In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. Convolution is similar to crosscorrelation. For realvalued functions, of a continuous or discrete variable, it differs from crosscorrelation only in that either f (x) or g(x) is reflected about the yaxis; thus it is a crosscorrelation of f (x) and g(−x), or f (−x) and g(x).^{ [note 1] } For continuous functions, the crosscorrelation operator is the adjoint of the convolution operator.
Mathematics includes the study of such topics as quantity, structure, space, and change.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations, such as addition and multiplication, and unary operations, such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is supposed to be finite. However, infinitary operations are sometimes considered, in which context the "usual" operations of finite arity are called finitary operations.
Convolution has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations.^{[ citation needed ]}
Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.
Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.
Computer vision is an interdisciplinary scientific field that deals with how computers can be made to gain highlevel understanding from digital images or videos. From the perspective of engineering, it seeks to automate tasks that the human visual system can do.
The convolution can be defined for functions on Euclidean space, and other groups. ^{[ citation needed ]} For example, periodic functions, such as the discretetime Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 13 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers.
In geometry, Euclidean space encompasses the twodimensional Euclidean plane, the threedimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.
Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.^{[ citation needed ]}
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Numerical linear algebra is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to mathematical questions. It is a subfield of numerical analysis, and a type of linear algebra. Because computers use floatingpoint arithmetic, they cannot exactly represent irrational data, and many algorithms increase that imprecision when implemented by a computer. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize computer error while retaining efficiency and precision.
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.
Computing the inverse of the convolution operation is known as deconvolution.
In mathematics, deconvolution is an algorithmbased process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing. Because these techniques are in turn widely used in many scientific and engineering disciplines, deconvolution finds many applications.
The convolution of f and g is written f∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:
An asterisk (*); from Late Latin asteriscus, from Ancient Greek ἀστερίσκος, asteriskos, "little star", is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star.
In mathematics, an integral transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily than in the original domain. The solution is then mapped back to the original domain using the inverse of the integral transform.
An equivalent definition is (see commutativity):
While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function f (τ) at the moment t where the weighting is given by g(–τ) simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.
For functions f, g supported on only [0, ∞) (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:
For the multidimensional formulation of convolution, see domain of definition (below).
A common engineering convention is:^{ [1] }
which has to be interpreted carefully to avoid confusion. For instance, f (t)∗g(t − t_{0}) is equivalent to (f ∗g)(t − t_{0}), but f (t − t_{0})∗g(t − t_{0}) is in fact equivalent to (f ∗g)(t − 2t_{0}).^{ [2] }
Convolution describes the output (in terms of the input) of an important class of operations known as linear timeinvariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.
Visual explanations of convolution  

 
 

One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754.^{ [3] }
Also, an expression of the type:
is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.^{ [4] } Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, JeanBaptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 60s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral.^{ [5] } Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses.^{ [6] }^{ [7] }
The operation:
is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913.^{ [8] }
When a function g_{T} is periodic, with period T, then for functions, f, such that f∗g_{T} exists, the convolution is also periodic and identical to:
where t_{o} is an arbitrary choice. The summation is called a periodic summation of the function f.
When g_{T} is a periodic summation of another function, g, then f∗g_{T} is known as a circular or cyclic convolution of f and g.
And if the periodic summation above is replaced by f_{T}, the operation is called a periodic convolution of f_{T} and g_{T}.
For complexvalued functions f, g defined on the set Z of integers, the discrete convolution of f and g is given by:^{ [9] }
or equivalently (see commutativity) by:
The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the Cauchy product of the coefficients of the sequences.
Thus when g has finite support in the set (representing, for instance, a finite impulse response), a finite summation may be used:^{ [10] }
When a function g_{N} is periodic, with period N, then for functions, f, such that f∗g_{N} exists, the convolution is also periodic and identical to:
The summation on k is called a periodic summation of the function f.
If g_{N} is a periodic summation of another function, g, then f∗g_{N} is known as a circular convolution of f and g.
When the nonzero durations of both f and g are limited to the interval [0, N−1], f∗g_{N} reduces to these common forms:
 (Eq.1) 
The notation (f ∗_{N}g) for cyclic convolution denotes convolution over the cyclic group of integers modulo N.
Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (FFT) algorithm.
In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation in multiplication of multidigit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997 , §4.3.3.C; von zur Gathen & Gerhard 2003 , §8.2).
Eq.1 requires N arithmetic operations per output value and N^{2} operations for N outputs. That can be significantly reduced with any of several fast algorithms. Digital signal processing and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.
The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem. Specifically, the circular convolution of two finitelength sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zeroextension and/or discarding portions of the output. Other fast convolution algorithms, such as the Schönhage–Strassen algorithm or the Mersenne transform,^{ [11] } use fast Fourier transforms in other rings.
