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Family of functions
a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such
Summability_kernel
Family of functions in mathematics
mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise
Fejér_kernel
Mathematical concept
D. Convolutions with this approximate unit gives an example of a summability kernel for the Fourier series of a function in L1(T) (Katznelson 1976). Let
Poisson_kernel
Class of algorithms for pattern analysis
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These
Kernel_method
Technique in statistics
In statistics, kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. The objective is to find a
Kernel_regression
Concept in statistics
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method
Kernel_density_estimation
Matrix used in image processing to alter an image
In image processing, a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is
Kernel_(image_processing)
In functional analysis, a Hilbert space
In functional analysis, a discipline within mathematics, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Concept in mathematical analysis
In mathematical analysis, the Dirichlet kernel, is the collection of periodic functions defined as D n ( x ) = ∑ k = − n n e i k x = 1 + 2 ∑ k = 1 n cos
Dirichlet_kernel
Infinite series that is not convergent
(Cesàro) summability implies Ingham summability, and Ingham summability implies (C,δ) summability. The series a1 + ... is called Lambert summable to s if
Divergent_series
The Landau kernel is named after the German number theorist Edmund Landau. The kernel is a summability kernel defined as: L n ( t ) = { ( 1 − t 2 ) n c
Landau_kernel
Mathematical theorem
be a positive-definite kernel if and only if ∑ i = 1 n ∑ j = 1 n K ( x i , x j ) c i c j ≥ 0 {\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}K(x_{i},x_{j})c_{i}c_{j}\geq
Mercer's_theorem
Technique in signal processing
Lanczos kernel, which is a sinc function windowed by the central lobe of a second, longer, sinc function. The sum of these translated and scaled kernels is
Lanczos_resampling
Mathematical problem in classical harmonic analysis
the partial sum SN is replaced by a suitable summability kernel (for example the Fejér sum obtained by convolution with the Fejér kernel), basic functional
Convergence_of_Fourier_series
Vectors mapped to 0 by a linear map
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of
Kernel_(linear_algebra)
Statistical technique
A kernel smoother is a statistical technique to estimate a real valued function f : R p → R {\displaystyle f:\mathbb {R} ^{p}\to \mathbb {R} } as the weighted
Kernel_smoother
Net in a normed algebra
module there is some λ with m = meλ. Mollifier Nascent delta function Summability kernel Dales, H. Garth (2000). Banach Algebras and Automatic Continuity.
Approximate_identity
Machine learning kernel function
learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular,
Radial_basis_function_kernel
Multivariate statistical technique
statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using
Kernel principal component analysis
Kernel_principal_component_analysis
Generalization of a positive-definite matrix
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix
Positive-definite_kernel
Class of nonparametric methods
In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which
Kernel embedding of distributions
Kernel_embedding_of_distributions
Type of kernel induced by artificial neural networks
study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks
Neural_tangent_kernel
Construction for n-dimensional noise functions
determine d2, the squared distance to the point. From there, each vertex's summed kernel contribution is determined using the expression ( max ( 0 , r 2 − d
Simplex_noise
Algorithmic technique
time is the sum of the (polynomial time) kernelization step and the (non-polynomial but bounded by the parameter) time to solve the kernel. Indeed, every
Kernelization
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
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
Fredholm_kernel
Fundamental solution to the heat equation, given boundary values
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate
Heat_kernel
Elements taken to zero by a homomorphism
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism
Kernel_(algebra)
Mathematical function between groups that preserves multiplication structure
homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category
Group_homomorphism
Model for approximating non-linear effects, similar to a Taylor series
lower-order kernels, will affect each diagonal element of order p by means of the summation ∑ m = 0 p − 1 G m x ( n ) {\displaystyle \sum \limits _{m=0}^{p-1}G_{m}x(n)}
Volterra_series
Concept in probability theory
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes
Markov_kernel
In linear algebra, relation between 3 dimensions
dimension of the image of f) and the nullity of f (the dimension of the kernel of f). It follows that for linear transformations of vector spaces of equal
Rank–nullity_theorem
Summability method used in harmonic analysis
The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier integrals. It
Bochner–Riesz_mean
Summability method in physics
regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used
Zeta_function_regularization
Machine learning kernel function
learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the
