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Mathematical concept
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given
Poisson_kernel
French mathematician and physicist (1781–1840)
they led to the discovery of the Poisson kernel. Thanks to the works of Dirichlet and Hermann Schwarz, the Poisson kernel is now typically presented in the
Siméon_Denis_Poisson
Generalized function whose value is zero everywhere except at zero
the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. This semigroup
Dirac_delta_function
Topics referred to by the same term
probability Poisson summation formula in Fourier analysis Poisson kernel in complex or harmonic analysis Poisson–Jensen formula in complex analysis This disambiguation
Poisson_formula
Types of wavelets
are connected with the derivatives of the Poisson integral kernel. For each positive integer n the Poisson wavelet ψ n ( t ) {\displaystyle \psi _{n}(t)}
Poisson_wavelet
Notion of boundary associated with a group
boundary circle of the hyperbolic plane, and the Poisson-like integral is the usual Poisson kernel for the upper half-plane. Let G {\displaystyle G}
Furstenberg_boundary
Mapping involving integration between function spaces
two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated inverse kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u
Integral_transform
Integral transform and linear operator
{y}{(x-s)^{2}+y^{2}}}\;\mathrm {d} s} which is the convolution of f with the Poisson kernel P ( x , y ) = y π ( x 2 + y 2 ) {\displaystyle P(x,y)={\frac {y}{\pi
Hilbert_transform
Type of random mathematical object
statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of
Poisson_point_process
solve Poisson's differential equation Poisson differential operator Dirichlet–Poisson problem Discrete Poisson equation Poisson kernel Poisson integral
List of things named after Siméon Denis Poisson
List_of_things_named_after_Siméon_Denis_Poisson
Probability distribution
moment generating function. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper
Cauchy_distribution
Mathematical measure space associated to a random walk
{\displaystyle K(z,\xi )={\frac {1-|z|^{2}}{|\xi -z|^{2}}}} is the Poisson kernel, holds for all z ∈ D {\displaystyle z\in \mathbb {D} } . One way to
Poisson_boundary
Number, approximately 3.14
theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane. Conjugate harmonic
Pi
proof analyzes the representation of harmonic functions provided by the Poisson kernel, applied to an interior tangent sphere. In modern presentations, Kellogg's
Kellogg's_theorem
Problem of solving a partial differential equation subject to prescribed boundary values
and the solution to the problem (at least for the ball) using the Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the Prussian
Dirichlet_problem
Mathematical formula in complex analysis
}r^{|n|}e^{in\omega }} is the Poisson kernel on the unit disk. If the function f {\displaystyle f} has no zeros in the unit disk, the Poisson-Jensen formula reduces
Jensen's_formula
Second-order partial differential equation
problem with continuous boundary data f {\displaystyle f} is given by the Poisson kernel formula u ( r e i θ ) = 1 2 π ∫ 0 2 π 1 − r 2 1 − 2 r cos ( θ − φ
Laplace's_equation
Type of singular integral operator
well-defined on tempered distributions modulo polynomials. Hilbert Transform Poisson kernel Riesz potential Strictly speaking, the definition (1) may only make
Riesz_transform
Family of functions
|t|>\delta } . The Fejér kernel The Poisson kernel (continuous index) The Landau kernel The Dirichlet kernel is not a summability kernel, since it fails the
Summability_kernel
d H 1 {\displaystyle d\omega (X,\mathbb {D} )/dH^{1}} is called the Poisson kernel. More generally, if n ≥ 2 {\displaystyle n\geq 2} and B n = { X ∈ R
Harmonic_measure
Concept within complex analysis
can regain a (harmonic) function f on the unit disk by means of the Poisson kernel Pr: f ( r e i θ ) = 1 2 π ∫ 0 2 π P r ( θ − ϕ ) f ~ ( e i ϕ ) d ϕ ,
Hardy_space
Theoretical framework in harmonic analysis
{\displaystyle u(x,y)=\int _{\mathbb {R} ^{n}}P_{y}(t)f(x-t)\,dt} where the Poisson kernel P on the upper half space { ( y ; x ) ∈ R n + 1 ∣ y > 0 } {\displaystyle
Littlewood–Paley_theory
Rudin, Real and Complex Analysis, p. 335. The proof given uses the Poisson kernel and the existence of boundary values for the Hardy space H1. Expansions
F._and_M._Riesz_theorem
of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic
Szegő_kernel
Type of vector space in math
mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions
Hilbert_space
Class of mathematical functions
)(e^{i\theta })=\sup _{0\leq r<1}\varphi (re^{i\theta }).} If Pr denotes the Poisson kernel, it follows from the subharmonicity that 0 ≤ φ ( r e i θ ) ≤ 1 2 π ∫
Subharmonic_function
Mathematical transformation
