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Notion of convergence in mathematics
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than
Pointwise_convergence
Applying operations to functions in terms of values for each input "point"
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f ( x ) {\displaystyle f(x)}
Pointwise
Information Theory
In statistics, probability theory and information theory, pointwise mutual information (PMI), or point mutual information, is a measure of association
Pointwise_mutual_information
Mode of convergence of a function sequence
uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n ) {\displaystyle (f_{n})} converges
Uniform_convergence
Theorem stating that pointwise boundedness implies uniform boundedness
operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The
Uniform_boundedness_principle
Theorem in measure theory
is almost everywhere pointwise convergent to a function then the sequence converges in L 1 {\displaystyle L_{1}} to its pointwise limit, and in particular
Dominated_convergence_theorem
Semantic similarity measure
In computational linguistics, second-order co-occurrence pointwise mutual information (SOC-PMI) is a method used to measure semantic similarity, or how
Second-order co-occurrence pointwise mutual information
Second-order_co-occurrence_pointwise_mutual_information
Lemma in measure theory
Then: the sequence { g n ( x ) } n {\displaystyle \{g_{n}(x)\}_{n}} is pointwise non-decreasing at any x and g n ≤ f n {\displaystyle g_{n}\leq f_{n}}
Fatou's_lemma
Use of machine learning to rank items
Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents
Learning_to_rank
In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, the function whose value
Lower_envelope
1966 result in mathematical analysis
fundamental result in mathematical analysis establishing the (Lebesgue) pointwise almost everywhere convergence of Fourier series of L2 functions, proved
Carleson's_theorem
Property of a sequence or series
define pointwise Cauchy convergence, uniform convergence, and uniform Cauchy convergence of the sequence. Pointwise convergence implies pointwise Cauchy
Modes_of_convergence
Spacetime modeled by four pointwise-orthonormal vector fields
relativity, a frame field (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on
Frame fields in general relativity
Frame_fields_in_general_relativity
Relation among continuous functions
equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence
Equicontinuity
Curves whose limit does not preserve length
provides an analogous example showing that polyhedral surfaces that converge pointwise to a curved surface do not necessarily converge to its area, even when
Staircase_paradox
Mathematical theorem in real analysis
well. This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒn : [0, 1] → R be the sequence of functions
Uniform_limit_theorem
Mathematical problem in classical harmonic analysis
must be met. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability
Convergence_of_Fourier_series
Result in probability theory
convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. This theorem is the basis
Lévy's_continuity_theorem
Decomposition of periodic functions
{\tfrac {n}{P}}x}\,dx.} The series does not necessarily converge (in the pointwise sense) and, even if it does, it is not necessarily equal to s ( x ) {\displaystyle
Fourier_series
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
characteristic functions ( φ n ) n {\displaystyle (\varphi _{n})_{n}} converges pointwise to the characteristic function φ {\displaystyle \varphi } of X {\displaystyle
Convergence of random variables
Convergence_of_random_variables
Whenever certain curvatures are pointwise constant then they must be globally constant
is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially
Schur's lemma (Riemannian geometry)
Schur's_lemma_(Riemannian_geometry)
Chess variant
bishop moves pointwise. It can also move one step edgewise. The queen moves as a rook and bishop. The king moves one step edgewise or pointwise. There is
Rhombic_chess
Class of confidence intervals around statistical functionals of a distribution
producing bounds on the CDF, we must differentiate between pointwise and simultaneous bands. A pointwise CDF bound is one which only guarantees their coverage
CDF-based nonparametric confidence interval
CDF-based_nonparametric_confidence_interval
Point to which functions converge in analysis
) = g ( y ) pointwise . {\displaystyle \lim _{x\to p}f(x,y)=g(y)\;\;{\text{pointwise}}.} Alternatively, we may say f tends to g pointwise as x approaches
Limit_of_a_function
Condition in thermodynamics
{\displaystyle h_{\nu }=-\nabla \cdot \mathbf {F} _{\nu }} . They define (pointwise) monochromatic radiative equilibrium by ∇ ⋅ F ν = 0 {\displaystyle \nabla
Radiative_equilibrium
Type of Riemannian manifold with constant Jacobi operator spectrum
Riemann curvature tensor. A manifold M n {\displaystyle M^{n}} is called pointwise Osserman if, for every p ∈ M n {\displaystyle p\in M^{n}} , the spectrum
Osserman_manifold
Vector space of infinite sequences
turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear
Sequence_space
-topology on F {\displaystyle F} is called the topology of pointwise convergence. The topology of pointwise convergence on F {\displaystyle F} is identical to
Topologies on spaces of linear maps
Topologies_on_spaces_of_linear_maps
Theorem concerning uniform convergence
Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff
Egorov's_theorem
Mathematical concept
on the circle, Hεf converges uniformly to Hf, so in particular pointwise. The pointwise limit is a Cauchy principal value, written H f = P . V . 1 π ∫
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
Hence ρ ( n ) − 1 {\displaystyle \rho (n)-1} is the exact number of pointwise linearly independent vector fields that exist on an ( n − 1 {\displaystyle
Vector_fields_on_spheres
In mathematics, vector space of linear forms
on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined
Dual_space
features that differentiate BMDs from BDDs are using linear instead of pointwise diagrams, and having weighted edges. The rules that ensure the canonicity
Binary_moment_diagram
System of complete and orthogonal polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number
Legendre_polynomials
Problem in natural language processing and information retrieval
In natural language processing and information retrieval, cluster labeling is the problem of picking descriptive, human-readable labels for the clusters
Cluster_labeling
Theorem in measure theory
Fatou and Henri Léon Lebesgue. If the sequence of functions converges pointwise, the inequalities turn into equalities and the theorem reduces to Lebesgue's
Fatou–Lebesgue_theorem
On convergent subsequences of functions that are locally of bounded total variation
that a ≤ fn ≤ b for every n ∈ N. Then the sequence (fn)n ∈ N admits a pointwise convergent subsequence. The proof requires the basic facts about monotonic
Helly's_selection_theorem
Measure of dependence between two variables
{\displaystyle X} and Y {\displaystyle Y} . MI is the expected value of the pointwise mutual information (PMI). The quantity was defined and analyzed by Claude
Mutual_information
Value to which an infinite sequence tends
called pointwise limit, denoted x n , m → y m pointwise {\displaystyle x_{n,m}\to y_{m}\quad {\text{pointwise}}} , or lim n → ∞ x n , m = y m pointwise {\displaystyle
Limit_of_a_sequence
introduced by Vitali (1907), Hahn (1922), and Saks (1933), proves that a pointwise convergent sequence of finite measures is uniformly absolutely continuous
Vitali–Hahn–Saks_theorem
Mathematical concept
Non-UI sequence of RVs. The area under the strip is always equal to 1, but X n → 0 {\displaystyle X_{n}\to 0} pointwise.