Polynomial_kernel
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
the Dirichlet kernel: D n ( θ ) = 1 + 2 ∑ k = 1 n cos k θ = sin ( ( n + 1 2 ) θ ) sin 1 2 θ . {\displaystyle D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos
List of trigonometric identities
List_of_trigonometric_identities
signal fD with a sampled Gaussian kernel: L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,G(n
Scale_space_implementation
Set of methods for supervised statistical learning
term in the sum measures the degree of closeness of the test point x to the corresponding data base point xi. In this way, the sum of kernels above can
Support_vector_machine
Mathematical function
answer is to sample the continuous Gaussian, yielding the sampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the
Gaussian_function
Form of kernel density estimation in which the size of the kernels used is varied
adaptive or "variable-bandwidth" kernel density estimation is a form of kernel density estimation in which the size of the kernels used in the estimate are varied
Variable kernel density estimation
Variable_kernel_density_estimation
Complex-valued function
The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. It was first discovered by Mehler in 1866
Mehler_kernel
algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure. Let G {\displaystyle
Positive-definite function on a group
Positive-definite_function_on_a_group
Type of image blur produced by a Gaussian function
values can be normalized by dividing each term in the kernel by the sum of all terms in the kernel. A much better and theoretically more well-founded approach
Gaussian_blur
Integral expressing the amount of overlap of one function as it is shifted over another
probability distribution of the sum of two independent random variables is the convolution of their individual distributions. In kernel density estimation, a distribution
Convolution
Idempotent linear transformation from a vector space to itself
eigenspaces are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique. Therefore, given a
Projection_(linear_algebra)
Machine learning technique
learning to approximate kernel methods, introduced by Ali Rahimi and Ben Recht in their 2007 paper "Random Features for Large-Scale Kernel Machines", and extended
Random_feature
Language and compiler for GPU kernels
for writing compute kernels for graphics processing units (GPUs). Triton is used in deep learning frameworks including vLLM. Kernels are written as decorated
Triton_(programming_language)
Generalized function whose value is zero everywhere except at zero
introduction of a variety of summability methods to produce convergence. The method of Cesàro summation leads to the Fejér kernel F N ( x ) = 1 N ∑ n = 0 N
Dirac_delta_function
Concept in mathematics
the Frobenius kernel and the Frobenius complement have very restricted structures. J. G. Thompson (1960) proved that the Frobenius kernel K is a nilpotent
Frobenius_group
Distributed operating system by Huawei
5.x+ both discards the common Unix-like Linux kernel and replaces the previous multiple-kernel, kernel agnostic system from OpenHarmony with its own bespoke
HarmonyOS
Continuous generalization of cellular automata
convolution kernel K {\displaystyle \mathbf {K} } . The final kernel is the composition of a kernel shell K C {\displaystyle K_{C}} and a kernel skeleton
Lenia
Mathematical function
mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define
Transition_kernel
Bayesian interpretation of kernel regularization examines how kernel methods in machine learning can be understood through the lens of Bayesian statistics
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Statistical learning theory
functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points
Representer_theorem
All bases of a vector space have equally many elements
to the dimension of the transformation's range plus the dimension of the kernel. See rank–nullity theorem for a fuller discussion. This uses the axiom of
Dimension theorem for vector spaces
Dimension_theorem_for_vector_spaces
statistics, kernel Fisher discriminant analysis (KFD), also known as generalized discriminant analysis and kernel discriminant analysis, is a kernelized version
Kernel Fisher discriminant analysis
Kernel_Fisher_discriminant_analysis
Concept in statistics mathematics
Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental
Multivariate kernel density estimation
Multivariate_kernel_density_estimation
Concept in regression analysis mathematics
f(x)=\sum _{i=1}^{n}\alpha _{i}K_{x_{i}}(x),\,f\in {\mathcal {H}}} , where all α i {\displaystyle \alpha _{i}} are real numbers. Some commonly used kernels
Regularized_least_squares
Kernel methods are a well-established tool to analyze the relationship between input data and the corresponding output of a function. Kernels encapsulate
Kernel methods for vector output
Kernel_methods_for_vector_output
Sequence of homomorphisms such that each kernel equals the preceding image
of an abelian category) such that the image of one morphism equals the kernel of the next. In the context of group theory, a sequence G0→ f1 G1→ f2 G2→ f3 ⋯→ fn Gn
Exact_sequence
Vectorizing features using a hash function
learning, feature hashing, also known as the hashing trick (by analogy to the kernel trick), is a fast and space-efficient way of vectorizing features, i.e.
