Lagrange interpolation for continuous piecewise smooth functions (Q950072)

Given \(y\in(a,b)\), let \(\gamma:[a,b]\to \mathbb{R}\) be an increasing continuous function with finite derivative for all \(x\in(a,b)\backslash \{y\}\), and \(\gamma'(y)=+\infty\). The following class of continuous piecewise smooth functions is constructed in the paper. \(\mathcal{F}_\gamma([a,b])\) denotes the class of functions \(f\) which are continuous in a finite interval \([a,b]\), and such that the following limit exists for all \(x\in[a,b]\). \[ \mathcal{D}(f)(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{\gamma(x+h)-\gamma(x)}. \] Taking into account the design of the derivative \(\mathcal{D}(f)\), the authors have extended the notion of divided differences, to obtain an explicit formulation for the error when the Lagrange interpolation is applied to \(f\in \mathcal{F}_\gamma([a,b])\). Let \(\gamma=\sum_{i=0}^m \gamma_i\), where \(\gamma_i\) is associated with \(y_i\in(a,b)\), \(i=0,...,m\). If \(f\) is smooth at all points \(x\in (y_i,y_{i+1})\), and has finite lateral derivatives at \(y_i\), \(i=0,...,m\), then the Lagrange interpolation formula holds true for \(f\).