{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T20:11:28Z","timestamp":1773778288466,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2012,3,14]],"date-time":"2012-03-14T00:00:00Z","timestamp":1331683200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2012,4]]},"DOI":"10.1007\/s10208-012-9116-x","type":"journal-article","created":{"date-parts":[[2012,3,13]],"date-time":"2012-03-13T14:08:53Z","timestamp":1331647733000},"page":"139-172","source":"Crossref","is-referenced-by-count":8,"title":["An Exact Order of Discrepancy of the Smolyak Grid and Some General Conclusions in the Theory of Numerical Integration"],"prefix":"10.1007","volume":"12","author":[{"given":"Nurlan","family":"Nauryzbayev","sequence":"first","affiliation":[]},{"given":"Nurlan","family":"Temirgaliyev","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,3,14]]},"reference":[{"key":"9116_CR1","first-page":"3","volume":"4","author":"N.S. Bakhvalov","year":"1959","unstructured":"N.S. Bakhvalov, On the approximate calculation of integrals, Vestnik MGU Ser. Math. Mech. 4, 3\u201318 (1959) (in Russian).","journal-title":"Vestnik MGU Ser. Math. Mech."},{"key":"9116_CR2","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511565984","volume-title":"Irregularities of Distribution","author":"J. Beck","year":"1987","unstructured":"J. Beck, W.W.L. Chen, Irregularities of Distribution (Cambridge University Press, Cambridge, 1987)."},{"key":"9116_CR3","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/978-3-642-18743-8_10","volume-title":"Monte Carlo and Quasi-Monte Carlo Methods 2002","author":"J. Dick","year":"2004","unstructured":"J. Dick, F.Y. Kuo, Constructing good lattice rules with millions of points, in Monte Carlo and Quasi-Monte Carlo Methods 2002, ed. by H. Niederreiter (Springer, Berlin, 2004), pp. 181\u2013197."},{"key":"9116_CR4","doi-asserted-by":"crossref","first-page":"1967","DOI":"10.1090\/S0025-5718-03-01610-7","volume":"73","author":"J. Dick","year":"2004","unstructured":"J. Dick, F.Y. Kuo, Reducing the construction cost of the component-by-component construction of good lattice rules, Math. Comput. 73, 1967\u20131988 (2004).","journal-title":"Math. Comput."},{"key":"9116_CR5","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511761188","volume-title":"Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration","author":"J. Dick","year":"2010","unstructured":"J. Dick, F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration (Cambridge University Press, Cambridge, 2010)."},{"key":"9116_CR6","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s00211-005-0674-6","volume":"103","author":"J. Dick","year":"2006","unstructured":"J. Dick, I.H. Sloan, X. Wang, H. Wo\u017aniakowski, Good lattice rules in weighted Korobov spaces with general weights, Numer. Math. 103, 63\u201397 (2006).","journal-title":"Numer. Math."},{"key":"9116_CR7","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0093404","volume-title":"Sequences, Discrepancies and Applications","author":"M. Drmota","year":"1997","unstructured":"M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications (Springer, Berlin, 1997)."},{"key":"9116_CR8","volume-title":"Zahlentheoretische Methoden in der numerischen Mathematik","author":"E. Hlawka","year":"1981","unstructured":"E. Hlawka, F. Firneis, P. Zinterhof, Zahlentheoretische Methoden in der numerischen Mathematik (1981). Oldenbourg."},{"key":"9116_CR9","volume-title":"Applications of Number Theory to Numerical Analysis","author":"L.-K. Hua","year":"1981","unstructured":"L.-K. Hua, Y. Wang, Applications of Number Theory to Numerical Analysis (Springer, Berlin, 1981)."},{"key":"9116_CR10","volume-title":"Number-Theoretic Methods in Approximate Analysis","author":"N.M. Korobov","year":"1963","unstructured":"N.M. Korobov, Number-Theoretic Methods in Approximate Analysis (Fizmatgiz, Moscow, 1963) (in Russian)."