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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
or transform is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
from one space of functions to another. Operators occur commonly in
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
,
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and mathematics. Many are
integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involvi ...
s and
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s. In the following ''L'' is an operator :L:\mathcal\to\mathcal which takes a function y\in\mathcal to another function L in\mathcal. Here, \mathcal and \mathcal are some unspecified
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
s, such as
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) H^p are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real anal ...
, ''L''p space,
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
, or, more vaguely, the space of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s. {, class="wikitable" , - style="background:#eaeaea" ! style="text-align: center" , Expression ! style="text-align: center" , Curve
definition ! style="text-align: center" , Variables ! style="text-align: center" , Description , - ! style="background:#eafaea" colspan=4, Linear transformations , - , L y^{(n)} , , , , , , Derivative of ''n''th order , - , L \int_a^t y \,dt , , Cartesian, , y=y(x)
x=t, , Integral, area , - , L y\circ f, , , , , ,
Composition operator In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule C_\phi (f) = f \circ \phi where f \circ \phi denotes function composition. It is also encountered in composition of permutations in permutati ...
, - , L \frac{y\circ t+y\circ -t}{2}, , , , , , Even component , - , L \frac{y\circ t-y\circ -t}{2}, , , , , , Odd component , - , L y\circ (t+1) - y\circ t = \Delta y, , , , , ,
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 ...
, - , L y\circ (t) - y\circ (t-1) = \nabla y, , , , , , Backward difference (Nabla operator) , - , L \sum y=\Delta^{-1}y, , , , , ,
Indefinite sum In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x or \Delta^ , is the linear operator, inverse of the forward difference operator \Delta . It relates to the forward difference operato ...
operator (inverse operator of difference) , - , L =-(py')'+qy , , , , , , Sturm–Liouville operator , - ! style="background:#eafaea" colspan=4, Non-linear transformations , - , F y^{ 1 , , , , , ,
Inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, - , F t\,y'^{ 1 - y\circ y'^{ 1 , , , , , ,
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a rea ...
, - , F f\circ y, , , , , , Left composition , - , F \prod y, , , , , ,
Indefinite product In mathematics, the indefinite product operator is the inverse operator of Q(f(x)) = \frac. It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Thus :Q\left( \prod_x f(x) \right) = f(x) \, . ...
, - , F \frac{y'}{y}, , , , , ,
Logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the in ...
, - , F {\frac{ty'}{y, , , , , , Elasticity , - , F {y \over y'}-{3\over 2}\left({y''\over y'}\right)^2, , , , , ,
Schwarzian derivative In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms an ...
, - , F \int_a^t , y', \,dt , , , , , ,
Total variation In mathematics, the total variation identifies several slightly different concepts, related to the (local property, local or global) structure of the codomain of a Function (mathematics), function or a measure (mathematics), measure. For a real ...
, - , F \frac{1}{t-a}\int_a^t y\,dt , , , , , ,
Arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
, - , F \exp \left( \frac{1}{t-a}\int_a^t \ln y\,dt \right) , , , , , ,
Geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
, - , F -\frac{y}{y'}, , Cartesian, , y=y(x)
x=t, , rowspan=3, Subtangent , - , F ,y -\frac{yx'}{y'}, , Parametric
Cartesian, , x=x(t)
y=y(t) , - , F -\frac{r^2}{r'}, , Polar, , r=r(\phi)
\phi=t , - , F \frac{1}{2}\int_a^t r^2 dt, , Polar, , r=r(\phi)
\phi=t , , Sector area , - , F \int_a^t \sqrt { 1 + y'^2 }\, dt, , Cartesian, , y=y(x)
x=t, , rowspan=3,
Arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
, - , F ,y \int_a^t \sqrt { x'^2 + y'^2 }\, dt, , Parametric
Cartesian, , x=x(t)
y=y(t) , - , F \int_a^t \sqrt { r^2 + r'^2 }\, dt, , Polar, , r=r(\phi)
\phi=t , - , F = \int_a^t\sqrt y''}\, dt , , Cartesian, , y=y(x)
x=t, , rowspan=3, Affine arc length , - , F ,y= \int_a^t\sqrt x'y''-x''y'}\, dt , , Parametric
Cartesian, , x=x(t)
y=y(t) , - , F ,y,z\int_a^t\sqrt z(x'y''-y'x'')+z''(xy'-x'y)+z'(x''y-xy'')}dt, , Parametric
Cartesian, , x=x(t)
y=y(t)
z=z(t) , - , F \frac{y''}{(1+y'^2)^{3/2, , Cartesian, , y=y(x)
x=t, , rowspan=4,
Curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
, - , F ,y \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2, , Parametric
Cartesian, , x=x(t)
y=y(t) , - , F \frac{r^2+2r'^2-rr''}{(r^2+r'^2)^{3/2, , Polar, , r=r(\phi)
\phi=t , - , F ,y,z\frac{\sqrt{(z''y'-z'y'')^2+(x''z'-z''x')^2+(y''x'-x''y')^2{(x'^2+y'^2+z'^2)^{3/2, , Parametric
Cartesian, , x=x(t)
y=y(t)
z=z(t) , - , F \frac{1}{3}\frac{y'}{(y'')^{5/3-\frac{5}{9}\frac{y^2}{(y'')^{8/3, , Cartesian, , y=y(x)
x=t, , rowspan=2,
Affine curvature Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that p ...
, - , F ,y \frac{x''y-xy''}{(x'y''-x''y')^{5/3-\frac{1}{2}\left frac{1}{(x'y''-x''y')^{2/3\right', , Parametric
Cartesian, , x=x(t)
y=y(t) , - , F ,y,z\frac{z(x'y''-y'x'')+z''(xy'-x'y)+z'(x''y-xy'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)}, , Parametric
Cartesian, , x=x(t)
y=y(t)
z=z(t), ,
Torsion of curves In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvatu ...
, - , X ,y\frac{y'}{yx'-xy'}

