HOME
TheInfoList
Click Here for Items Related To -
Shift-equivariant
OR:
Shift-equivariant
on:  
[Wikipedia]  
[Google]  
[Amazon]
In
mathematics
, a delta operator is a shift-equivariant
linear operator
Q\colon\mathbb
\longrightarrow \mathbb
/math> on the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s in a variable
x
over a
field
Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
\mathbb
that reduces
degrees
by one. To say that
Q
is shift-equivariant means that if
g(x) = f(x + a)
, then :
.\,
In other words, if
f
is a "shift" of
g
, then
Qf
is also a shift of
Qg
, and has the same "shifting vector"
a
. To say that an operator ''reduces degree 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 when
\mathbb
has
characteristic
zero, since shift-equivariance is a fairly strong condition.
Examples
* The forward
difference operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
::
(\Delta f)(x) = f(x + 1) - f(x)\,
:is a delta operator. *
Differentiation
with respect to ''x'', written as ''D'', is also a delta operator. * Any operator of the form ::
\sum_^\infty c_k D^k
: (where ''D''
''n''
(ƒ) = ƒ
(''n'')
is the ''n''
th
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
) with
c_1\neq0
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 ::
\Delta=e^D-1=\sum_^\infty \frac.
* The generalized derivative of
time scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hy ...
which unifies the forward difference operator with the derivative of
standard calculus
is a delta operator. * In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
and
cybernetics
, the term "discrete-time delta operator" (δ) is generally taken to mean a difference operator ::
,
: the
Euler approximation
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit met ...
of the usual derivative with a discrete sample time
\Delta t
. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.
Basic polynomials
Every delta operator ''
Q
'' has a unique sequence of "basic polynomials", a
polynomial sequence
defined by three conditions: *
p_0(x)=1 ;
*
p_(0)=0;
*
(Qp_n)(x)=np_(x) \text n \in \mathbb N.
Such a sequence of basic polynomials is always of
binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities :p_ ...
, 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
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are ...
—a more general concept.
See also
*
Pincherle derivative
In mathematics, the Pincherle derivative T' of a linear operator T: \mathbb \to \mathbb /math> on the vector space of polynomials in the variable ''x'' over a field \mathbb is the commutator of T with the multiplication by ''x'' in the algebra of ...
*
Shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift ...
*
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
References
*
External links
* {{MathWorld, title=Delta Operator, urlname=DeltaOperator
Linear algebra
Polynomials
Finite differences