TY - JOUR
T1 - Generalized Taylor Series and Peano Kernel Theorem
AU - Zürnacı-Yetiş, Fatma
AU - Dişibüyük, Çetin
N1 - Publisher Copyright:
© 2024 John Wiley & Sons Ltd.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - divided differences
KW - generalized Taylor series
KW - non-polynomial divided differences
KW - Peano kernel theorem
UR - http://www.scopus.com/inward/record.url?scp=85211131414&partnerID=8YFLogxK
U2 - 10.1002/mma.10616
DO - 10.1002/mma.10616
M3 - Article
AN - SCOPUS:85211131414
SN - 0170-4214
JO - Mathematical Methods in the Applied Sciences
JF - Mathematical Methods in the Applied Sciences
ER -