Y ,y\frac{x'}{xy'-yx'}, , Parametric
Cartesian, , x=x(t)
y=y(t), ,
Dual curve In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. I ...

(tangent coordinates) , - , X ,yx+\frac{ay'}{\sqrt {x'^2+y'^2

Y ,yy-\frac{ax'}{\sqrt {x'^2+y'^2, , Parametric
Cartesian, , x=x(t)
y=y(t), ,
Parallel curve A parallel of a curve is the envelope (mathematics), envelope of a family of Congruence (geometry), congruent circles centered on the curve. It generalises the concept of ''parallel (geometry), parallel (straight) lines''. It can also be defi ...
, - , X ,yx+y'\frac{x'^2+y'^2}{x''y'-y''x'}

Y ,yy+x'\frac{x'^2+y'^2}{y''x'-x''y'}, , Parametric
Cartesian, , x=x(t)
y=y(t), , rowspan=2,
Evolute In the differential geometry of curves, the evolute of a curve is the locus (mathematics), locus of all its Center of curvature, centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the result ...
, - , F t (r'\circ r^{ 1), , Intrinsic, , r=r(s)
s=t , - , X ,yx-\frac{x'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2

Y ,yy-\frac{y'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 , , Parametric
Cartesian, , x=x(t)
y=y(t), , ,
Involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the Locus (mathematics), locus of a point on a piece of taut string as the string is eith ...
, - , X ,y\frac{(xy'-yx')y'}{x'^2 + y'^2}

Y ,y\frac{(yx'-xy')x'}{x'^2 + y'^2}, , Parametric
Cartesian, , x=x(t)
y=y(t), , ,
Pedal curve A pedal (from the Latin ''wikt:pes#Latin, pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is us ...
with pedal point (0;0) , - , X ,y\frac{(x'^2-y'^2)y'+2xyx'}{xy'-yx'}

Y ,y\frac{(x'^2-y'^2)x'+2xyy'}{xy'-yx'}, , Parametric
Cartesian, , x=x(t)
y=y(t), , , Negative pedal curve with pedal point (0;0) , - , X = \int_a^t \cos \left int_a^t \frac{1}{y} \,dt\rightdt

Y = \int_a^t \sin \left int_a^t \frac{1}{y} \,dt\rightdt, , Intrinsic, , y=r(s)
s=t, , Intrinsic to
Cartesian
transformation , - ! style="background:#eafaea" colspan=4, Metric functionals , - , F \, y\, =\sqrt{\int_E y^2 \, dt}, , , , , ,
Norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...
, - , F ,y\int_E xy \, dt, , , , , ,
Inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
, - , F ,y\arccos \left frac{\int_E xy \, dt}{\sqrt{\int_E x^2 \, dt}\sqrt{\int_E y^2 \, dt\right/math>, , , , , ,
Fubini–Study metric In mathematics, the Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study. A ...

(inner angle) , - ! style="background:#eafaea" colspan=4, Distribution functionals , - , F ,y= x * y = \int_E x(s) y(t - s)\, ds, , , , , ,
Convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, - , F = \int_E y \ln y \, dt, , , , , ,
Differential entropy Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy (a measure of average surprisal) of a random variable, to continu ...
, - , F = \int_E yt\,dt, , , , , ,
Expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
, - , F = \int_E \left(t-\int_E yt\,dt\right)^2y\,dt, , , , , ,
Variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...


See also

*
List of transforms This is a list of transforms in mathematics. Integral transforms *Abel transform * Aboodh transform * Bateman transform *Fourier transform ** Short-time Fourier transform **Gabor transform * Hankel transform * Hartley transform * Hermite transf ...
*
List of Fourier-related transforms This is a list of linear transformations of function (mathematics), functions related to Fourier analysis. Such transformations Map (mathematics), map a function to a set of coefficients of basis functions, where the basis functions are trigonomet ...
*
Transfer operator In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1 ...
*
Fredholm operator In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' :  ...
* Borel transform *
Glossary of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula ...
Operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
Operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
Operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...