Moduli of smoothness using discrete data (Q1099018)

An interesting approximation theorem on specific cardinal B spline approximation satisfying estimates on its rate of convergence and derivatives is proved. The main result is: For \(f\in C(R)\) satisfyng \(| \Delta_ hf(x)| \leq Kh^{\beta},\) for some \(\beta\) and K, the condition \(| \Delta \quad m_{h_ n}f(\xi_ n+kh_ n)| \leq Kh^{\alpha}_ n,\) where \(\alpha\leq m\) for all k, a sequence \(h_ n\) satisfying \(h_ n=o(1)\) and \(1\leq (h_ n/h_{n+1})\leq M\), and some sequence \(\xi_ n\) of reals implies \(| \Delta \quad m_ hf(x)| \leq K_ 1h^{\alpha},\) for all x and all h. The conclusions are generalized relaxing conditions. For differences given at all points, an extension theorem is achieved by the authors using Steklov type integrals and the Whitney extension theorem. The paper contains several other illuminating ideas on moduli of smoothness with discrete data and some detailed discussions as well.