Incomplete Delta Functions

Abstract

By applying projection operators to state vectors of coordinates we obtain subspaces in which these states are no longer normalized according to Dirac's delta function but normalized according to what we call "incomplete delta functions". We show that this class of functions satisfy identities similar to those satisfied by the Dirac delta function. The incomplete delta functions may be employed advantageously in projected subspaces and in the link between functions defined on the whole space and the projected subspace. We apply a similar procedure to finite dimensional vector spaces for which we define incomplete Kronecker deltas. Dispersion relations for the momenta are obtained and ''sums over poles'' are defined and obtained with the aid of differences of incomplete delta functions.