{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,16]],"date-time":"2026-10-16T18:39:05Z","timestamp":1792175945385,"version":"4.4.0"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We define a Walsh space which contains all functions whose partial mixed derivatives up to order $\\delta \\ge 1$ exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi\u2013Monte Carlo rules based on digital $(t,\\alpha,s)$-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital $(t,\\alpha,s)$-sequences are given, hence providing explicit quasi\u2013Monte Carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions.<\/jats:p>","DOI":"10.1137\/060666639","type":"journal-article","created":{"date-parts":[[2008,4,4]],"date-time":"2008-04-04T18:08:07Z","timestamp":1207332487000},"page":"1519-1553","source":"Crossref","is-referenced-by-count":98,"title":["Walsh Spaces Containing Smooth Functions and Quasi\u2013Monte Carlo Rules of Arbitrary High Order"],"prefix":"10.1137","volume":"46","author":[{"given":"Josef","family":"Dick","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1955.5.17"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/060658916"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01742-4"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2004.07.003"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.4064\/aa117-4-4"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2003.06.002"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-005-0674-6"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176345462"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.4064\/aa-41-4-337-351"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1949-0032833-2"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-03-01531-X"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02415361"},{"key":"R13","unstructured":"J. F. Koksma,\n                      Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo\n                      1, Mathematica B (Zutphen), 11 (1942\/1943), pp. 7\u201311."},{"key":"R14","unstructured":"L. Kuipers and H. Niederreiter,\n                      Uniform Distribution of Sequences\n                      , Pure Appl. Math. (N.Y.), John Wiley, New York, 1974."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01298952"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1999.189.75"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"J. Matou\u0161ek,\n                      Geometric Discrepancy\n                      , Algorithms Combin. 18, Springer, Berlin, 1999.","DOI":"10.1007\/978-3-642-03942-3"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1978-14532-7"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01490206"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"H. Niederreiter,\n                      Random Number Generation and Quasi-Monte Carlo Methods\n                      , CBMS-NSF Regional Conf. Ser. in Appl. Math. 63, SIAM, Philadelphia, 1992.","DOI":"10.1137\/1.9781611970081"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2005.01.001"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.4064\/aa97-2-5"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"H. Niederreiter and C. P. Xing,\n                      Quasirandom points and global function fields\n                      , in Finite Fields and Applications, S. Cohen and H. Niederreiter, eds., London Math. Soc. Lecture Note Ser. 233, Cambridge University Press, Cambridge, UK, 1996, pp. 269\u2013296.","DOI":"10.1017\/CBO9780511525988.022"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"A. B. Owen,\n                      Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo\n                      , in Monte Carlo and Quasi-Monte Carlo Methods 1998 (Claremont, CA), Springer, Berlin, 2000, pp. 86\u201397.","DOI":"10.1007\/978-3-642-59657-5_5"},{"key":"R25","unstructured":"G. Pirsic,\n                      Embedding Theorems and Numerical Integration of Walsh Series over Groups\n                      , Ph.D. thesis, University of Salzburg, 1997."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/050622638"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01458040"},{"key":"R28","first-page":"45","volume":"33","author":"Rosenbloom M. Yu.","year":"1997","journal-title":"Problemi Peredachi Inf.","ISSN":"https:\/\/id.crossref.org\/issn\/0555-2923","issn-type":"print"},{"key":"R29","first-page":"370","volume":"3","author":"Sharygin I. F.","year":"1963","journal-title":"Zh. Vychisl. Mat. i Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1006\/jcom.1997.0463"},{"key":"R31","first-page":"784","volume":"7","author":"Sobol I. M.","year":"1967","journal-title":"Zh. Vychisl. Mat. i Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"key":"R32","doi-asserted-by":"crossref","unstructured":"J. Stoer and R. Bulirsch,\n                      Introduction to Numerical Analysis\n                      , Texts Appl. Math. 12, Springer, New York, 2002.","DOI":"10.1007\/978-0-387-21738-3"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.2307\/2387224"},{"key":"R34","unstructured":"A. Zygmund,\n                      Trigonometric Series\n                      , Cambridge University Press, Cambridge, UK, 1959."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060666639","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:48Z","timestamp":1787325228000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060666639"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060666639"],"URL":"https:\/\/doi.org\/10.1137\/060666639","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}