If one sequence is much longer than the other, zeroextension of the shorter sequence and fast circular convolution is not the most computationally efficient method available.^{ [12] } Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the Overlap–save method and Overlap–add method.^{ [13] } A hybrid convolution method that combines block and FIR algorithms allows for a zero inputoutput latency that is useful for realtime convolution computations.^{ [14] }
The convolution of two complexvalued functions on R^{d} is itself a complexvalued function on R^{d}, defined by:
is welldefined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blowup in g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g:
If f and g are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous ( Hörmander 1983 , Chapter 1). More generally, if either function (say f) is compactly supported and the other is locally integrable, then the convolution f∗g is welldefined and continuous.
Convolution of f and g is also well defined when both functions are locally square integrable on R and supported on an interval of the form [a, +∞) (or both supported on [−∞, a]).
The convolution of f and g exists if f and g are both Lebesgue integrable functions in L^{1}(R^{d}), and in this case f∗g is also integrable ( Stein & Weiss 1971 , Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L^{1}, under the discrete convolution, or more generally for the convolution on any group.
Likewise, if f ∈ L^{1}(R^{d}) and g ∈ L^{p}(R^{d}) where 1 ≤ p ≤ ∞, then f∗g ∈ L^{p}(R^{d}), and
In the particular case p = 1, this shows that L^{1} is a Banach algebra under the convolution (and equality of the two sides holds if f and g are nonnegative almost everywhere).
More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable L^{p} spaces. Specifically, if 1 ≤ p, q, r ≤ ∞ satisfy:
then
so that the convolution is a continuous bilinear mapping from L^{p}×L^{q} to L^{r}. The Young inequality for convolution is also true in other contexts (circle group, convolution on Z). The preceding inequality is not sharp on the real line: when 1 < p, q, r < ∞, there exists a constant B_{p,q} < 1 such that:
The optimal value of B_{p,q} was discovered in 1975.^{ [15] }
A stronger estimate is true provided 1 < p, q, r < ∞ :
where is the weak L^{q} norm. Convolution also defines a bilinear continuous map for , owing to the weak Young inequality:^{ [16] }
In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly. In particular, if f and g are rapidly decreasing functions, then so is the convolution f∗g. Combined with the fact that convolution commutes with differentiation (see Properties), it follows that the class of Schwartz functions is closed under convolution ( Stein & Weiss 1971 , Theorem 3.3).
Under some circumstances, it is possible to define the convolution of a function with a distribution, or of two distributions. If f is a compactly supported function and g is a distribution, then f∗g is a smooth function defined by a distributional formula analogous to
More generally, it is possible to extend the definition of the convolution in a unique way so that the associative law
remains valid in the case where f is a distribution, and g a compactly supported distribution ( Hörmander 1983 , §4.2).
The convolution of any two Borel measures μ and ν of bounded variation is the measure λ defined by ( Rudin 1962 )
This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L^{1} functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.
The convolution of measures also satisfies the following version of Young's inequality
where the norm is the total variation of a measure. Because the space of measures of bounded variation is a Banach space, convolution of measures can be treated with standard methods of functional analysis that may not apply for the convolution of distributions.
The convolution defines a product on the linear space of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative associative algebra without identity ( Strichartz 1994 , §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, are closed under the convolution, and so also form commutative associative algebras.
Proof: By definition
Changing the variable of integration to and the result follows.
Proof: This follows from using Fubini's theorem (i.e., double integrals can be evaluated as iterated integrals in either order).
Proof: This follows from linearity of the integral.
for any real (or complex) number .
No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a delta distribution or, at the very least (as is the case of L^{1}) admit approximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,
where δ is the delta distribution.
Some distributions have an inverse element for the convolution, S^{(−1)}, which is defined by
The set of invertible distributions forms an abelian group under the convolution.
Proof:
If f and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:
This follows from Fubini's theorem. The same result holds if f and g are only assumed to be nonnegative measurable functions, by Tonelli's theorem.
In the onevariable case,
where d/dx is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:
A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of f and g is differentiable as many times as f and g are in total.
These identities hold under the precise condition that f and g are absolutely integrable and at least one of them has an absolutely integrable (L^{1}) weak derivative, as a consequence of Young's convolution inequality. For instance, when f is continuously differentiable with compact support, and g is an arbitrary locally integrable function,
These identities also hold much more broadly in the sense of tempered distributions if one of f or g is a compactly supported distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.
In the discrete case, the difference operator Df(n) = f(n + 1) −f(n) satisfies an analogous relationship:
The convolution theorem states that
where denotes the Fourier transform of , and is a constant that depends on the specific normalization of the Fourier transform. Versions of this theorem also hold for the Laplace transform, twosided Laplace transform, Ztransform and Mellin transform.
See also the less trivial Titchmarsh convolution theorem.
The convolution commutes with translations, meaning that
where τ_{x}f is the translation of the function f by x defined by
If f is a Schwartz function, then τ_{x}f is the convolution with a translated Dirac delta function τ_{x}f = f ∗ τ_{x}δ. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.
Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds
Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of timeinvariant systems, and especially LTI system theory. The representing function g_{S} is the impulse response of the transformation S.