Feature_hashing
Period when a computer system is available
to access the same raw monotonic clock. On FreeBSD (but not macOS), the kernel exports three symbols that can be used as extern variables in C to obtain
Uptime
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
Integral transform and linear operator
New York. Grafakos, Loukas (1994). "An elementary proof of the square summability of the discrete Hilbert transform". American Mathematical Monthly. 101
Hilbert_transform
Estimate of an unobservable underlying probability density function
estimates are kernel density estimates using a Gaussian kernel. That is, a Gaussian density function is placed at each data point, and the sum of the density
Density_estimation
English mathematician
possible. Her 1953 paper established several important results on summability kernels and is referenced in two textbooks on functional analysis. Her papers
Winifred_Sargent
Statistical model
isometric. From now on, let ℋ(R) be a reproducing kernel Hilbert space with positive definite kernel R. Driscoll's zero-one law is a result characterizing
Gaussian_process
Infinite sum
{\displaystyle \sum _{n=0}^{\infty }a_{n}=\sum _{n\in \mathbb {N} }a_{n}.} By nature, the definition of unconditional summability is insensitive to
Series_(mathematics)
Transmission Control Protocol technology
the Linux kernel. The currently available implementations are: Linux kernel (new reference implementation) introduced in the mainlined kernel in v5.6 Linux
Multipath_TCP
Statistical formula
H(k)\times \dots \times H(k):\sum _{i=1}^{d}\|f_{i}\|_{H(k)}^{2}\leq 1\right\},} associated to the matrix-valued reproducing kernel K ( x , x ′ ) = k ( x ,
Stein_discrepancy
Tree-based ensemble machine learning methods
adaptive kernel estimates. Davies and Ghahramani proposed Kernel Random Forest (KeRF) and showed that it can empirically outperform state-of-art kernel methods
Random_forest
Statistical tool
requires the user to specify the bandwidth and usage of the Bartlett kernel from Kernel density estimation Regression models estimated with time series data
Newey–West_estimator
Fatty seed of Theobroma cacao
roast gives a more intense, bitter flavor lacking complex flavor notes. The sum of all ingredients derived from dried, shelled cocoa beans is often expressed
Cocoa_bean
Monster and modular connection
using Rademacher sums to produce the McKay–Thompson series as (2 + 1)-dimensional gravity partition functions by a regularized sum over global torus-isogeny
Monstrous_moonshine
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
Recursive algorithm in applied mathematics
windowing is to derive a kernel from the desired time domain window, and to apply that kernel to the output of the SDFT. Since the kernel depends on the window
Sliding_DFT
Type of vector space in math
The Hardy space H2(D) also admits a reproducing kernel, known as the Szegő kernel. Reproducing kernels are common in other areas of mathematics as well
Hilbert_space
matrices, and these can be generalized to the nonparametric case of multiple kernel learning. Consider a matrix W to be learned from a set of examples, S=(Xit
Matrix_regularization
East Asian jellied dessert made from almonds
dau6 fu6; rōmaji: annindōfu) is a soft, jellied dessert made of apricot kernel milk, agar, and sugar popular throughout East Asia. The name "tofu" here
Almond_tofu
Identity for a sequence of orthogonal polynomials
y):=\sum _{j=0}^{n}f_{j}(x)f_{j}(y)/h_{j},\quad n=0,1,\dots } which are called the Christoffel–Darboux kernels. By the orthogonality, the kernel satisfies
Christoffel–Darboux_formula
Game where groups of players may enforce cooperative behaviour
the nucleolus is in the core. The nucleolus is always in the kernel, and since the kernel is contained in the bargaining set, it is always in the bargaining