{\varepsilon /\pi }{(t-t_{0})^{2}+\varepsilon ^{2}}}} is also known as the Poisson kernel (for the half-plane). The denominator ( t − t 0 ) 2 + ε 2 {\displaystyle
Stieltjes_transformation
Polynomial sequence
potential. Similar expressions are available for the expansion of the Poisson kernel in a ball. It follows that the quantities C k ( ( n − 2 ) / 2 ) ( x
Gegenbauer_polynomials
Area of mathematical analysis
and of the Poisson kernel, is often more delicate than Fourier methods alone can resolve. Thus questions about the convergence of Poisson integrals, the
Harmonic_analysis
Theorem in complex analysis
H^{p}(\mathbb {D} )} is the Hardy space. The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle. The analogous
Fatou's_theorem
Mathematical concept
Lebesgue point of f. In fact the operator T1 − εHf has kernel Qr + i, where the conjugate Poisson kernel Qr is defined by Q r ( θ ) = 2 r sin θ 1 − 2 r cos
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
}^{1}(1-t^{2})^{n}\,dt\leq (n+1)(1-\delta ^{2})^{n}} Poisson kernel Fejér kernel Dirichlet kernel Terras, Audrey (May 25, 2009). "Lecture 8. Dirac and
Landau_kernel
Differential equation for the description of waves or standing wave
{\omega }}.} The integral can be solved by analytically continuing the Poisson kernel, giving G ( t , x ) = lim ϵ → 0 + C D D − 1 Im [ ‖ x ‖ 2 − ( t − i
Wave_equation
Stochastic process generalizing Brownian motion
x-y\rVert ^{d}}},\quad x\in B_{1}(0),\,y\in \mathbb {S} _{d-1},} is the Poisson kernel for the unit ball. Let D ⊂ R d {\displaystyle D\subset \mathbb {R} ^{d}}
Wiener_process
Special mathematical functions defined on the surface of a sphere
The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector
Spherical_harmonics
Probabilistic concept
context, the function q {\displaystyle q} happens to be the well-known Poisson kernel, which has the explicit formula q ( x ∣ y ) = 1 σ ( ∂ B ) 1 − ‖ y ‖
Doob's_h-transform
f_{out}(x+1)} where β > 0 {\displaystyle \beta >0} , the one-sided Poisson kernel p ( n , t ) = e − t t n n ! {\displaystyle p(n,t)=e^{-t}{\frac {t^{n}}{n
Multi-scale_approaches
x k ) {\displaystyle K(x,x_{k})} is the Radial basis function kernel (Gaussian kernel) as formulated below. K ( x , x k ) = e − d k / 2 σ 2 , d k = (
General regression neural network
General_regression_neural_network
Equation in Fourier analysis
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values
Poisson_summation_formula
d\mu (\theta ).} This follows from the previous theorem because: the Poisson kernel is the real part of the integrand above the real part of a holomorphic
Positive_harmonic_function
^{n-1}} . The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors, 1 ω n − 1 1 − r 2
Zonal_spherical_harmonics
Mathematical function
derive the following interesting[clarification needed] identity from the Poisson summation formula: ∑ k ∈ Z exp ( − π ⋅ ( k c ) 2 ) = c ⋅ ∑ k ∈ Z exp
Gaussian_function
important role in harmonic analysis on the boundary, in the theory of Poisson kernel, and in the study of invariants such as the Maslov index. A bounded
Shilov_boundary
Vector field defined for any energy function
Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding
Hamiltonian_vector_field
Green's function for Laplacian
function having a mathematical singularity at the origin, the Newtonian kernel Γ {\displaystyle \Gamma } which is the fundamental solution of the Laplace
Newtonian_potential
Family of three random counting measures
Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning
Poisson-type_random_measure
Random set of points on a space with random number and random position
\lambda (y)=\sum _{X\in \Phi }h(X,y)} for a Poisson point process Φ ( ⋅ ) {\displaystyle \Phi (\cdot )} and kernel h ( ⋅ , ⋅ ) {\displaystyle h(\cdot ,\cdot
Point_process
Wrapped probability distribution
(See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.) The circular Cauchy distribution expressed in
Wrapped_Cauchy_distribution
|^{2}}\right),} where the term in brackets on the right hand side is the Poisson kernel for the unit disk and ζ {\displaystyle \zeta } corresponds to the radial
Busemann_function
} and points z , w {\displaystyle z,w} in U {\displaystyle U} . Poisson Poisson kernel power series A power series is informally a polynomial of infinite
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Integral expressing the amount of overlap of one function as it is shifted over another
individual distributions. In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian
Convolution
corresponding Poisson kernel. For a fixed a in G, the Dirichlet problem with boundary value log |z − a| can be solved using the Poisson kernels. It yields
Planar_Riemann_surface
Mexican mathematician
J; Guzmán–Partida, Martha; Skórnik, U. "S'–convolvability with the Poisson kernel in the Euclidean case and the product domain case." Studia Mathematica
Martha_Guzmán_Partida
Metric on a smooth statistical manifold
