Bounding the Lebesgue constant for Berrut's rational interpolant at general nodes (Q387252)

The authors study interpolations with \(n+1\) distinct real nodes in the interval \([a,b]\) given by \(X_n=\{x_0,x_1,\dots,x_n\}\) and the corresponding set of cardinal basis functions \(B_n=\{b_0,b_1,\dots,b_n\}\) (i.e. \(b_j(x_k)=\delta_{jk}\)).NEWLINENEWLINENEWLINEThe Lebesgue constant is the norm of the operator mapping \(f\in{\mathcal C}^0[a,b]\) to \(g=\sum_{j=0}^n\,b_jf(x_j)\) and given by NEWLINE\[NEWLINE\Lambda_n(X_n,B_n)=\max_{a\leq x\leq b}\,\sum_{j=0}^n\,|b_j(x)|.NEWLINE\]NEWLINENEWLINENEWLINEThe specific type of interpolant used in this paper is Berrut's first rational interpolant with basis functions NEWLINE\[NEWLINEb_i(x)={(-1)^i\over x-x_i}/ \sum_{j=0}^n\,{(-1)^j\over x-x_j},\quad i=0,\dots,n,NEWLINE\]NEWLINE and the interest of the authors focuses on the growth of the corresponding Lebesgue constant NEWLINE\[NEWLINE\Lambda(X_n)=\max_{a\leq x\leq b}\,{ \sum_{j=0}^n\,{1\over |x-x_j|}\over\left| \sum_{j=0}^n\,{(-1)^j\over x-x_j}\right|}.NEWLINE\]NEWLINENEWLINENEWLINEThe main result isNEWLINENEWLINE{ Theorem 2.2.} If \((X_n)_{n\in\mathbf{N}}\) is a family of well-spaced nodes, then there exists a constant \(c>0\) such that NEWLINE\[NEWLINE\Lambda(X_n)\leq c\ln{n}\text{ for any }n\geq 2.NEWLINE\]NEWLINE The concept of a well-spaced family is introduced by the requirement of the existence of constants \(C,R\geq 1\), independent of \(n\), such that each of the sets of nodes \(X_n\) satisfies NEWLINE\[NEWLINE\begin{aligned} &{x_{k+1}-x_k\over x_{k+1}-x_j}\leq {C\over k+1-j}\;(0\leq j\leq k,\,0\leq k\leq n-1),\\& {x_{k+1}-x_k\over x_j-x_k}\leq {C\over j-k}\;(k+1\leq j\leq n,\,0\leq k\leq n-1),\\& {1\over R}\leq {x_{k+1}-x_k\over x_k-x_{k-1}}\;(1\leq k\leq n-1).\end{aligned}NEWLINE\]NEWLINENEWLINENEWLINEFurthermore, the authors give a method to construct well-spaced families using regular distribution functions:NEWLINENEWLINE- distribution function: strictly increasing bijection on the interval \([0,1]\),NEWLINENEWLINE- regular: \(F\in{\mathcal C}^1[0,1],\;F'\) has a finite number of zeros \(T=[t_1,t_2,\dots,t_{\ell}]\subset [0,1]\) with finite multiplicitiesNEWLINENEWLINEand there exist positive real numbers \(r_1,r_2,\dots,r_{\ell}\) with NEWLINE\[NEWLINEG_j(t)=F'(t)/|t-t_j|^{r_j}\text{ is continuous with }\lim_{t\rightarrow t_j}\;G_j(t)>0\text{ for }j=1,\dots,\ell.NEWLINE\]NEWLINE \(X_n=X_n(F)\) is then defined by \(x_k=F(k/n)\), \(0\leq k\leq n\).