{"title":"三点有理插值中样条函数在半轴上逼近$ \\ exp (-x) $","authors":"A. Ramazanov, V. Magomedova","doi":"10.31029/demr.11.4","DOIUrl":null,"url":null,"abstract":"For the function $f(x)=\\exp(-x)$, $x\\in [0,+\\infty)$ on grids of nodes $\\Delta: 0=x_0<x_1<\\dots $ with $x_n\\to +\\infty$ we construct rational spline-functions such that $R_k(x,f, \\Delta)=R_i(x,f)A_{i,k}(x)\\linebreak+R_{i-1}(x, f)B_{i,k}(x)$ for $x\\in[x_{i-1}, x_i]$ $(i=1,2,\\dots)$ and $k=1,2,\\dots$ Here $A_{i,k}(x)=(x-x_{i-1})^k/((x-x_{i-1})^k+(x_i-x)^k)$, $B_{i,k}(x)=1-A_{i,k}(x)$, $R_j(x,f)=\\alpha_j+\\beta_j(x-x_j)+\\gamma_j/(x+1)$ $(j=1,2,\\dots)$, $R_j(x_m,f)=f(x_m)$ при $m=j-1,j,j+1$; we take $R_0(x,f)\\equiv R_1(x,f)$.\n\nBounds for the convergence rate of $R_k(x,f, \\Delta)$ with $f(x)=\\exp(-x)$, $x\\in [0,+\\infty)$, are found.","PeriodicalId":431345,"journal":{"name":"Daghestan Electronic Mathematical Reports","volume":"9 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the approximation of $ \\\\ exp (-x) $ on the half-axis by spline functions in three-point rational interpolants\",\"authors\":\"A. Ramazanov, V. Magomedova\",\"doi\":\"10.31029/demr.11.4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For the function $f(x)=\\\\exp(-x)$, $x\\\\in [0,+\\\\infty)$ on grids of nodes $\\\\Delta: 0=x_0<x_1<\\\\dots $ with $x_n\\\\to +\\\\infty$ we construct rational spline-functions such that $R_k(x,f, \\\\Delta)=R_i(x,f)A_{i,k}(x)\\\\linebreak+R_{i-1}(x, f)B_{i,k}(x)$ for $x\\\\in[x_{i-1}, x_i]$ $(i=1,2,\\\\dots)$ and $k=1,2,\\\\dots$ Here $A_{i,k}(x)=(x-x_{i-1})^k/((x-x_{i-1})^k+(x_i-x)^k)$, $B_{i,k}(x)=1-A_{i,k}(x)$, $R_j(x,f)=\\\\alpha_j+\\\\beta_j(x-x_j)+\\\\gamma_j/(x+1)$ $(j=1,2,\\\\dots)$, $R_j(x_m,f)=f(x_m)$ при $m=j-1,j,j+1$; we take $R_0(x,f)\\\\equiv R_1(x,f)$.\\n\\nBounds for the convergence rate of $R_k(x,f, \\\\Delta)$ with $f(x)=\\\\exp(-x)$, $x\\\\in [0,+\\\\infty)$, are found.\",\"PeriodicalId\":431345,\"journal\":{\"name\":\"Daghestan Electronic Mathematical Reports\",\"volume\":\"9 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1900-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Daghestan Electronic Mathematical Reports\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31029/demr.11.4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Daghestan Electronic Mathematical Reports","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31029/demr.11.4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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