Mitsuhiro; Shishido, Yuichi (2008). "Fisher information metric and Poisson kernels" (PDF). Differential Geometry and Its Applications. 26 (4): 347–356
Fisher_information_metric
Infinite series summing alternating 1 and -1 terms
Abel sums of this series involve limits of the Dirichlet, Fejér, and Poisson kernels, respectively. Multiplying the terms of Grandi's series by 1/nz yields
Grandi's_series
Stochastic way of assigning quantities across a space
important point processes such as Poisson point processes and Cox processes. Random measures can be defined as transition kernels or as random elements. Both
Random_measure
discrete Fourier series Gibbs phenomenon Sigma approximation Dini test Poisson summation formula Spectrum continuation analysis Convergence of Fourier
List of Fourier analysis topics
List_of_Fourier_analysis_topics
Dobiński's formula represents the n {\displaystyle n} th moment of the Poisson distribution with mean 1. Sometimes Dobiński's formula is stated as saying
Dobiński's_formula
Function used in signal processing
of Bayesian analysis and curve fitting, this is often referred to as the kernel. When analyzing a transient signal in modal analysis, such as an impulse
Window_function
excitatory weight J_in = -0.5 # inhibitory weight p_rate = 20000.0 # external Poisson rate neuron_params= {"C_m": 1.0, "tau_m": 20.0, "t_ref": 2.0, "E_L": 0
NEST_(software)
The linear-nonlinear-Poisson (LNP) cascade model is a simplified functional model of neural spike responses. It has been successfully used to describe
Linear-nonlinear-Poisson cascade model
Linear-nonlinear-Poisson_cascade_model
Statistical Markov model
determination Confusion matrix Learning curve ROC curve Mathematical foundations Kernel machines Bias–variance tradeoff Computational learning theory Empirical
Hidden_Markov_model
process Poisson binomial distribution Poisson distribution Poisson hidden Markov model Poisson limit theorem Poisson process Poisson regression Poisson random
List_of_statistics_articles
Trigonometric function Trigonometric polynomial Exponential sum Dirichlet kernel Fejér kernel Gibbs phenomenon Parseval's identity Parseval's theorem Weyl differintegral
List of harmonic analysis topics
List_of_harmonic_analysis_topics
Smoothing filler for images
simple trick to efficiently implement a bilateral filter is to exploit Poisson-disk subsampling. OpenCV implements the function: bilateralFilter( source
Bilateral_filter
Mathematical theory of integral equations
given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. It therefore forms a branch of operator theory and functional
Fredholm_theory
Method of plotting numeric data
a box plot, but has enhanced information with the addition of a rotated kernel density plot on each side. The violin plot was proposed in 1997 by Jerry
Violin_plot
Estimate of an unobservable underlying probability density function
distribution Kernel density estimation Mean integrated squared error Histogram Multivariate kernel density estimation Spectral density estimation Kernel embedding
Density_estimation
Type of mathematical function
said to be a radial kernel centered at c ∈ V {\textstyle \mathbf {c} \in V} . A radial function and the associated radial kernels are said to be radial
Radial_basis_function
Provides integral formulas for all derivatives of a holomorphic function
uniformly. The analog of the Cauchy integral formula in real analysis is the Poisson integral formula for harmonic functions; many of the results for holomorphic
Cauchy's_integral_formula
Statistical method
sampling from a kernel density estimate of the data. Assume K to be a symmetric kernel density function with unit variance. The standard kernel estimator f
Bootstrapping_(statistics)
Partial differential equations
particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form ∇ 2 u ( x
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
Graphical representation of the distribution of numerical data
_{i=1}^{k}{m_{i}}.} A histogram can be thought of as a simplistic kernel density estimation, which uses a kernel to smooth frequencies over the bins. This yields a smoother
Histogram
Category of regression analysis
Bayes. The hyperparameters typically specify a prior covariance kernel. In case the kernel should also be inferred nonparametrically from the data, the critical
Nonparametric_regression
Examples of the probabilistic construct
Unlike the standard Poisson process, the rate of popping changes depending on the current state. If i {\displaystyle i} kernels have already popped,
Examples_of_Markov_chains
Concept in regression analysis mathematics
notation, the i , j {\displaystyle i,j} entry of kernel matrix K {\displaystyle K} (as opposed to kernel function K ( ⋅ , ⋅ ) {\displaystyle K(\cdot ,\cdot
Regularized_least_squares
Supergeometric generalization of a manifold
are either +1 or −1.) Given an odd symplectic 2-form ω one may define a Poisson bracket known as the antibracket of any two functions F and G on a supermanifold
