Integral transform
Encyclopedia
In mathematics
, an integral transform is any transform T of the following form:
The input of this transform is a function
f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator.
There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables
, the kernel function or nucleus of the transform.
Some kernels have an associated inverse kernel
which (roughly speaking) yields an inverse transform:
A symmetric kernel is one that is unchanged when the two variables are permuted
.
to express functions in finite intervals. Later the Fourier transform
was developed to remove the requirement of finite intervals.
Using the Fourier series, just about any practical function of time (the voltage
across the terminals of an electronic device for example) can be represented as a sum of sine
s and cosines, each suitably scaled (multiplied by a constant factor), shifted (advanced or retarded in time) and "squeezed" or "stretched" (increasing or decreasing the frequency). The sines and cosines in the Fourier series are an example of an orthonormal basis.
The normal Cartesian graph per se of a function can be thought of as an orthonormal expansion. Indeed, each point just reflects the contribution of a given orthonormal basis function to the sum. Intuitively, the point (3,5) on the graph means that the orthonormal basis function δ(x-3), where "δ" is the Dirac delta function
, is scaled up by a factor of five to contribute to the sum in this form. In this way, the graph of a continuous real
-valued function in the plane corresponds to an infinite set of basis functions; if the number of basis functions were finite, the curve would consist of a discrete set of points rather than a continuous contour.
or integro-differential equation
s in the "time" domain
into polynomial equations in what is termed the "complex frequency" domain
. (Complex frequency is similar to actual, physical frequency but rather more general. Specifically, the imaginary component ω of the complex frequency s = -σ + iω corresponds to the usual concept of frequency, viz., the speed at which a sinusoid cycles, whereas the real component σ of the complex frequency corresponds to the degree of "damping". ) The equation cast in terms of complex frequency is readily solved in the complex frequency domain (roots of the polynomial equations in the complex frequency domain correspond to eigenvalues in the time domain), leading to a "solution" formulated in the frequency domain. Employing the inverse transform, i.e., the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator) correspond to power series in the time domain, while axial shifts in the complex frequency domain correspond to damping by decaying exponentials in the time domain.
The Laplace transform finds wide application in physics and particularly in electrical engineering, where the characteristic equation
s that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially damped, scaled, and time-shifted sinusoids in the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines.
In the limits of integration for the inverse transform, c is a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform, c must be greater than the largest real part of the zeroes of the transform function.
then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem
).
The general theory of such integral equation
s is known as Fredholm theory
. In this theory, the kernel is understood to be a compact operator
acting on a Banach space
of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm operator
, the nuclear operator
or the Fredholm kernel
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an integral transform is any transform T of the following form:
The input of this transform is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator.
There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
, the kernel function or nucleus of the transform.
Some kernels have an associated inverse kernel
which (roughly speaking) yields an inverse transform:A symmetric kernel is one that is unchanged when the two variables are permuted
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
.
Motivation
Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. An integral transform "maps" an equation from its original "domain" into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.History
The precursor of the transforms were the Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
to express functions in finite intervals. Later the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
was developed to remove the requirement of finite intervals.
Using the Fourier series, just about any practical function of time (the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
across the terminals of an electronic device for example) can be represented as a sum of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s and cosines, each suitably scaled (multiplied by a constant factor), shifted (advanced or retarded in time) and "squeezed" or "stretched" (increasing or decreasing the frequency). The sines and cosines in the Fourier series are an example of an orthonormal basis.
Importance of orthogonality
The individual basis functions have to be orthogonal. That is, the product of two dissimilar basis functions—integrated over their domain—must be zero. An integral transform, in actuality, just changes the representation of a function from one orthogonal basis to another. Each point in the representation of the transformed function in the image of the transform corresponds to the contribution of a given orthogonal basis function to the expansion. The process of expanding a function from its "standard" representation to a sum of a number of orthonormal basis functions, suitably scaled and shifted, is termed "spectral factorization." This is similar in concept to the description of a point in space in terms of three discrete components, namely, its x, y, and z coordinates. Each axis correlates only to itself and nothing to the other orthogonal axes. Note the terminological consistency: the determination of the amount by which an individual orthonormal basis function must be scaled in the spectral factorization of a function, F, is termed the "projection" of F onto that basis function.The normal Cartesian graph per se of a function can be thought of as an orthonormal expansion. Indeed, each point just reflects the contribution of a given orthonormal basis function to the sum. Intuitively, the point (3,5) on the graph means that the orthonormal basis function δ(x-3), where "δ" is the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
, is scaled up by a factor of five to contribute to the sum in this form. In this way, the graph of a continuous real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function in the plane corresponds to an infinite set of basis functions; if the number of basis functions were finite, the curve would consist of a discrete set of points rather than a continuous contour.