A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a continuous linear operator with respect to the appropriate topology. It is known, for instance, that every continuous translation invariant continuous linear operator on L^{1} is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear operator on L^{p} for 1 ≤ p < ∞ is the convolution with a tempered distribution whose Fourier transform is bounded. To wit, they are all given by bounded Fourier multipliers.
If G is a suitable group endowed with a measure λ, and if f and g are real or complex valued integrable functions on G, then we can define their convolution by
It is not commutative in general. In typical cases of interest G is a locally compact Hausdorff topological group and λ is a (left) Haar measure. In that case, unless G is unimodular, the convolution defined in this way is not the same as . The preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:
Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former.
On locally compact abelian groups, a version of the convolution theorem holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The circle group T with the Lebesgue measure is an immediate example. For a fixed g in L^{1}(T), we have the following familiar operator acting on the Hilbert space L^{2}(T):
The operator T is compact. A direct calculation shows that its adjoint T* is convolution with
By the commutativity property cited above, T is normal: T* T = TT* . Also, T commutes with the translation operators. Consider the family S of operators consisting of all such convolutions and the translation operators. Then S is a commuting family of normal operators. According to spectral theory, there exists an orthonormal basis {h_{k}} that simultaneously diagonalizes S. This characterizes convolutions on the circle. Specifically, we have
which are precisely the characters of T. Each convolution is a compact multiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above.
A discrete example is a finite cyclic group of order n. Convolution operators are here represented by circulant matrices, and can be diagonalized by the discrete Fourier transform.
A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finitedimensional unitary representations form an orthonormal basis in L^{2} by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.
Let G be a (multiplicatively written) topological group. If μ and ν are finite Borel measures on G, then their convolution μ∗ν is defined as the pushforward measure of the group action and can be written as
for each measurable subset E of G. The convolution is also a finite measure, whose total variation satisfies
In the case when G is locally compact with (left)Haar measure λ, and μ and ν are absolutely continuous with respect to a λ, so that each has a density function, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions.
If μ and ν are probability measures on the topological group (R,+), then the convolution μ∗ν is the probability distribution of the sum X + Y of two independent random variables X and Y whose respective distributions are μ and ν.
Let (X, Δ, ∇, ε, η) be a bialgebra with comultiplication Δ, multiplication ∇, unit η, and counit ε. The convolution is a product defined on the endomorphism algebra End(X) as follows. Let φ, ψ ∈ End(X), that is, φ,ψ : X → X are functions that respect all algebraic structure of X, then the convolution φ∗ψ is defined as the composition
The convolution appears notably in the definition of Hopf algebras ( Kassel 1995 , §III.3). A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that
Convolution and related operations are found in many applications in science, engineering and mathematics.
Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.
In probability theory, the normaldistribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent realvalued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.
The Fourier transform (FT) decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform of a function of time is itself a complexvalued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest, one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time. In image processing the notion of a time domain is replaced by that of a spatial domain where the intensity of a signal is identified by its spatial position rather than at any point in time.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, convolution in one domain equals pointwise multiplication in the other domain. Versions of the convolution theorem are true for various Fourierrelated transforms. Let and be two functions with convolution .
The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum.
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For realvalued functions on the real line two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as , the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function :
Linear timeinvariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and timeinvariant system to an arbitrary input signal. Trajectories of these systems are commonly measured and tracked as they move through time, but in applications like image processing and field theory, the LTI systems also have trajectories in spatial dimensions. Thus, these systems are also called linear translationinvariant to give the theory the most general reach. In the case of generic discretetime systems, linear shiftinvariant is the corresponding term. A good example of LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.
In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The AmbroseKakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary S^{1} = G / AN and G / A = S^{1} × S^{1} \ diag S^{1}. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III_{0} is an ergodic flow on a measure space.
In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a widesensestationary random process has a spectral decomposition given by the power spectrum of that process.
The circular convolution, also known as cyclic convolution, of two aperiodic functions occurs when one of them is convolved in the normal way with a periodic summation of the other function. That situation arises in the context of the circular convolution theorem. The identical operation can also be expressed in terms of the periodic summations of both functions, if the infinite integration interval is reduced to just one period. That situation arises in the context of the discretetime Fourier transform (DTFT) and is also called periodic convolution. In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences.
The Wigner distribution function (WDF) is used in signal processing as a transform in timefrequency analysis.
A Modified Wigner distribution function is a variation of the Wigner distribution function (WD) with reduced or removed crossterms.
Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a subfield of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time; this subfield used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signalprocessing problems.
In calculus, interchange of the order of integration is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.
In mathematics, lifting theory was first introduced by John von Neumann in his (1931) pioneering paper, followed later by Dorothy Maharam’s (1958) paper, and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper. Lifting theory was motivated to a large extent by its striking applications; for its development up to 1969, see the Ionescu Tulceas' work and the monograph, now a standard reference in the field. Lifting theory continued to develop after 1969, yielding significant new results and applications.
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