q-Peano Kernel and its Applications

TL;DR

This paper introduces a q-analogue of the Peano kernel theorem, extending classical results into quantum calculus, and demonstrates its applications in polynomial interpolation and relations to q-B-splines.

Contribution

It presents the first q-analogue of the Peano kernel theorem, bridging quantum calculus with classical remainder theory and spline analysis.

Findings

q-Peano kernel reduces to classical kernel as q approaches 1

Applications improve polynomial interpolation where classical methods fail

Establishes a relation between q-B-splines and divided differences

Abstract

We introduce a q-analogue of the Peano kernel theorem by replacing ordinary derivatives and integrals by quantum derivatives and quantum integrals. In the limit q \to 1, the q-Peano kernel reduces to the classical Peano kernel. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works. Furthermore we derive a relation between q-B-splines and divided differences via the q-Peano kernel.