Usage example
As an example of an application of integral transforms, consider the Laplace transform. This is a technique that maps differentialDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
or integro-differential equation
Integro-differential equation
An integro-differential equation is an equation which involves both integrals and derivatives of a function.The general first-order, linear integro-differential equation is of the form...
s in the "time" domain
Time domain
Time domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...
into polynomial equations in what is termed the "complex frequency" domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
. (Complex frequency is similar to actual, physical frequency but rather more general. Specifically, the imaginary component ω of the complex frequency s = -σ + iω corresponds to the usual concept of frequency, viz., the speed at which a sinusoid cycles, whereas the real component σ of the complex frequency corresponds to the degree of "damping". ) The equation cast in terms of complex frequency is readily solved in the complex frequency domain (roots of the polynomial equations in the complex frequency domain correspond to eigenvalues in the time domain), leading to a "solution" formulated in the frequency domain. Employing the inverse transform, i.e., the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator) correspond to power series in the time domain, while axial shifts in the complex frequency domain correspond to damping by decaying exponentials in the time domain.
The Laplace transform finds wide application in physics and particularly in electrical engineering, where the characteristic equation
Characteristic equation
Characteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...
s that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially damped, scaled, and time-shifted sinusoids in the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines.
Table of transforms
| Transform | Symbol |  |
t1 | t2 |  |
u1 | u2 |
|---|---|---|---|---|---|---|---|
| Fourier transform Fourier transform In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions... |
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| Fourier sine transform |  |
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| Fourier cosine transform |  |
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| Hartley transform Hartley transform In mathematics, the Hartley transform is an integral transform closely related to the Fourier transform, but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by R. V. L. Hartley in 1942, and is one of many known... |
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| Mellin transform Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform... |
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| Two-sided Laplace transform Two-sided Laplace transform In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform... |
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| Laplace transform |  |
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| Weierstrass transform Weierstrass transform In mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is the function F defined by... |
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| Hankel transform Hankel transform In mathematics, the Hankel transform expresses any given function f as the weighted sum of an infinite number of Bessel functions of the first kind Jν. The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis... |
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| Abel transform Abel transform In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions... |
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| Hilbert transform Hilbert transform In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the... |
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| Poisson kernel Poisson kernel In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation... |
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| Identity transform Dirac delta function The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical... |
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In the limits of integration for the inverse transform, c is a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform, c must be greater than the largest real part of the zeroes of the transform function.
Different domains
Here integral transforms are defined for functions on the real numbers, but they can be defined more generally for functions on a group.- If instead one uses functions on the circle (periodic functions), integration kernels are then biperiodic functions; convolution by functions on the circle yields circular convolutionCircular convolutionThe 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...
. - If one uses functions on the cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order n (
or 
), one obtains n × n matrices as integration kernels; convolution corresponds to circulant matrices.
General theory
Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized functionGeneralized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem
Schwartz kernel theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz himself have a two-variable theory that includes all reasonable...
).
The general theory of such integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s is known as Fredholm theory
Fredholm theory
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm...
. In this theory, the kernel is understood to be a compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
acting on a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
, the nuclear operator
Nuclear operator
In mathematics, a nuclear operator is roughly a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis .Nuclear operators are essentially the same as trace class operators, though most authors reserve the term "trace...
or the Fredholm kernel
Fredholm kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory....
.
See also
- Reproducing kernel
- Convolution kernel
- Circular convolutionCircular convolutionThe 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...
- Circulant matrixCirculant matrixIn linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence...
- List of transforms
- List of operators
- List of Fourier-related transforms
- Kernel trickKernel trickFor machine learning algorithms, the kernel trick is a way of mapping observations from a general set S into an inner product space V , without ever having to compute the mapping explicitly, in the hope that the observations will gain meaningful linear structure in V...
- Kernel methodsKernel methodsIn computer science, kernel methods are a class of algorithms for pattern analysis, whose best known elementis the support vector machine...
- Nachbin's theoremNachbin's theoremIn mathematics, in the area of complex analysis, Nachbin's theorem is commonly used to establish a bound on the growth rates for an analytic function. This article will provide a brief review of growth rates, including the idea of a function of exponential type...
- Bateman transformBateman transformIn the mathematical study of partial differential equation, the Bateman transform is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral of a holomorphic function in three complex variables...
- Symbolic integrationSymbolic integrationIn calculus symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f, i.e...