Supermanifold
Construction analogous to that of a dual vector space
mathematics more broadly. For instance, it is used in the statement of the Poisson summation formula, transference theorems provide connections between the
Dual_lattice
Group in group theory and physics
functions. The span of these functions does not form a Lie algebra under the Poisson bracket, however, because { x i , p j } = δ i , j . {\displaystyle \{x_{i}
Heisenberg_group
Statistical technique
special case of this setting when the kernel function is chosen to be the linear kernel. In general, under the kernel machine setting, the vector of covariates
Principal component regression
Principal_component_regression
Type of statistical measure over subsets of a dataset
low-pass filter called sinc-in-frequency (or, in discrete time, is a Dirichlet kernel). The continuous moving average of an integrable function f : R → R {\displaystyle
Moving_average
Interactive theorem prover software
Implementation language Features Higher-order logic Dependent types Small kernel Proof automation Proof by reflection Code generation ACL2 8.3 Matt Kaufmann
Proof_assistant
Differential operator in mathematics
occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion
Laplace_operator
Mapping between functions in the quantum phase space
space is a symplectic manifold, or possibly a Poisson manifold. Related structures include the Poisson–Lie groups and Kac–Moody algebras. The following
Wigner–Weyl_transform
Software used in mathematical applications
in the West GNU MPFR GNU Multi-Precision Library GNU Scientific Library hypre Integer set library Libfixmath OpenBLAS UMFPACK Intel Math Kernel Library
Mathematical_software
French econometrician
Trognon, A. (1984). "Pseudo Maximum Likelihood Methods: Applications to Poisson Models". Econometrica. 52 (3): 701–720. doi:10.2307/1913472. JSTOR 1913472
Christian_Gouriéroux
interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin
Weil–Brezin_Map
Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups. F is a global field, such as the field
Arthur–Selberg_trace_formula
Role of coherent states
runs through H {\displaystyle {\mathfrak {H}}} , forming a reproducing kernel Hilbert space. The objective in both cases is to ensure that an arbitrary
Coherent states in mathematical physics
Coherent_states_in_mathematical_physics
Type of statistical model
many other statistic methods. In 1988, Robinson applied Nadaraya-Waston kernel estimator to test the nonparametric element to build a least-squares estimator
Partially_linear_model
Scientific application
incorporates the effects of ionic strength mediated screening by evaluating the Poisson-Boltzmann equation at a finite number of points within a three-dimensional
DelPhi
French mathematician
orbits of the Lie algebras. She proved a generalized Poisson summation formula (called the Poisson-Plancherel formula), which is the integral of a function
Michèle_Vergne
Mathematical model for stochastic processes
function β {\displaystyle \beta } . Functional Linear Regression, Functional Poisson Regression and Functional Binomial Regression, with the important Functional
Generalized functional linear model
Generalized_functional_linear_model
Non-local formulation of continuum mechanics
Later, to overcome bond-based framework limitations for the material Poisson's ratio ( 1 / 3 {\displaystyle 1/3} for plane stress and 1 / 4 {\displaystyle
Peridynamics
Operator equation in the style of Fredholm theory
is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel of the form K (
Volterra_integral_equation
Random process independent of past history
discovered long before his work in the early 20th century in the form of the Poisson process. Markov was interested in studying an extension of independent
Markov_chain
Generates a forecast of future values of a time series
low-pass filters to remove high-frequency noise. This method is preceded by Poisson's use of recursive exponential window functions in convolutions from the
Exponential_smoothing
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
decreases. This result also follows easily using Fourier series and the Poisson integral. For continuous functions f on the whole plane which are smooth
Neumann–Poincaré_operator
Statistical matching technique
propensity score. One example is the Epanechnikov kernel. Radius matching is a special case where a uniform kernel is used. Mahalanobis metric matching in conjunction
Propensity_score_matching
addition there are Markov chain Monte Carlo routines for fitting Poisson-Gamma and Poisson-Lognormal models, including where these have a conditional spatial
CrimeStat
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POISSON KERNEL
POISSON KERNEL
Girl/Female
Arabic, Farsi, Iranian
Poison
Male
English
Variant spelling of English unisex Addison, ADISSON means "son of Adam."
