{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T06:57:53Z","timestamp":1762066673567,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,24]],"date-time":"2022-09-24T00:00:00Z","timestamp":1663977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals \u222b02\u03c0ei\u03c9x\u03c6(x)dx in the Sobolev space H\u02dc2m. Here, H\u02dc2m is the Hilbert space of periodic and complex-valued functions whose m-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the considered quadrature formula is used. By minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is then obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. The convergence of the constructed optimal quadrature formula is investigated, and it is shown that the rate of convergence of the optimal quadrature formula is O(hm) for |\u03c9|&lt;N and O(|\u03c9|\u2212m) for |\u03c9|\u2265N. Finally, we present numerical results of comparison for absolute errors of the optimal quadrature formula with the exp(i\u03c9x) weight in the case m=2 and the Midpoint formula. There, one can see the advantage of the optimal quadrature formulas.<\/jats:p>","DOI":"10.3390\/a15100344","type":"journal-article","created":{"date-parts":[[2022,9,25]],"date-time":"2022-09-25T21:14:28Z","timestamp":1664140468000},"page":"344","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On an Optimal Quadrature Formula in a Hilbert Space of Periodic Functions"],"prefix":"10.3390","volume":"15","author":[{"given":"Kholmat","family":"Shadimetov","sequence":"first","affiliation":[{"name":"Department of Informatics and Computer Graphycs, Tashkent State Transport University, 1, Odilkhodjaev Str., Tashkent 100167, Uzbekistan"},{"name":"V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b, University Str., Tashkent 100174, Uzbekistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2756-9542","authenticated-orcid":false,"given":"Abdullo","family":"Hayotov","sequence":"additional","affiliation":[{"name":"V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b, University Str., Tashkent 100174, Uzbekistan"},{"name":"Department of Computational Mathematics and Information Systems, National University of Uzbekistan named after M.Ulugbek, 4, University Str., Tashkent 100174, Uzbekistan"}]},{"given":"Botir","family":"Abdikayimov","sequence":"additional","affiliation":[{"name":"Department of Informatics and Computer Graphycs, Tashkent State Transport University, 1, Odilkhodjaev Str., Tashkent 100167, Uzbekistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1017\/S0370164600026262","article-title":"On a quadrature formula for trigonometric integrals","volume":"49","author":"Filon","year":"1928","journal-title":"Proc. Roy. Soc. Edinb."},{"key":"ref_2","first-page":"441","article-title":"An optimal formula for computation of linear functionals","volume":"10","year":"1965","journal-title":"APL Mater. (Ger.)"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s11075-016-0150-7","article-title":"Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L2(m)(0,1)","volume":"74","author":"Boltaev","year":"2017","journal-title":"Numer. Algorithms"},{"key":"ref_4","first-page":"1233","article-title":"Optimal quadrature formulas for numerical evaluation of Fourier coefficients in W2(m,m\u22121)","volume":"7","author":"Boltaev","year":"2017","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1145\/321021.321029","article-title":"A modification of Filon\u2019s method of numerical integration","volume":"7","author":"Filin","year":"1960","journal-title":"J. Assoc. Comput. 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The Theory of Cubature Formulas, Siberian Division of the Russia Academy of Sciences Novosibirsk.","DOI":"10.1007\/978-94-015-8913-0"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/344\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:38:38Z","timestamp":1760143118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/344"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,24]]},"references-count":18,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["a15100344"],"URL":"https:\/\/doi.org\/10.3390\/a15100344","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,9,24]]}}}