{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T07:03:01Z","timestamp":1762066981080,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,25]],"date-time":"2022-09-25T00:00:00Z","timestamp":1664064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"AIMet4AI","award":["CZ.02.1.01\/0.0\/0.0\/17-049\/0008414"],"award-info":[{"award-number":["CZ.02.1.01\/0.0\/0.0\/17-049\/0008414"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this paper is to reduce the complexity of computing the components of the integral Fm-transform, m\u22650, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton\u2013Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the Fm-transform. Taking into account this fact and the fact of exact integration of orthogonal polynomials, we obtain exact analytic expressions for the denominators of the components of the Fm-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the Fm-transformation for 0\u2264m\u22643. As a side result, we obtain a new method of numerical integration, which can be recommended not only for continuous functions, but also for strongly oscillating functions. The advantage of the proposed calculation method is shown by examples.<\/jats:p>","DOI":"10.3390\/axioms11100501","type":"journal-article","created":{"date-parts":[[2022,9,25]],"date-time":"2022-09-25T21:14:28Z","timestamp":1664140468000},"page":"501","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Quadrature Rules for the Fm-Transform Polynomial Components"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1531-1111","authenticated-orcid":false,"given":"Irina","family":"Perfilieva","sequence":"first","affiliation":[{"name":"Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tam","family":"Pham","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Petr","family":"Ferbas","sequence":"additional","affiliation":[{"name":"Advanced Engineering Department, Varroc Lighting Systems, Suvorovova 195, 742 42 \u0160enov u Nov\u00e9ho Ji\u010d\u00edna, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,25]]},"reference":[{"key":"ref_1","unstructured":"Bakhvalov, I.S., Zhidkov, I.P., and Kobelkov, G.M. 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