{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T06:15:47Z","timestamp":1764828947673},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The problem of numerical differentiation for periodic bivariate functions with finite smoothness is studied.\nTo achieve stable approximations, we investigate some variants of the Fourier truncation method.\nEstimates of the accuracy and volume of the used Fourier coefficients are found for the constructed methods.\nWe perform numerical experiments that confirm correctness of our theoretical conclusions.<\/jats:p>","DOI":"10.1515\/cmam-2020-0138","type":"journal-article","created":{"date-parts":[[2022,2,7]],"date-time":"2022-02-07T18:17:00Z","timestamp":1644257820000},"page":"477-491","source":"Crossref","is-referenced-by-count":11,"title":["Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions"],"prefix":"10.1515","volume":"22","author":[{"given":"Evgeniya V.","family":"Semenova","sequence":"first","affiliation":[{"name":"Institute of Mathematics , National Academy of Sciences of Ukraine , Kyiv , Ukraine"}]},{"given":"Sergiy G.","family":"Solodky","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , National Academy of Sciences of Ukraine , Kyiv , Ukraine"}]},{"given":"Serhii A.","family":"Stasyuk","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , National Academy of Sciences of Ukraine , Kyiv , Ukraine"}]}],"member":"374","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"key":"2023033111281523706_j_cmam-2020-0138_ref_001","doi-asserted-by":"crossref","unstructured":"S. Ahn, U. J. Choi and A. G. Ramm,\nA scheme for stable numerical differentiation,\nJ. Comput. Appl. Math. 186 (2006), no. 2, 325\u2013334.","DOI":"10.1016\/j.cam.2005.02.002"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_002","unstructured":"K. I. Babenko,\nApproximation of periodic functions of many variables by trigonometric polynomials (in Russian),\nDokl. Akad. Nauk SSSR 132 (1960), 247\u2013250."},{"key":"2023033111281523706_j_cmam-2020-0138_ref_003","doi-asserted-by":"crossref","unstructured":"F. Cobos, T. K\u00fchn and W. Sickel,\nOptimal approximation of multivariate periodic Sobolev functions in the sup-norm,\nJ. Funct. Anal. 270 (2016), no. 11, 4196\u20134212.","DOI":"10.1016\/j.jfa.2016.03.018"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_004","doi-asserted-by":"crossref","unstructured":"D. D\u0169ng, V. Temlyakov and T. Ullrich,\nHyperbolic Cross Approximation,\nAdv. Courses Math. CRM Barcelona,\nBirkh\u00e4user\/Springer, Cham, 2018.","DOI":"10.1007\/978-3-319-92240-9"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_005","doi-asserted-by":"crossref","unstructured":"T. F. Dolgopolova and V. K. Ivanov,\nOn numerical differentiation,\nUSSR Comput. Math. Math. Phys. 6 (1966), no. 3, 223\u2013232.","DOI":"10.1016\/0041-5553(66)90145-5"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_006","unstructured":"I. S. Gradshteyn and I. M. Ryzhik,\nTable of Integrals, Series, and Products, 6th ed.,\nAcademic Press, San Diego, 2000."},{"key":"2023033111281523706_j_cmam-2020-0138_ref_007","doi-asserted-by":"crossref","unstructured":"C. W. Groetsch,\nOptimal order of accuracy in Vasin\u2019s method for differentiation of noisy functions,\nJ. Optim. Theory Appl. 74 (1992), no. 2, 373\u2013378.","DOI":"10.1007\/BF00940901"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_008","doi-asserted-by":"crossref","unstructured":"M. Hanke and O. Scherzer,\nInverse problems light: Numerical differentiation,\nAmer. Math. Monthly 108 (2001), no. 6, 512\u2013521.","DOI":"10.1080\/00029890.2001.11919778"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_009","doi-asserted-by":"crossref","unstructured":"O. V. Lepski\u012d,\nA problem of adaptive estimation in Gaussian white noise,\nTheory Probab. Appl. 35 (1990), no. 3, 454\u2013466.","DOI":"10.1137\/1135065"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_010","doi-asserted-by":"crossref","unstructured":"Z. Meng, Z. Zhao, D. Mei and Y. Zhou,\nNumerical differentiation for two-dimensional functions by a Fourier extension method,\nInverse Probl. Sci. Eng. 28 (2020), no. 1, 