},{"key":"9116_CR11","volume-title":"Number-Theoretic Methods in Approximate Analysis","author":"N.M. Korobov","year":"2004","unstructured":"N.M. Korobov, Number-Theoretic Methods in Approximate Analysis (MCCME, Moscow, 2004) (in Russian)."},{"key":"9116_CR12","volume-title":"Uniform Distribution of Sequences","author":"L. Kuipers","year":"1974","unstructured":"L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Wiley, New York, 1974)."},{"key":"9116_CR13","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1134\/S0001434609050320","volume":"85","author":"N.Z. Nauryzbaev","year":"2009","unstructured":"N.Z. Nauryzbaev, N. Temirgaliev, On the order of discrepancy of the Smolyak grid, Math. Notes 85, 897\u2013901 (2009).","journal-title":"Math. Notes"},{"key":"9116_CR14","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10208-002-0074-6","volume":"4","author":"E. Novak","year":"2004","unstructured":"E. Novak, I.H. Sloan, H. Wo\u017aniakowski, Tractability of approximation for weighted Korobov spaces on classical and quantum computers, Found. Comput. Math. 4, 121\u2013155 (2004).","journal-title":"Found. Comput. Math."},{"key":"9116_CR15","series-title":"Linear Information","doi-asserted-by":"crossref","DOI":"10.4171\/026","volume-title":"Tractability of Multivariate Problems: Volume I","author":"E. Novak","year":"2008","unstructured":"E. Novak, H. Wo\u017aniakowski, Tractability of Multivariate Problems: Volume I, Linear Information (2008). European Mathematical Society."},{"key":"9116_CR16","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1006\/jcom.1993.1019","volume":"9","author":"S.H. Paskov","year":"1993","unstructured":"S.H. Paskov, Average case complexity for multivariate integration for smooth functions, J. Complex. 9, 291\u2013312 (1993).","journal-title":"J. Complex."},{"key":"9116_CR17","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1112\/S0025579300000541","volume":"1","author":"K.F. Roth","year":"1954","unstructured":"K.F. Roth, On irregularities of distribution, Mathematika 1, 73\u201379 (1954).","journal-title":"Mathematika"},{"key":"9116_CR18","doi-asserted-by":"crossref","first-page":"80","DOI":"10.2307\/2372095","volume":"71","author":"A. Sard","year":"1949","unstructured":"A. Sard, Best approximate integration formulas, best approximation formulas, Am. J. Math. 71, 80\u201391 (1949).","journal-title":"Am. J. Math."},{"key":"9116_CR19","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/0041-5553(63)90033-8","volume":"3","author":"I.F. Sharygin","year":"1963","unstructured":"I.F. Sharygin, A lower estimate for the error of quadrature formulas for certain classes of functions, U.S.S.R. Comput. Math. Math. Phys. 3, 489\u2013497 (1963).","journal-title":"U.S.S.R. Comput. Math. Math. Phys."},{"key":"9116_CR20","first-page":"387","volume":"13","author":"W. Sickel","year":"2007","unstructured":"W. Sickel, T. Ullrich, The Smolyak algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness, East J. Approx. 13, 387\u2013425 (2007).","journal-title":"East J. Approx."},{"key":"9116_CR21","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1134\/S0001434610050330","volume":"87","author":"M.B. Sikhov","year":"2010","unstructured":"M.B. Sikhov, N. Temirgaliev, On an algorithm for construction uniformly distribution Korobov grids, Math. Notes 87, 916\u2013917 (2010).","journal-title":"Math. Notes"},{"key":"9116_CR22","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1006\/jcom.2001.0599","volume":"17","author":"I.H. Sloan","year":"2001","unstructured":"I.H. Sloan, H. Wo\u017aniakowski, Tractability of multivariate integration for weighted Korobov classes, J. Complex. 17, 697\u2013721 (2001).","journal-title":"J. Complex."},{"key":"9116_CR23","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1006\/jcom.2001.0626","volume":"18","author":"I.H. Sloan","year":"2002","unstructured":"I.H. Sloan, H. Wo\u017aniakowski, Tractability of integration in non-periodic and periodic weighted tensor product Hilbert spaces, J. Complex. 