On the Concept of a Filtered Bundle

We present the notion of a filtered bundle as a generalization of a graded bundle. In
particular, we weaken the necessity of the transformation laws for local coordinates
to exactly respect the weight of the coordinates by allowing more general polynomial
transformation laws. The key examples of such bundles include affine bundles and various
jet bundles, both of which play fundamental roles in geometric mechanics and classical
field theory. We also present the notion of double filtered bundles which provide natural
generalizations of double vector bundles and double affine bundles. Furthermore, we show
that the linearization of a filtered bundle — which can be seen as a partial polarization
of the admissible changes of local coordinates — is well defined.