Generalized Taylor Series and Peano Kernel Theorem

Abstract

ABSTRACTAs in the polynomial case, non‐polynomial divided differences can be viewed as a discrete analog of derivatives. This link between non‐polynomial divided differences and derivatives is defined by a generalization of the derivative operator. In this study, we obtain a generalization of Taylor series using the link between non‐polynomial divided differences and derivatives, and state generalized Taylor theorem. With the definition of a definite integral, the relation between the non‐polynomial divided difference and non‐polynomial B‐spline functions is given in terms of integration. Also, we derive a general form of the Peano kernel theorem based on a generalized Taylor expansion with the integral remainder. As in the polynomial case, it is shown that the non‐polynomial B‐splines are in fact the Peano kernels of non‐polynomial divided differences.MSC2020 Classification: 65D05, 65D07