A., Markhasin, L., Oettershagen, J., Ullrich, T.: Optimal quasi-Monte Carlo rules on higher order digital nets for the numerical integration of multivariate periodic functions. e-prints (2015)
Lee, C.-L., Wong, K.B.: On Chebyshev’s polynomials and certain combinatorial identities. Bull. Malays. Math. Sci. Soc. 2(34), 279–286 (2011)
Nguyen, V.K., Ullrich, M. Ullrich, T.: Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube (2015) (preprint)
Novak, E., Woźniakowaski, H.: Tractability of Multivariate Problems, Volume II: Standard Information for Functionals EMS Tracts in Mathematics, Vol. 12, Eur. Math. Soc. Publ. House, Zürich (2010)
Skriganov, M.M.: Constructions of uniform distributions in terms of geometry of numbers. Algebra i Analiz 6, 200–230 (1994)
Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford Science Publications, New York (1994)
Temlyakov, V.N.: Approximation of Periodic Functions. Computational Mathematics and Analysis Series. Nova Science Publishers Inc, NY (1993)
Temlyakov, V.N.: Cubature formulas, discrepancy, and nonlinear approximation. J. Complex. 19, 352–391 (2003)
Ullrich, M., Ullrich, T.: The role of Frolov’s cubature formula for functions with bounded mixed derivative, SIAM J. Numer. Anal. 54(2), 969–993 (2016)
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Ullrich, M. (2016). On “Upper Error Bounds for Quadrature Formulas on Function Classes” by K.K. Frolov. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_31
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DOI: https://doi.org/10.1007/978-3-319-33507-0_31
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