Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components
The purpose of this paper is to reduce the complexity of computing the components of the integral <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup>&l...
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MDPI AG
2022-09-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/10/501 |
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| author | Irina Perfilieva Tam Pham Petr Ferbas |
| author_facet | Irina Perfilieva Tam Pham Petr Ferbas |
| author_sort | Irina Perfilieva |
| collection | DOAJ |
| description | The purpose of this paper is to reduce the complexity of computing the components of the integral <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-transform, <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-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 <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-transformation for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>3</mn></mrow></semantics></math></inline-formula>. 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. |
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| spelling | doaj.art-ccf4a429af2f43ec8f02ca783d6558202023-11-23T22:53:25ZengMDPI AGAxioms2075-16802022-09-01111050110.3390/axioms11100501Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial ComponentsIrina Perfilieva0Tam Pham1Petr Ferbas2Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech RepublicAdvanced Engineering Department, Varroc Lighting Systems, Suvorovova 195, 742 42 Šenov u Nového Jičína, Czech RepublicThe purpose of this paper is to reduce the complexity of computing the components of the integral <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-transform, <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-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 <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>F</mi><mi>m</mi></msup></semantics></math></inline-formula>-transformation for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>3</mn></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2075-1680/11/10/501Fm-transformfuzzy partitiongenerating functionGaussian quadrature rule |
| spellingShingle | Irina Perfilieva Tam Pham Petr Ferbas Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components Axioms Fm-transform fuzzy partition generating function Gaussian quadrature rule |
| title | Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components |
| title_full | Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components |
| title_fullStr | Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components |
| title_full_unstemmed | Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components |
| title_short | Quadrature Rules for the <em>F</em><sup>m</sup>-Transform Polynomial Components |
| title_sort | quadrature rules for the em f em sup m sup transform polynomial components |
| topic | Fm-transform fuzzy partition generating function Gaussian quadrature rule |
| url | https://www.mdpi.com/2075-1680/11/10/501 |
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