Delta operator


In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one.

To say that is shift-equivariant means that if , then

In other words, if is a "shift" of , then is also a shift of , and has the same "shifting vector" .

To say that an operator reduces degree by one means that if is a polynomial of degree , then is either a polynomial of degree , or, in case , is 0.

Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in that maps to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when has characteristic zero, since shift-equivariance is a fairly strong condition.


is a delta operator.
  • Differentiation with respect to x, written as D, is also a delta operator.
  • Any operator of the form
(where Dn(ƒ) = ƒ(n) is the nth derivative) with   is a delta operator. It can be shown that all delta operators can be written in this form. For example, the difference operator given above can be expanded as
the Euler approximation of the usual derivative with a discrete sample time  . The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.

Basic polynomials


Every delta operator   has a unique sequence of "basic polynomials", a polynomial sequence defined by three conditions:


Such a sequence of basic polynomials is always of binomial type, and it can be shown that no other sequences of binomial type exist. If the first two conditions above are dropped, then the third condition says this polynomial sequence is a Sheffer sequence—a more general concept.

See also



  • Nikol'Skii, Nikolai Kapitonovich (1986), Treatise on the shift operator: spectral function theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-15021-5