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The role of Frolov’s cubature formula for functions with bounded mixed derivative. (English) Zbl 1336.65023

##### MSC:
 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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##### References:
 [1] T. I. Amanov, Spaces of Differentiable Functions with Dominating Mixed Derivatives, Nauka Kaz. SSR, Alma-Ata, 1976. [2] N. S. Bakhvalov, Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions, Zh. Vychisl. Mat. Mat. Fiz., 4 (1963), pp. 5–63. [3] N. S. Bakhvalov, Lower estimates of asymptotic characteristics of classes of functions with dominant mixed derivative, Mat. Zametki, 12 (1972), pp. 655–664. [4] H.-Q. Bui, M. Paluszyński, and M. H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel–Lizorkin spaces, Stud. Math., 119 (1996), pp. 219–246. · Zbl 0861.42009 [5] A.-P. Calderón, An atomic decomposition of distributions in parabolic $$H^{p}$$ spaces, Adv. Math., 25 (1977), pp. 216–225. · Zbl 0379.46050 [6] J. Dick and F. Pillichshammer, Discrepancy theory and quasi-Monte Carlo integration, in Panorama in Discrepancy Theory, W. W. L. Chen, A. Srivastav, end G. Travaglini, eds., Springer, Berlin, 2013, to appear. · Zbl 1358.11086 [7] V. V. Dubinin, Cubature formulas for classes of functions with bounded mixed difference, Math. USSR Sbornik, 76 (1993), pp. 283–292. · Zbl 0786.41025 [8] V. V. Dubinin, Cubature formulae for Besov classes, Izv. Math, 61 (1997), pp. 259–283. · Zbl 0882.41019 [9] D. Du͂ng and T. Ullrich, Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square, Math. Nachr., 288 (2015), pp. 743–762. · Zbl 1317.65067 [10] J. Franke, On the spaces $$F^s_{p,q}$$ of Triebel–Lizorkin type: Pointwise multipliers and spaces on domains, Math. Nachr., 12 (1986), pp. 29–68. · Zbl 0617.46036 [11] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 93 (1990), pp. 34–170. · Zbl 0716.46031 [12] K. K. Frolov, Upper error bounds for quadrature formulas on function classes, Dokl. Akad. Nauk SSSR, 231 (1976), pp. 818–821. · Zbl 0358.65014 [13] P. Glasserman, Monte Carlo Methods in Financial Engineering, Stoch. Model. Appl. Probab., Springer, Berlin, 2004. · Zbl 1038.91045 [14] M. Hansen, Nonlinear Approximation and Function Spaces of Dominating Mixed Smoothness, Thesis, Friedrich-Schiller-Universität Jena, Jena, 2010. [15] M. Hansen and J. Vybíral, The Jawerth–Franke embedding of spaces with dominating mixed smoothness, Georgian Math. J., 16 (2009), pp. 667–682. · Zbl 1187.42022 [16] A. Hinrichs, Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness, Math. Nachr., 283 (2010), pp. 478–488. · Zbl 1198.11073 [17] A. Hinrichs, L. Markhasin, J. Oettershagen, and T. Ullrich, Optimal quasi-Monte Carlo rules on order $$2$$ digital nets for the numerical integration of multivariate periodic functions, Numer. Math., 2015, \tt doi:10.1007/s00211-015-0765-y. · Zbl 1358.65004 [18] A. Hinrichs, E. Novak, M. Ullrich, and H. Woźniakowski, The curse of dimensionality for numerical integration of smooth functions, Math. Comp., 83 (2014), pp. 2853–2863. · Zbl 1345.65014 [19] A. Hinrichs, E. Novak, M. Ullrich, and H. Woźniakowski, The curse of dimensionality for numerical integration of smooth functions II, J. Complexity, 30 (2014), pp. 117–143. · Zbl 1286.65040 [20] A. Hinrichs, E. Novak, and M. Ullrich, On weak tractability of the Clenshaw–Curtis Smolyak algorithm, J. Approx. Theory, 183 (2014), pp. 31–44. · Zbl 1295.41028 [21] A. Hinrichs and J. Oettershagen, Optimal Point Sets for Quasi–Monte Carlo Integration of Bivariate Periodic Functions with Bounded Mixed Derivatives, preprint, arXiv:1409.5894, 2014. · Zbl 1356.65007 [22] E. Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math., 66 (1962), pp. 140–151. [23] S. Janson and M. H. Taibleson, Calderón’s representation theorems, Rend. Semin. Mat. Univ. Politec. Torino, 39 (1982), pp. 27–35. [24] B. Jawerth, Some observations on Besov and Lizorkin–Triebel spaces, Math. Scand., 40 (1977), pp. 94–104. · Zbl 0358.46023 [25] N. M. Korobov, Approximate evaluation of repeated integrals, Dokl. Akad. Nauk SSSR, 124 (1959), pp. 1207–1210. · Zbl 0089.04201 [26] D. Krieg and E. Novak, A Universal Algorithm for Multivariate Integration, preprint, arXiv:1507.06853, 2015. · Zbl 1384.65003 [27] U. Luther and K. Rost, Matrix exponentials and inversion of confluent Vandermonde matrices, Electron. Trans. Numer. Anal., 18 (2004), pp. 91–100. · Zbl 1065.34001 [28] L. Markhasin, Discrepancy and integration in function spaces with dominating mixed smoothness, Dissertationes Math. (Rozprawy Mat.), 494 (2013), pp. 1–81. · Zbl 1284.46030 [29] L. Markhasin, Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension, J. Complexity, 29 (2013), pp. 370–388. · Zbl 1286.65005 [30] L. Markhasin, Discrepancy of generalized Hammersley type point sets in Besov spaces with dominating mixed smoothness, Unif. Distrib. Theory, 8 (2013), pp. 135–164. · Zbl 1349.11113 [31] V. K. Nguyen, M. Ullrich, and T. Ullrich, Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube, preprint, arXiv:1511.02036, 2015. · Zbl 1376.65028 [32] S. M. Nikol’skij, Approximation of functions of several variables and embedding theorems, Nauka Moskva, 1977. [33] V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications, Russian Math. Surveys, 45 (1990), pp. 87–120. · Zbl 0704.34090 [34] V. S. Rychkov, On a theorem of Bui, Paluszyński and Taibleson, Proc. Steklov Inst. Math., 227 (1999), pp. 280–292. · Zbl 0979.46019 [35] V. S. Rychkov, On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains, J. Lond. Math. Soc., 60 (1999), pp. 237–257. · Zbl 0940.46017 [36] A. Seeger and T. Ullrich, Haar Projection Numbers and Failure of Unconditional Convergence in Sobolev Spaces, preprint, arXiv e-prints, arXiv:1507.01211 [math.CA], 2015. [37] A. Seeger and T. Ullrich, Lower Bounds for Haar Projections: Deterministic Examples, arXiv e-prints, arXiv:1511.01470 [math.CA], 2015. [38] H.-J. Schmeisser and H. Triebel, Topics in Fourier analysis and function spaces, John Wiley & Sons, Chichester, 1987. · Zbl 0661.46024 [39] M. M. Skriganov, Constructions of uniform distributions in terms of geometry of numbers, Algebra i Analiz, 6 (1994), pp. 200–230. · Zbl 0840.11041 [40] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, NJ, 1971. · Zbl 0232.42007 [41] J.-O. Strömberg and A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989. [42] V. N. Temlyakov, Approximation of periodic functions of several variables by trigonometric polynomials, and widths of some classes of functions, Izv. SSSR, 49 (1985), pp. 986–1030. [43] V. N. Temlyakov, On reconstruction of multivariate periodic functions based on their values at the knots of number-theoretical nets, Anal. Math., 12 (1986), pp. 287–305. · Zbl 0621.41004 [44] V. N. Temlyakov, Approximation of functions with bounded mixed derivative, Tr. MIAN, 178 (1986), pp. 1–112. [45] V. N. Temlyakov, On a way of obtaining lower estimates for the errors of quadrature formulas, Mat. Sb., 181 (1990), pp. 1403–1413. [46] V. N. Temlyakov, Error estimates for Fibonacci quadrature formulas for classes of functions with a bounded mixed derivative, Tr. Mat. Inst. Steklova, 200 (1991), pp. 327–335. · Zbl 0813.41023 [47] V. N. Temlyakov, Approximation of Periodic Functions, Comput. Math. Anal. Ser., Nova Science Publishers, Commack, NY, 1993. · Zbl 0899.41001 [48] V. N. Temlyakov, On error estimates of cubature formulas, Tr. Mat. Inst. Steklova, 207 (1994), pp. 326–338. [49] V. N. Temlyakov, Cubature formulas, discrepancy, and nonlinear approximation, J. Complexity, 19 (2003), pp. 352–391. · Zbl 1031.41016 [50] V. N. Temlyakov, Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness, Mtem. Sb., 206 (2015), pp. 131–160. · Zbl 1362.41009 [51] M. F. Timan, Imbedding classes of functions in $$L_p$$, Izv. Vyssh. Uchebn. Zaved., Mat., 10 (1974), pp. 61–74. [52] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. [53] H. Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, EMS Tracts Math., European Mathematical Society, Zürich, 2010. · Zbl 1202.46002 [54] M. Ullrich, On “Upper error bounds for quadrature formulas on function classes” by K. K. Frolov, arXiv e-prints, arXiv:1404.5457, 2014. [55] T. Ullrich, Local Mean Characterization of Besov–Triebel–Lizorkin Type Spaces with Dominating Mixed Smoothness on Rectangular Domains, preprint, 2008. [56] T. Ullrich, Optimal cubature in Besov spaces with dominating mixed smoothness on the unit square, J. Complexity, 30 (2014), pp. 72–94. · Zbl 1321.65006 [57] P. L. Ul’yanov, Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics, Mat. Sb. (N.S.), 81 (1970), pp. 104–131. [58] J. Vybíral, Function spaces with dominating mixed smoothness, Dissertationes Math., 436 (2006), 73 pp. [59] J. Vybíral, A new proof of the Jawerth–Franke embedding, Rev. Mat. Complut., 21 (2008), pp. 75–82.
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