Sufficient conditions for convergence of generalized sinc-approximations on segment
The classical approximation by translates of the sinc-function is a special case of quasi-interpolation using kernel functions that may be generalised in many ways e.g. by radial basis functions or spline approximations. In this paper the quasi-interpolation with the sinc-function is generalised by presenting this kernel as a special case of quasi-interpolation with solutions \(y(\cdot)\) of the Cauchy problem. The mentioned kernels are computed from \(y(\cdot)=y(\cdot,\lambda)\) with \(y''(x)+(\lambda-q_\lambda(x))y(x)=0\) using Hermite boundary conditions. \(x\) is between \(0\) and \(\pi\). If \(x_{i,\lambda}\) are the \(\lambda\)-dependent zeros of \(y(\cdot,\lambda)\), then the said kernels, within \([0,\pi]\), are \[ \frac{y(\cdot,\lambda)}{y'(x_{i,\lambda},\lambda)(\cdot-x_{i,\lambda})}\] and the quasi-interpolants \(S_\lambda(f,\cdot)\) are formed simply by point evaluations of the approximants \(f\) at the aforementioned zeros, employing these kernels. For this and another variant of the approximation which is called \(T_\lambda\) and which takes special care of the boundary values at the origin and at \(\pi\) of the approximant, uniform convergence theorems are derived by the author.
scientific article; zbMATH DE number 7351528
scientific article