18, 479\u2013499 (2002).","journal-title":"J. Complex."},{"key":"9116_CR24","first-page":"1042","volume":"148","author":"S.A. Smolyak","year":"1963","unstructured":"S.A. Smolyak, Quadrature and interpolation formulas for tensor product of certain classes of functions, Dokl. Akad. Nauk SSSR 148, 1042\u20131045 (1963).","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"9116_CR25","unstructured":"S.A. Smolyak, On optimal restoration of functions and functional of them, Candidate Dissertation, Moscow (1965)."},{"key":"9116_CR26","volume-title":"Multidimensional Quadrature Formulae and Haar Functions","author":"I.M. Sobol","year":"1969","unstructured":"I.M. Sobol, Multidimensional Quadrature Formulae and Haar Functions (Nauka, Moscow, 1969) (in Russian)."},{"key":"9116_CR27","first-page":"490","volume":"181","author":"N. Temirgaliev","year":"1990","unstructured":"N. Temirgaliev, Application of divisor theory to the numerical integration of periodic functions of several variables, Math. USSR Sb. 181, 490\u2013505 (1990).","journal-title":"Math. USSR Sb."},{"key":"9116_CR28","first-page":"90","volume":"3","author":"N. Temirgaliev","year":"1997","unstructured":"N. Temirgaliev, Number-theoretical methods and probability-theoretical approach to problems in analysis: theory of embeddings and approximations, absolute convergence and Fourier series transforms, Vestnik ENU 3, 90\u2013144 (1997).","journal-title":"Vestnik ENU"},{"key":"9116_CR29","first-page":"414","volume":"68","author":"N. Temirgaliev","year":"2003","unstructured":"N. Temirgaliev, Classes U s (\u03b2,\u03b8,\u03b1,\u03c8) and quadrature formulas, Dokl. Math. 68, 414\u2013417 (2003).","journal-title":"Dokl. Math."},{"key":"9116_CR30","first-page":"1","volume":"Special issue","author":"N. Temirgaliev","year":"2009","unstructured":"N. Temirgaliev, Computer (numerical) width. Algebraic theory of functions and harmonic analysis in restoration problems (quasi-Monte Carlo method). Theory of embeddings and approximation, Fourier series, Vestnik ENU, Special issue, 1\u2013194 (2009).","journal-title":"Vestnik ENU"},{"key":"9116_CR31","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1134\/S1064562410010229","volume":"81","author":"N. Temirgaliev","year":"2010","unstructured":"N. Temirgaliev, Tensor products of functionals and their application, Dokl. Math. 81, 78\u201382 (2010).","journal-title":"Dokl. Math."},{"key":"9116_CR32","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1134\/S1064562407050110","volume":"76","author":"N. Temirgaliev","year":"2007","unstructured":"N. Temirgaliev, Y. Bailov, A. Zhubanisheva, General algorithm for the numerical integration of periodic function of several variables, Dokl. Math. 76, 681\u2013685 (2007).","journal-title":"Dokl. Math."},{"key":"9116_CR33","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1070\/IM2009v073n02ABEH002451","volume":"73","author":"N. Temirgaliev","year":"2009","unstructured":"N. Temirgaliev, S.S. Kudaibergenov, A.A. Shomanova, An application of tensor products of functionals in problems of numerical integration, Izv. RAN Ser. Math. 73, 393\u2013434 (2009).","journal-title":"Izv. RAN Ser. Math."},{"key":"9116_CR34","doi-asserted-by":"crossref","first-page":"45","DOI":"10.3103\/S1066369X10030084","volume":"54","author":"N. Temirgaliev","year":"2010","unstructured":"N. Temirgaliev, S.S. Kudaibergenov, A.A. Shomanova, Applications of Smolyak quadrature formulas to the numerical integration of Fourier coefficients and in function recovery problems, Russ. Math. (Izv. VUZ) 54, 45\u201362 (2010).","journal-title":"Russ. Math. (Izv. VUZ)"},{"key":"9116_CR35","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1070\/RM1985v040n04ABEH003658","volume":"40","author":"V.N. Temlyakov","year":"1985","unstructured":"V.N. Temlyakov, Quadrature formulae and recovery of number-theoretical nets from nodal values for classes of functions with small degree of smoothness, Russ. Math. Surv. 40, 223\u2013224 (1985).","journal-title":"Russ. Math. Surv."