  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.
