A generalization of the peano kernel and its applications

Abstract

Abstract. Based on the q-Taylor Theorem, we introduce a more general form of the Peano kernel (q-Peano) which is also applicable to non-di¤erentiable functions. Then we show that quantum B-splines are the q-Peano kernels of divided di¤erences. We also give applications to polynomial interpolation and construct examples in which classical remainder theory fails whereas q-Peano kernel works.