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List Of Transforms
This is a list of transforms in mathematics. Integral transforms *Abel transform *Bateman transform *Fourier transform **Short-time Fourier transform **Gabor transform *Hankel transform *Hartley transform *Hermite transform *Hilbert transform **Hilbert–Schmidt integral operator *Jacobi transform *Laguerre transform *Laplace transform **Inverse Laplace transform **Two-sided Laplace transform ** Inverse two-sided Laplace transform *Laplace–Carson transform *Laplace–Stieltjes transform *Legendre transform *Linear canonical transform *Mellin transform **Inverse Mellin transform ** Poisson–Mellin–Newton cycle *N-transform *Radon transform *Stieltjes transformation * Sumudu transform *Wavelet transform (integral) *Weierstrass transform Discrete transforms *Binomial transform *Discrete Fourier transform, DFT **Fast Fourier transform, a popular implementation of the DFT * Discrete cosine transform **Modified discrete cosine transform *Discrete Hartley transform *Discrete sine tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transform (mathematics)
In mathematics, a transformation is a function ''f'', usually with some geometrical underpinning, that maps a set ''X'' to itself, i.e. . Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations. Partial transformations While it is common to use the term transformation for any function of a set into itself (especially in terms like "transformation semigroup" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to partial functions, then a partial transformation is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' are subsets of some set ''X''. Algebraic structures The set of all transformations on a given base set, together with function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-sided Laplace Transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If ''f''(''t'') is a real- or complex-valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral :\mathcal\(s) = F(s) = \int_^\infty e^ f(t)\, dt. The integral is most commonly understood as an improper integral, which converges if and only if both integrals :\int_0^\infty e^ f(t) \, dt,\quad \int_^0 e^ f(t)\, dt exist. There seems to be no generally accepted notation for the two-sided transform; the \mathcal used here recalls "bilateral". The two-sided transform used by some authors is :\mathcal\(s) = s\mathcal\(s) = sF(s) = s \int_^\infty e^ f(t)\, dt. In pure mathematics the argum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Transform
In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''. Specifically, it is the function defined by :F(x)=\frac\int_^\infty f(y) \; e^ \; dy = \frac\int_^\infty f(x-y) \; e^ \; dy~, the convolution of with the Gaussian function :\frac e^~. The factor 1/√(4 π) is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform. Instead of one also writes . Note that need not exist for every real number , when the defining integral fails to converge. The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet Transform
In mathematics, a wavelet series is a representation of a square-integrable (real number, real- or complex number, complex-valued) function (mathematics), function by a certain orthonormal series (mathematics), series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. Definition A function \psi \,\in\, L^2(\mathbb) is called an orthonormal wavelet if it can be used to define a Hilbert space#Orthonormal bases, Hilbert basis, that is a orthonormal basis, complete orthonormal system, for the Hilbert space L^2\left(\mathbb\right) of Square-integrable function, square integrable functions. The Hilbert basis is constructed as the family of functions \ by means of Dyadic transformation, dyadic translation (geometry), translations and dilation (operator theory), dilations of \psi\,, :\psi_(x) = 2^\frac \psi\left(2^jx - k\right)\, for integers j,\, k \,\in\, \mathbb. If under the standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stieltjes Transformation
In mathematics, the Stieltjes transformation of a measure of density on a real interval is the function of the complex variable defined outside by the formula S_(z)=\int_I\frac, \qquad z \in \mathbb \setminus I. Under certain conditions we can reconstitute the density function starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density is continuous throughout , one will have inside this interval \rho(x)=\lim_ \frac. Connections with moments of measures If the measure of density has moments of any order defined for each integer by the equality m_=\int_I t^n\,\rho(t)\,dt, then the Stieltjes transformation of admits for each integer the asymptotic expansion in the neighbourhood of infinity given by S_(z)=\sum_^\frac+o\left(\frac\right). Under certain conditions the complete expansion as a Laurent series can be obtained: S_(z) = \sum_^\frac. Relationships to orthogonal polynomials The correspondence (f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon Transform
In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform). It was later generalized to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object. Explanation If a function f represents an unknown density, then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-transform
In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan in 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence to the Laplace and Sumudu transforms, the N-transform inherits all the applied aspects of the both transforms. Most recently, F. B. M. Belgacem has renamed it the natural transform and has proposed a detail theory and applications. Formal definition The natural transform of a function ''f''(''t''), defined for all real numbers ''t'' ≥ ''0'', is the function ''R''(''u'', ''s''), defined by: : R(u, s) = \mathcal\ = \int_0^\infty f(ut)e^\,dt.\qquad(1) Khan showed that the above integral converges to Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a function is :\left\(s) = \varphi(s)=\int_0^\infty x^ f(x) \, dx. The inverse transform is :\left\(x) = f(x)=\frac \int_^ x^ \varphi(s)\, ds. The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part ''c'' need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Canonical Transform
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Transform (integral Transform)
In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials P_n(x) as kernels of the transform. Legendre transform is a special case of Jacobi transform In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a cla .... The Legendre transform of a function f(x) isChurchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100. :\mathcal_n\ = \tilde f(n) = \int_^1 P_n(x)\ f(x) \ dx The inverse Legendre transform is given by :\mathcal_n^\ = f(x) = \sum_^\infty \frac \tilde f(n) P_n(x) Associated Legendre transform Associated Legendre transform is defined as :\mathcal_\ = \tilde f(n,m) = \int_^1 (1-x^2)^P_n^m(x) \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
