Generalized Taylor Series and Peano Kernel Theorem

OpenAlex İngilizce

ABSTRACT As 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