Boy/Male
Australian, British, English
Son of Adam
Surname or Lastname
English
English : patronymic from Middle English prest ‘priest’, i.e. ‘son of the priest’.French : occupational name for a presser of wine or oil, from a derivative of presser ‘to press’.
Surname or Lastname
English
English : patronymic from Middle English Pole or Poul, vernacular forms of Paul.Americanized spelling of Scandinavian Poulsen.
Girl/Female
Indian, Telugu
Poison
Girl/Female
Tamil
Poison
Boy/Male
Hindu, Indian
Poison
Boy/Male
Tamil
Poison
Surname or Lastname
English
English : patronymic from Phil, a short form of the personal name Philip.
Surname or Lastname
English
English : variant spelling of Pierson.
Surname or Lastname
English
English : variant of Grissom.
Surname or Lastname
English and French
English and French : from Old French pinson ‘finch’, perhaps a nickname applied to a bright and cheerful person.English and French : metonymic occupational name for someone who made pincers or forceps or who used them in their work, from Old French pinson ‘pincers’ (a derivative of pincier ‘to pinch’).
Boy/Male
Indian, Sanskrit
Poison Spewing
Girl/Female
Biblical
Poison, tricks.
Surname or Lastname
English
English : topographic name for someone who lived by a postern gate, from Old French posterne; in some cases it would have been a metonymic occupational name for a gatekeeper.English : habitational name from Poston in Herefordshire or Poston in Shropshire, which is named with an Old English personal name Possa + þorn ‘thorn tree’.
Surname or Lastname
English (Midlands)
English (Midlands) : habitational name from Pointon in Lincolnshire, Poynton in Cheshire, or Poynton Green in Shropshire. The first is named from Old English Pohhingtūn ‘settlement (Old English tūn) associated with Pohha’, a byname apparently meaning ‘bag’; the others have as the first element the Old English personal names Pofa and Pēofa respectively.
Boy/Male
Hindu
Poison
Boy/Male
Indian
Poison
Girl/Female
Gujarati, Hindu, Indian
Poison; Earth
POISSON KERNEL
POISSON KERNEL
POISSON KERNEL
POISSON KERNEL
POISSON KERNEL
POISSON KERNEL
POISSON KERNEL
imp. & p. p.
of Poison
pl.
of Cornet-a-piston
p. pr. & vb. n.
of Poison
n.
Rat poison; white arsenic.
n.
Poison; venom.
v. i.
To act as, or convey, a poison.
n.
Poison spittle; poison ejected from the mouth.
n.
To taint; to corrupt; to vitiate; as, vice poisons happiness; slander poisoned his mind.
v. t.
To poison; to drug.
n.
Venom; poison.
n.
Any agent which, when introduced into the animal organism, is capable of producing a morbid, noxious, or deadly effect upon it; as, morphine is a deadly poison; the poison of pestilential diseases.
n.
To put poison upon or into; to infect with poison; as, to poison an arrow; to poison food or drink.
n.
A four-wheeled carriage for conveying ammunition, consisting of two parts, a body and a limber. In light field batteries there is one caisson to each piece, having two ammunition boxes on the body, and one on the limber.
v. t.
To poison; to infect with poison.
n.
A kind of antidote for poisons; a counter poison formerly in vogue.
v. t.
To imprison; to shut up in, or as in, a prison; to confine; to restrain from liberty.
n.
The California poison oak (Rhus diversiloba). See under Poison, a.
n.
Poison.
n.
That which taints or destroys moral purity or health; as, the poison of evil example; the poison of sin.
n.
To injure or kill by poison; to administer poison to.
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