Difference operator

Difference operator

In mathematics, a difference operator maps a function, "f"("x"), to another function, "f"("x + a") − "f"("x + b").

The forward difference operator :Delta f(x)=f(x+1)-f(x),occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the derivative, but used in discrete circumstances. Difference equations can often be solved with techniques very similar to those for solving differential equations.This similarity led to the development of time scale calculus. Analogously we can have the backward difference operator

: abla f(x)=f(x)-f(x-1).,

When restricted to polynomial functions "f", the forward difference operator is a delta operator, i.e., a shift-equivariant linear operator on polynomials that reduces degree by 1.

"n"-th difference

The "n"th forward difference of a function "f"("x") is given by

:Delta^n [f] (x)= sum_{k=0}^n {n choose k} (-1)^{n-k} f(x+k)

where {n choose k} is the binomial coefficient. Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and, as such, have a number of interesting combinatorial properties.

Forward differences may be evaluated using the Nörlund-Rice integral. The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large "n".

Newton series

The Newton series or Newton forward difference equation, named after Isaac Newton, is the relationship

:f(x+a)=sum_{k=0}^inftyfrac{Delta^k [f] (a)}{k!}(x)_k= sum_{k=0}^infty {x choose k} Delta^k [f] (a)

which holds for any polynomial function "f" and for some, but not all, analytic functions. Here,

:{x choose k}

is the binomial coefficient, and

:(x)_k=x(x-1)(x-2)cdots(x-k+1)

is the "falling factorial" or "lower factorial" and the empty product ("x")0 defined to be 1. Note also the formal similarity of this result to Taylor's theorem; this is one of the observations that lead to the idea of umbral calculus.

In analysis with p-adic numbers, Mahler's theorem states that the assumption that "f" is a polynomial function can be weakened all the way to the assumption that "f" is merely continuous.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not, in general, exist.

The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of scaled forward differences.

Rules for finding the difference applied to a combination of functions

Analogous to rules for finding the derivative, we have:
* Constant rule: If "c" is a constant, then : riangle c = 0
* Linearity: if "a" and "b" are constants,: riangle (a f + b g) = a , riangle f + b , riangle g

All of the above rules apply equally well to any difference operator, including abla as to riangle.
* Product rule: : riangle (f g) = f , riangle g + g , riangle f + riangle f , riangle g : abla (f g) = f , abla g + g , abla f - abla f , abla g
* Quotient rule:: abla left( frac{f}{g} ight) = frac{1}{g} det egin{bmatrix} abla f & abla g \ f & g end{bmatrix} left( det {egin{bmatrix} g & abla g \ 1 & 1 end{bmatrix ight)^{-1} ::or: ablaleft( frac{f}{g} ight)= frac {g , abla f - f , abla g}{g cdot (g - abla g)}: riangleleft( frac{f}{g} ight)= frac {g , riangle f - f , riangle g}{g cdot (g + riangle g)}

* Summation rules::sum_{n=a}^{b} riangle f(n) = f(b+1)-f(a):sum_{n=a}^{b} abla f(n) = f(b)-f(a-1)

Generalizations

Difference operator generalizes to Möbius inversion over a partially ordered set.

As a convolution operator

Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function μ; for the difference operator, μ is the sequence (1,-1,0,dots).

ee also

* Newton polynomial
* Table of Newtonian series
* Lagrange polynomial
* Gilbreath's conjecture

References

*citation
first1 = Philippe | last1 = Flajolet
authorlink2 = Robert Sedgewick (computer scientist) | first2 = Robert | last2 = Sedgewick
url = http://www-rocq.inria.fr/algo/flajolet/Publications/mellin-rice.ps.gz
title = Mellin transforms and asymptotics: Finite differences and Rice's integrals
journal = Theoretical Computer Science
volume = 144 | issue = 1–2 | year = 1995 | pages = 101–124
doi = 10.1016/0304-3975(94)00281-M
.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Difference polynomials — In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg s polynomials, and the Stirling… …   Wikipedia

  • Operator messaging — is the term, similar to Text Messaging and Voice Messaging, applying to an answering service call center who focuses on one specific scripting style that has grown out of the alphanumeric pager history. Contents 1 Early history 2 Message Center… …   Wikipedia

  • Operator (sternwheeler) — Operator on Skeena River 1911 Career (Canada) …   Wikipedia

  • Difference of Gaussian — Helligkeitsänderung einer Kante Verlauf der 2. Ableitung an der Kante Der Marr Hildreth Operator oder Laplacian of Gaussian (LoG) ist eine spezielle Form eines diskreten Laplace Filters …   Deutsch Wikipedia

  • Difference of Gaussians — In computer vision, Difference of Gaussians is a grayscale image enhancement algorithm that involves the subtraction of one blurred version of an original grayscale image from another, less blurred version of the original. The blurred images are… …   Wikipedia

  • Operator product expansion — Contents 1 2D Euclidean quantum field theory 2 General 3 See also 4 External links 2D Euclidean quantum field theory …   Wikipedia

  • Difference Engine — ▪ calculating machine       an early calculating machine, verging on being the first computer, designed and partially built during the 1820s and 30s by Charles Babbage (Babbage, Charles). Babbage was an English mathematician and inventor; he… …   Universalium

  • Finite difference — A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences… …   Wikipedia

  • 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 …   Wikipedia

  • Lag operator — In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time series:X= {X 1, X 2, dots },then :, L X t = X {t 1} for all ; t > 1,where L is… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”