{"title":"椭圆上解析函数的高斯-洛巴托正交误差边界","authors":"Hiroshi Sugiura , Takemitsu Hasegawa","doi":"10.1016/j.cam.2024.116326","DOIUrl":null,"url":null,"abstract":"<div><div>For the (<span><math><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></math></span>)-point Gauss–Jacobi–Lobatto quadrature to integrals with the Jacobi weight function <span><math><mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>α</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>β</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></math></span>) over the interval <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>, we estimate the location where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. As in our previous work on the Gauss–Jacobi rule, when <span><math><mrow><mi>α</mi><mo>≠</mo><mi>β</mi></mrow></math></span>, the location is the intersection point of the ellipse with the real axis in the complex plane. When <span><math><mrow><mi>α</mi><mo>=</mo><mi>β</mi></mrow></math></span> (the Gegenbauer weight), it is the intersection points with the real axis for <span><math><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>0</mn></mrow></math></span> or for <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span>, and with the imaginary axis for <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>></mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> or for <span><math><mrow><mi>α</mi><mo>></mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Here, <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> (<span><math><mrow><mo>></mo><mn>0</mn></mrow></math></span>) is a monotonously decreasing function for <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span> with <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>α</mi><mo>→</mo><mo>+</mo><mn>0</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo><mi>∞</mi></mrow></math></span>. Some numerical results are given to confirm the locations.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116326"},"PeriodicalIF":2.1000,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Error bounds for Gauss–Lobatto quadrature of analytic functions on an ellipse\",\"authors\":\"Hiroshi Sugiura , Takemitsu Hasegawa\",\"doi\":\"10.1016/j.cam.2024.116326\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>For the (<span><math><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></math></span>)-point Gauss–Jacobi–Lobatto quadrature to integrals with the Jacobi weight function <span><math><mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup></mrow></math></span> (<span><math><mrow><mi>α</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>β</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></math></span>) over the interval <span><math><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span>, we estimate the location where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. As in our previous work on the Gauss–Jacobi rule, when <span><math><mrow><mi>α</mi><mo>≠</mo><mi>β</mi></mrow></math></span>, the location is the intersection point of the ellipse with the real axis in the complex plane. When <span><math><mrow><mi>α</mi><mo>=</mo><mi>β</mi></mrow></math></span> (the Gegenbauer weight), it is the intersection points with the real axis for <span><math><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>0</mn></mrow></math></span> or for <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span>, and with the imaginary axis for <span><math><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>></mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> or for <span><math><mrow><mi>α</mi><mo>></mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Here, <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> (<span><math><mrow><mo>></mo><mn>0</mn></mrow></math></span>) is a monotonously decreasing function for <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span> with <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>α</mi><mo>→</mo><mo>+</mo><mn>0</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>=</mo><mi>∞</mi></mrow></math></span>. Some numerical results are given to confirm the locations.</div></div>\",\"PeriodicalId\":50226,\"journal\":{\"name\":\"Journal of Computational and Applied Mathematics\",\"volume\":\"457 \",\"pages\":\"Article 116326\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-10-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0377042724005740\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724005740","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Error bounds for Gauss–Lobatto quadrature of analytic functions on an ellipse
For the ()-point Gauss–Jacobi–Lobatto quadrature to integrals with the Jacobi weight function (, ) over the interval , we estimate the location where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. As in our previous work on the Gauss–Jacobi rule, when , the location is the intersection point of the ellipse with the real axis in the complex plane. When (the Gegenbauer weight), it is the intersection points with the real axis for or for and , and with the imaginary axis for and or for . Here, () is a monotonously decreasing function for with and . Some numerical results are given to confirm the locations.
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