Oscillation matrices in spline-interpolation problems (Q1109251)

For a given strictly increasing sequence \(\Delta:...<x_{-1}<x_ 0<x_ 1<..\). such that \(\lim_{\nu \to -\infty}x_{\nu}=a\), \(\lim_{\nu \to \infty}x_{\nu}=b\), \(-\infty \leq a<b\leq \infty\), we define \({\mathcal S}_ n(\Delta)=\{S:S\in C^{n-1}(a,b)\), \(S|_{(x_{\nu},x_{\nu +1})}\in {\mathcal P}_ n\), \(\nu \in {\mathbb{Z}}=\{0,\pm 1,....\}\}\), where \({\mathcal P}_ n\) is the set of polynomials of degree at most n; \({\mathcal S}_ n(\Delta)\) is called the class of spline functions of degree n on the net \(\Delta\). We consider the problem of the construction of a spline \(S\in {\mathcal S}_ n(\Delta)\) such that \(S(x_{\nu})=y_{\nu}\), \(\nu\in Z\), for given values \(...,y_{-1},y_ 0,y_ 1,..\).