Cooperative_game_theory
Inverse of a finite difference
2011. Alabdulmohsin, Ibrahim M. (2018). Summability Calculus: A Comprehensive Theory of Fractional Finite Sums. Cham: Springer. ISBN 978-3-319-74647-0
Indefinite_sum
Algorithm for job scheduling
smallest sum is Θ ( 1 / n ) {\displaystyle \Theta (1/n)} . In the kernel partitioning problem, there are some m pre-specified jobs called kernels, and each
Longest-processing-time-first scheduling
Longest-processing-time-first_scheduling
Infinite series summing alternating 1 and -1 terms
(0, 1, 3, 4, 6, ...) is 1 3 {\displaystyle {\tfrac {1}{3}}} . The summability of 1 − 1 + 1 − 1 + ⋯ can be frustrated by separating its terms with exponentially
Grandi's_series
Mathematical function, in linear algebra
same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( ℵ 0 + 0 = ℵ 0 + 1 {\displaystyle \aleph
Linear_map
Extension of cubic spline interpolation
on a two-dimensional regular grid. The interpolated surface (meaning the kernel shape, not the image) is smoother than corresponding surfaces obtained by
Bicubic_interpolation
Concept in machine learning
express the kernel, data and inverse transform of the kernel. 𝒴=𝒜[(Cg)⊙(Bd)], where 𝒜,ℬ and 𝒞 are the inverse transform, data and kernel. The derivation
Tensor_(machine_learning)
Solving multiple machine learning tasks at the same time
f(x)=\sum _{i=1}^{N}k(x,x_{i})Ac_{i}} . The model output on the training data is then KCA , where K is the n × n {\displaystyle n\times n} empirical kernel
Multi-task_learning
Graphical representation of the distribution of numerical data
{\displaystyle n=\sum _{i=1}^{k}{m_{i}}.} A histogram can be thought of as a simplistic kernel density estimation, which uses a kernel to smooth frequencies
Histogram
expressing it in terms of the heat kernel K ( P , Q , t ) = ∑ n = 1 ∞ f n ( P ) f n ( Q ) e − λ n t {\displaystyle K(P,Q,t)=\sum _{n=1}^{\infty }f_{n}(P)f_{n}(Q)e^{-\lambda
Minakshisundaram–Pleijel zeta function
Minakshisundaram–Pleijel_zeta_function
Summability method for a class of divergent series
a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory. Define the Lambert kernel by
Lambert_summation
approximated as a sum over kernels, whose domain is the feature space. If this is done in a reproducing kernel Hilbert space, a kernel method can be a universal
Kernel_adaptive_filter
Population balance equation in statistical physics
There exists a unique solution for a chosen kernel function. The operator, K, is known as the coagulation kernel and describes the rate at which particles
Smoluchowski coagulation equation
Smoluchowski_coagulation_equation
Tool in homological algebra
topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category
Mapping cone (homological algebra)
Mapping_cone_(homological_algebra)
Smoothing filler for images
spatial closeness (using the spatial kernel g s {\displaystyle g_{s}} ) and the intensity difference (using the range kernel f r {\displaystyle f_{r}} ). Consider
Bilateral_filter
Theorem
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It
Schwartz_kernel_theorem
Subgroup invariant under conjugation
Furthermore, the normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms with domain G , {\displaystyle G,} which means that
Normal_subgroup
Means of constructing a group from two subgroups
In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract
Direct_sum_of_groups
Statistics named for Richard von Mises
V_{mn}={\frac {1}{n^{m}}}\sum _{i_{1}=1}^{n}\cdots \sum _{i_{m}=1}^{n}h(x_{i_{1}},x_{i_{2}},\dots ,x_{i_{m}}),} where h is a symmetric kernel function. Serfling
V-statistic
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SUMMABILITY KERNEL
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