{"id":2683,"date":"2024-05-15T08:49:55","date_gmt":"2024-05-15T05:49:55","guid":{"rendered":"https:\/\/fti.dp.ua\/conf\/?p=2683"},"modified":"2024-05-15T08:49:57","modified_gmt":"2024-05-15T05:49:57","slug":"05157-0846","status":"publish","type":"post","link":"https:\/\/fti.dp.ua\/conf\/2024\/05157-0846\/","title":{"rendered":"Estimates of the approximation errors of the classes of continuous"},"content":{"rendered":"\n

Estimates of the approximation errors of the classes of continuous<\/h1>\n\n\n\n
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Oleksandr Shchytov<\/strong><\/strong><\/strong><\/strong><\/h5>\n\n\n\n

ORCID: https:\/\/orcid.org\/0000-0002-1435-2918<\/a><\/em><\/p>\n\n\n\n

TEC-Lyceum No. 100<\/em><\/p>\n<\/div><\/div>\n\n\n\n

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Mykola Mormul<\/strong><\/strong><\/strong><\/strong><\/h5>\n\n\n\n

ORCID: https:\/\/orcid.org\/0000-0002-8036-3236<\/a><\/em><\/p>\n\n\n\n

University of Customs and Finance<\/em><\/p>\n<\/div><\/div>\n\n\n\n

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In the metrics of continuous and non-decreasing functions \u03c6(x), the following are obtained: a) estimates of the approximation of the classes of 1-periodic functions W^(2\u03bd+1) H\u03c9* (\u03bd \u2208 N), where \u03c9(t) is an upwardly convex modulus of continuity, and by interpolating the functions f(t) \u2208 W^(2\u03bd+1) H\u03c9*; b) by piecewise constant functions \u03c3_n (f,t) in the integral metric L_p (0 < p < \u221e); c) by piecewise constant functions \u03c3_n (f,t) in a uniform metric. Estimates of the approximation errors of the classes of 1-periodic functions from the classes W^(2\u03bd+1) H\u03c9* \u00a0(\u03bd \u2208 N), where \u03c9(t) is the convex upward modulus of continuity, by the piecewise constant functions \u03c3n(f, t) in the integral and uniform metrics Lp (0 < p < \u221e). Estimates are expressed in terms of the function \u03a9_2v(w, t). The accuracy of the error estimates of the obtained approximations has been clarified. The theorem on the connection between the continuous and monotonically increasing function \u03c6(x) \u2208 \u0424 on the interval [0, \u221e) and any function W^(2\u03bd+1) H\u03c9* (\u03bd\u2208N) \u00a0and n = 2, 3, \u2026, \u221e has been proved; as well as two lemmas and two consequences from the theorem. The results of the conducted research are a kind of extension of previously known results of approximation of functions to classes of 1-periodic functions \u00a0 \u00a0and more general spaces \u03c6(L). It is proved that the obtained estimates are non-improvable for n = 2m (m \u2208 N) on the entire class W^(2\u03bd+1) H\u03c9*. The new results of the function approximation theory obtained in the course of the study can be used for further practical applications, in particular, in the wavelet theory for the analysis of frequency components of signals (time-dependent functions) by methods similar to the Fourier transform. An applied aspect of the use of the obtained scientific results is also the possibility of applying estimates of approximation errors of the theory of numerical methods in the construction of numerical algorithms and signal processing in circuit engineering.<\/p>\n\n\n\n

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CIMS 2024 Vernal<\/a><\/div>\n\n
Information Technology and Cybersecurity<\/a><\/div><\/div>\n\n\n\n
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Estimates of the approximation errors of the classes of continuous Oleksandr Shchytov ORCID: https:\/\/orcid.org\/0000-0002-1435-2918 TEC-Lyceum No. 100 Mykola Mormul ORCID: https:\/\/orcid.org\/0000-0002-8036-3236 University of Customs and Finance In the metrics of continuous and non-decreasing functions \u03c6(x), the following are obtained: a) estimates of the approximation of the classes of 1-periodic functions W^(2\u03bd+1) H\u03c9* (\u03bd \u2208 N), where \u03c9(t) is an upwardly convex modulus of continuity, and by interpolating the functions f(t) \u2208 W^(2\u03bd+1) H\u03c9*; b) by piecewise constant functions \u03c3_n (f,t) … <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[35],"tags":[29],"class_list":["post-2683","post","type-post","status-publish","format-standard","hentry","category-info-tech-2","tag-cims-2024-vernal"],"_links":{"self":[{"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/posts\/2683","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/comments?post=2683"}],"version-history":[{"count":1,"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/posts\/2683\/revisions"}],"predecessor-version":[{"id":2684,"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/posts\/2683\/revisions\/2684"}],"wp:attachment":[{"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/media?parent=2683"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/categories?post=2683"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fti.dp.ua\/conf\/wp-json\/wp\/v2\/tags?post=2683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}