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In mathematics, an integral transform is any transform T of the following form: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
This is a list of transforms in mathematics, by Wikipedia page. ...
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. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
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 K − 1(u,t) which (roughly speaking) yields an inverse transform:
A symmetric kernel is one that is unchanged when the two variables are permuted.
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" (e.g., functions where time is the independent variable are said to be in the time domain) into another domain. Manipulating and solving the equation in the target domain is, ideally, 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. Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ...
Integral transforms work because they are based upon the concept of spectral factorization over orthonormal bases. What this means is that, other than a few, quite artificial exceptions, arbitrarily complicated functions can be represented as sums of much simpler functions. In mathematics, an orthonormal basis of an inner product space V(i. ...
History The precursor of the transforms were the Fourier series to express functions in finite intervals. Later the Fourier transform was developed to remove the requirement of finite intervals. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...
Using the Fourier series, just about any practical function of time (the voltage across the terminals of an electronic device, perhaps) can be represented as a sum of sines and cosines, each suitably scaled (multiplied by a constant factor) and shifted (advanced or retarded in time). The sines and cosines in the Fourier series are an example of an orthonormal basis. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...
Electronics is the study and use of electrical devices that operate by controlling the flow of electrons or other electrically charged particles in devices such as thermionic valves and semiconductors. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
==Importance of orthogonality== ha
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 target domain 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. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
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 Kronecker 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. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
Usage example As an example of an application of integral transforms, consider the Laplace transform. This is a technique that maps differential or integro-differential equations 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. In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t ≥ 0) that transforms it to a function F(s) (image) with a complex argument s. ...
A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...
An integro-differential equation is an equation which has both integrals and derivatives of an unknown function. ...
Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ...
Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...
In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...
The Laplace transform finds wide application in physics and particularly in electrical engineering, where the characteristic equations 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 linear algebra, the characteristic equation of a square matrix A is the equation in one variable λ where I is the identity matrix. ...
Table of transforms 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. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...
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. ...
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the 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. ...
In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t ≥ 0) that transforms it to a function F(s) (image) with a complex argument s. ...
In mathematics, the Hankel transform of order ν of a function f(r) is given by: where Jν is the Bessel function of the first kind of order ν with ν ≥ −1/2. ...
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. ...
The Hilbert transform, in red, of a square wave, in blue In mathematics and in signal processing, the Hilbert transform, here denoted , of a real-valued function, , is obtained by convolving signal with to obtain . ...
The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...
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 function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem). In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, generalized functions are objects generalizing the notion of functions. ...
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. ...
The general theory of such integral equations 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. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ...
In mathematics, Fredholm theory is a theory of integral equations. ...
In functional analysis, a compact operator (or completely continuous 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. Such an operator is necessarily a...
In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. ...
In mathematics, a nuclear operator or a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. ...
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. ...
See also In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...
This is a list of transforms in mathematics, by Wikipedia page. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
This is a list of linear transformations of functions related to the Fourier transform. ...
In machine learning, the kernel trick is a method for converting a linear classifier algorithm into a non-linear one by using a non-linear function to map the original observations into a higher-dimensional space; this makes a linear classification in the new space equivalent to non-linear classification...
Kernel Methods (KMs) are a class of algorithms for pattern analysis, whose best known element is the Support Vector Machine (SVM). ...
In mathematics, in the area of complex analysis, Nachbins theorem is commonly used to establish a bound on the growth rates for an analytic function. ...
References - A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
- Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
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