{"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}
引用次数: 0
Abstract
For the function $f(x)=\exp(-x)$, $x\in [0,+\infty)$ on grids of nodes $\Delta: 0=x_0