Delta operatorIn mathematics, a delta operator is a shift-equivariant linear operator Q on the vector space of polynomials in a variable x that reduces degrees by one.
To say that Q is shift-equivariant means that if f(x) = g(x + a), i.e., f is a "shift" of g, then (Qf)(x) = (Qg)(x + a), Qf is the same shift of Qg that f is of g. That the operator reduces degrees by one means that if f is a polynomial of degree n, then Qf is either a polynomial of degree n - 1, or, in case n = 0, Qf is 0.
Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in x that maps x to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition.
The name "delta operator" is due to F. Hildebrandt.