},{"key":"9116_CR36","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1070\/SM1987v056n01ABEH003034","volume":"56","author":"V.N. Temlyakov","year":"1987","unstructured":"V.N. Temlyakov, Approximate recovery of periodic functions of several variables, Math. USSR Sb. 56, 249\u2013261 (1987).","journal-title":"Math. USSR Sb."},{"key":"9116_CR37","unstructured":"T. Ullrich, Smolyak\u2019s algorithm, sparse grid approximation and periodic function spaces with dominating mixed smoothness, Doctoral (Ph.D.) thesis, Jena (2007)."},{"key":"9116_CR38","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1070\/IM1995v045n02ABEH001657","volume":"45","author":"S.M. Voronin","year":"1995","unstructured":"S.M. Voronin, On quadrature formulas, Izv. RAN Ser. Math. 45, 417\u2013422 (1995).","journal-title":"Izv. RAN Ser. Math."},{"key":"9116_CR39","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1070\/IM1995v059n04ABEH000028","volume":"59","author":"S.M. Voronin","year":"1995","unstructured":"S.M. Voronin, The construction of quadrature formulae, Izv. RAN Ser. Math. 59, 665\u2013670 (1995).","journal-title":"Izv. RAN Ser. Math."},{"key":"9116_CR40","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1070\/IM1996v060n05ABEH000084","volume":"60","author":"S.M. Voronin","year":"1996","unstructured":"S.M. Voronin, V.I. Skalyga, On numerical integration algorithms, Izv. RAN Ser. Math. 60, 887\u2013891 (1996).","journal-title":"Izv. RAN Ser. Math."},{"key":"9116_CR41","unstructured":"S.M. Voronin, Selected Works: Mathematics, Moscow State Technical University n.a. N.E. Bauman, Moscow (2006) (in Russian)."},{"key":"9116_CR42","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/BF01137622","volume":"46","author":"S.M. Voronin","year":"1989","unstructured":"S.M. Voronin, N. Temirgaliev, Quadrature Formulas Associated with Divisors of the Field of Gaussian Numbers, Math. Notes 46, 34\u201341 (1989).","journal-title":"Math. Notes"},{"key":"9116_CR43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcom.1995.1001","volume":"11","author":"G.W. Wasilkowski","year":"1995","unstructured":"G.W. Wasilkowski, H. Wo\u017aniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complex. 11, 1\u201356 (1995).","journal-title":"J. Complex."},{"key":"9116_CR44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3103\/S1066369X0904001X","volume":"53","author":"G.W. Wasilkowski","year":"2009","unstructured":"G.W. Wasilkowski, H. Wo\u017aniakowski, A survey of average case complexity for linear multivariate problems, Russ. Math. (Izv. VUZ) 53, 1\u201314 (2009).","journal-title":"Russ. Math. (Izv. VUZ)"},{"key":"9116_CR45","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1134\/S0965542509010023","volume":"49","author":"A.Z. Zhubanisheva","year":"2009","unstructured":"A.Z. Zhubanisheva, N. Temirgaliev, Z.N. Temirgalieva, Application of divisor theory to the construction of tables of optimal coefficients for quadrature formulas, Comput. Math. Math. Phys. 49, 12\u201322 (2009).","journal-title":"Comput. Math. Math. Phys."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-012-9116-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-012-9116-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-012-9116-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T07:05:11Z","timestamp":1559113511000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-012-9116-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3,14]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2012,4]]}},"alternative-id":["9116"],"URL":"https:\/\/doi.org\/10.1007\/s10208-012-9116-x","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3,14]]}}}