Category: Integral transforms

Y and H transforms
In mathematics, the Y transforms and H transforms are complementary pairs of integral transforms involving, respectively, the Neumann function (Bessel function of the second kind) Yν of order ν and th
Stieltjes transformation
In mathematics, the Stieltjes transformation Sρ(z) of a measure of density ρ on a real interval I is the function of the complex variable z defined outside I by the formula Under certain conditions we
In mathematics, the K transform (also called the Kemp-Macdonald Transform or Single-Pixel X-ray Transform) is an integral transform introduced by and in 2016. The transform allows a 3-dimensional inho
Continuous wavelet transform
In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of
Hermite transform
In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials as kernels of the transform. This was first introduced by Loken
S transform
S transform as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the S transform is a generalization of the short-time Fourier transform (STFT), extending
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly
Projection-slice theorem
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * Take a two-dimens
Riesz transform
In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension d > 1. They are a type of singular integ
Funk transform
In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integ
Weierstrass transform
In mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centere
Hankel transform
In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all o
Motions in the time-frequency distribution
Several techniques can be used to move signals in the time-frequency distribution. Similar to computer graphic techniques, signals can be subjected to horizontal shifting, vertical shifting, dilation
X-ray transform
In mathematics, the X-ray transform (also called John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very close
Fourier analysis
In mathematics, Fourier analysis (/ˈfʊrieɪ, -iər/) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from t
Constant-Q transform
In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier tra
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 a
Hartley transform
In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed
Laplace transform applied to differential equations
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear d
Fractional Fourier transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier
Quadratic Fourier transform
In mathematical physics and harmonic analysis, the quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transfo
Legendre transform (integral transform)
In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials as kernels of the transform. Legendre transform is a sp
In mathematics and statistical analysis, bicoherence (also known as bispectral coherency) is a squared normalised version of the bispectrum. The bicoherence takes values bounded between 0 and 1, which
Riemann–Liouville integral
In mathematics, the Riemann–Liouville integral associates with a real function another function Iα f of the same kind for each value of the parameter α > 0. The integral is a manner of generalization
List of Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinuso
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
Nørlund–Rice integral
In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory o
Sine and cosine transforms
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fo
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually , in the time domain)
Pricing kernel
No description available.
In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.
Post's inversion formula
No description available.
Kontorovich–Lebedev transform
In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike othe
Laplace–Stieltjes transform
The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace tr
Mellin inversion theorem
In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defi
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
Integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more
Nachbin's theorem
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provi
Hilbert transform
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). Th
Gabor transform
The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal a
Inverse Laplace transform
In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property: where denotes the Laplace transform
Triple correlation
The triple correlation of an ordinary function on the real line is the integral of the product of that function with two independently shifted copies of itself: The Fourier transform of triple correla
Oscillatory integral operator
In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form where the function S(x,y) is called the phase of the operator and the function a
Fourier operator
The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional function when it corresponds to the Fourier transf
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. The Abel transform of a fu
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 tr
Laguerre transform
In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials as kernels of the transform. The Laguerre transfo
Multidimensional transform
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.
Perron's formula
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transfo
Gabor–Wigner transform
The Gabor transform, named after Dennis Gabor, and the Wigner distribution function, named after Eugene Wigner, are both tools for time-frequency analysis. Since the Gabor transform does not have high
Linear canonical transformation
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-di
Jacobi transform
In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials as kernels of the transform . The Jacobi transform of a
Mehler–Fock transform
In mathematics, the Mehler–Fock transform is an integral transform introduced by Mehler and rediscovered by Fock. It is given by where P is a Legendre function of the first kind. Under appropriate con
Laplace–Carson transform
In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications in the field of physics and engineering,