126\u2013143.","DOI":"10.1080\/17415977.2019.1661410"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_011","doi-asserted-by":"crossref","unstructured":"G. L. Mile\u012dko and S. G. Solodki\u012d,\nHyperbolic cross and the complexity of various classes of linear ill-posed problems,\nUkrainian Math. J. 69 (2017), no. 7, 1107\u20131122.","DOI":"10.1007\/s11253-017-1418-3"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_012","doi-asserted-by":"crossref","unstructured":"G. Nakamura, S. Wang and Y. Wang,\nNumerical differentiation for the second order derivatives of functions of two variables,\nJ. Comput. Appl. Math. 212 (2008), no. 2, 341\u2013358.","DOI":"10.1016\/j.cam.2006.11.035"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_013","doi-asserted-by":"crossref","unstructured":"S. Pereverzev and E. Schock,\nOn the adaptive selection of the parameter in regularization of ill-posed problems,\nSIAM J. Numer. Anal. 43 (2005), no. 5, 2060\u20132076.","DOI":"10.1137\/S0036142903433819"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_014","doi-asserted-by":"crossref","unstructured":"S. V. Pereverzev,\nOptimization of projection methods for solving ill-posed problems,\nComputing 55 (1995), no. 2, 113\u2013124.","DOI":"10.1007\/BF02238096"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_015","doi-asserted-by":"crossref","unstructured":"S. V. Pereverzev and S. G. Solodki\u012d,\nOptimal discretization of ill-posed problems,\nUkrainian Math. J. 52 (2000), no. 1, 115\u2013132.","DOI":"10.1007\/BF02514141"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_016","doi-asserted-by":"crossref","unstructured":"Z. Qian, C.-L. Fu, X.-T. Xiong and T. Wei,\nFourier truncation method for high order numerical derivatives,\nAppl. Math. Comput. 181 (2006), no. 2, 940\u2013948.","DOI":"10.1016\/j.amc.2006.01.057"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_017","unstructured":"A. G. Ramm,\nNumerical differentiation,\nIzv. Vyssh. Uchebn. Zaved. Mat. 11 (1968), 131\u2013134."},{"key":"2023033111281523706_j_cmam-2020-0138_ref_018","unstructured":"K. Sharipov,\nOn the recovery of continuous functions from noisy Fourier coefficients,\nJ. Numer. Appl. Math. 109 (2012), 116\u2013124."},{"key":"2023033111281523706_j_cmam-2020-0138_ref_019","doi-asserted-by":"crossref","unstructured":"S. G. Solodky and K. K. Sharipov,\nSummation of smooth functions of two variables with perturbed Fourier coefficients,\nJ. Inverse Ill-Posed Probl. 23 (2015), no. 3, 287\u2013297.","DOI":"10.1515\/jiip-2013-0076"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_020","doi-asserted-by":"crossref","unstructured":"S. G. Solodky and S. A. Stasyuk,\nEstimates of efficiency for two methods of stable numerical summation of smooth functions,\nJ. Complexity 56 (2020), Article ID 101422.","DOI":"10.1016\/j.jco.2019.101422"},{"key":"2023033111281523706_j_cmam-2020-0138_ref_021","unstructured":"V. Temlyakov,\nMultivariate Approximation,\nCambridge Monogr. Appl. Comput. Math. 32,\nCambridge University, Cambridge, 2018."},{"key":"2023033111281523706_j_cmam-2020-0138_ref_022","unstructured":"V. V. Vasin,\nRegularization of a numerical differentiation problem (in Russian),\nUral. Gos. Univ. Mat. Zap. 7 (1969\/1970), no. 2, 29\u201333."},{"key":"2023033111281523706_j_cmam-2020-0138_ref_023","doi-asserted-by":"crossref","unstructured":"Z. Zhao, Z. Meng, L. Zhao, L. You and O. Xie,\nA stabilized algorithm for multi-dimensional numerical differentiation,\nJ. Algorithms Comput. Technol. 10 (2016), no. 2, 73\u201381.","DOI":"10.1177\/1748301816640450"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0138\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0138\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T15:00:44Z","timestamp":1680274844000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0138\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,8]]},"references-count":23,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,11,5]]},"published-print":{"date-parts":[[2022,4,1]]}},"alternative-id":["10.1515\/cmam-2020-0138"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2020-0138","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,8]]}}}