{"title":"平滑类的近似问题","authors":"Yongping Liu, Man Lu","doi":"10.1007/s10473-024-0505-4","DOIUrl":null,"url":null,"abstract":"<p>This paper investigates the relative Kolmogorov <i>n</i>-widths of 2<i>π</i>-periodic smooth classes in <span>\\(\\widetilde{L}_{q}\\)</span>. We estimate the relative widths of <span>\\(\\widetilde{W}^{r}H^{\\omega}_{p}\\)</span> and its generalized class <i>K</i><sub><i>p</i></sub><i>H</i><sup><i>ω</i></sup> (<i>P</i><sub><i>r</i></sub>), where <i>K</i><sub><i>p</i></sub><i>H</i><sup><i>ω</i></sup> (<i>P</i><sub><i>r</i></sub>) is defined by a self-conjugate differential operator <i>P</i><sub><i>r</i></sub> (<i>D</i>) induced by</p><span>$$P_{r}(t):= t^{\\sigma} \\Pi_{j=1}^{l}(t^{2}- t_{j}^{2}),~t_{j} > 0,~j=1, 2,\\cdots, l,~l \\geq 1,~\\sigma \\geq 1,~r=2l+\\sigma.$$</span><p>Also, the modulus of continuity of the <i>r</i>-th derivative, or <i>r</i>-th self-conjugate differential, does not exceed a given modulus of continuity <i>ω</i>. Then we obtain the asymptotic results, especially for the case <i>p</i> = ∞, 1 ≤ <i>q</i> ≤ ∞.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"166 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approximation problems on the smoothness classes\",\"authors\":\"Yongping Liu, Man Lu\",\"doi\":\"10.1007/s10473-024-0505-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper investigates the relative Kolmogorov <i>n</i>-widths of 2<i>π</i>-periodic smooth classes in <span>\\\\(\\\\widetilde{L}_{q}\\\\)</span>. We estimate the relative widths of <span>\\\\(\\\\widetilde{W}^{r}H^{\\\\omega}_{p}\\\\)</span> and its generalized class <i>K</i><sub><i>p</i></sub><i>H</i><sup><i>ω</i></sup> (<i>P</i><sub><i>r</i></sub>), where <i>K</i><sub><i>p</i></sub><i>H</i><sup><i>ω</i></sup> (<i>P</i><sub><i>r</i></sub>) is defined by a self-conjugate differential operator <i>P</i><sub><i>r</i></sub> (<i>D</i>) induced by</p><span>$$P_{r}(t):= t^{\\\\sigma} \\\\Pi_{j=1}^{l}(t^{2}- t_{j}^{2}),~t_{j} > 0,~j=1, 2,\\\\cdots, l,~l \\\\geq 1,~\\\\sigma \\\\geq 1,~r=2l+\\\\sigma.$$</span><p>Also, the modulus of continuity of the <i>r</i>-th derivative, or <i>r</i>-th self-conjugate differential, does not exceed a given modulus of continuity <i>ω</i>. Then we obtain the asymptotic results, especially for the case <i>p</i> = ∞, 1 ≤ <i>q</i> ≤ ∞.</p>\",\"PeriodicalId\":50998,\"journal\":{\"name\":\"Acta Mathematica Scientia\",\"volume\":\"166 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-08-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Scientia\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10473-024-0505-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0505-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
This paper investigates the relative Kolmogorov n-widths of 2π-periodic smooth classes in \(\widetilde{L}_{q}\). We estimate the relative widths of \(\widetilde{W}^{r}H^{\omega}_{p}\) and its generalized class KpHω (Pr), where KpHω (Pr) is defined by a self-conjugate differential operator Pr (D) induced by
Also, the modulus of continuity of the r-th derivative, or r-th self-conjugate differential, does not exceed a given modulus of continuity ω. Then we obtain the asymptotic results, especially for the case p = ∞, 1 ≤ q ≤ ∞.
期刊介绍:
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.