Uniform_integrability
functions. For a function f {\displaystyle f} defined on [0,1), the first pointwise dyadic derivative of f {\displaystyle f} at a point x {\displaystyle x}
Dyadic_derivative
Set of functions between two fixed sets
set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space
Function_space
Sufficient criterion for uniform convergence
theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then
Dini's_theorem
n } , {\displaystyle \beta _{i}\in \{1,2,\ldots ,n\},} such that the pointwise stabilizer of B {\displaystyle B} is trivial (i.e., let B {\displaystyle
Strong_generating_set
Algorithm for multiplying large numbers
Marco Bodrato. The algorithm has five main steps: Splitting Evaluation Pointwise multiplication Interpolation Recomposition In a typical large integer
Toom–Cook_multiplication
Linear map from a vector space to its field of scalars
a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic
Linear_form
Mathematical function with no sudden changes
microcontinuity). The formal definitions for, and distinction between, pointwise continuity and uniform continuity were first given by Bolzano in the 1830s
Continuous_function
Theorem in complex analysis
statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. If we have a holomorphic function
Fatou's_theorem
Sequence space that is Fréchet
to turn a sequence space into a Fréchet space, namely the topology of pointwise convergence. Thus the name coordinate space because a sequence in an FK-space
FK-space
African-American mathematician
his research contributions include pointwise convergence of averages along cubes, being “the first complete pointwise convergence result obtained in the
Idris_Assani
Finite or infinite ordered list of elements
turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear
Sequence
Indicator function of positive numbers
kx\right)\end{aligned}}} These limits hold pointwise and in the sense of distributions. In general, however, pointwise convergence need not imply distributional
Heaviside_step_function
group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of G
Multiplicative_character
Vector space with a partial order
the reals) of real-valued functions on S , {\displaystyle S,} then the pointwise order on X {\displaystyle X} is given by, for all f , g ∈ X , {\displaystyle
Ordered_vector_space
Theorem
sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability
Skorokhod's representation theorem
Skorokhod's_representation_theorem
-tuples of natural numbers has finitely many minimal elements, for the pointwise partial order. This simple fact from combinatorics has become attributed
Dickson's_lemma
Theorems on the convergence of bounded monotonic sequences
Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions 0 ≤ f 1 ( x ) ≤ f 2 ( x ) ≤ ⋯ {\displaystyle
Monotone_convergence_theorem
Non-monotonic logic created by John McCarthy
) Pointwise circumscription is a variant of first-order circumscription that has been introduced by Vladimir Lifschitz. The rationale of pointwise circumscription
Circumscription_(logic)
than being uniformly Cauchy. In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly
Uniformly_Cauchy_sequence
left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure. The dual module is typically
Dual_module
Measure in mathematical analysis
{\displaystyle I(u_{n})\to 0} . The pointwise limit lim u n {\displaystyle \lim u_{n}} is identically zero, but the pointwise limit lim n u n ′ {\displaystyle
Young_measure
Tools to represent statistical uncertainty
these confidence intervals constitute a 95% pointwise confidence band for f(x). In mathematical terms, a pointwise confidence band f ^ ( x ) ± w ( x ) {\displaystyle
Confidence and prediction bands
Confidence_and_prediction_bands
Mathematics of real numbers and real functions
functions, pointwise convergence often fails to preserve operations on the limit function. For example, it is not generally true that the pointwise limit of
Real_analysis
Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence. The convergence was defined by
Wijsman_convergence
{\displaystyle {\mathcal {C}}(X),} is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover
Space of continuous functions on a compact space
Space_of_continuous_functions_on_a_compact_space
German mathematician (1815–1897)
Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement that is false
Karl_Weierstrass
Area of mathematical analysis
Hardy–Littlewood maximal function. Maximal functions are used to control pointwise convergence, differentiation of integrals, and boundary limits of harmonic
Harmonic_analysis
theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It
Classification of electromagnetic fields
Classification_of_electromagnetic_fields
Average value of a random variable
[X_{n}]\to \operatorname {E} [X]} even if X n → X {\displaystyle X_{n}\to X} pointwise. Thus, one cannot interchange limits and expectation, without additional
Expected_value
continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire
Baire_function
Branch of mathematics that studies dynamical systems
distribution of probabilities on the unit interval. More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the
Ergodic_theory
Theorem in functional analysis
\left(X^{\prime },X\right).} The weak-* topology is also called the topology of pointwise convergence because given a map f {\displaystyle f} and a net of maps
Banach–Alaoglu_theorem
difference between a nonlocal and a local response. In a local medium a pointwise intensity peak such as a Dirac delta gives rise to an equally sharp spatial
Nematicon
Integer
takes the inverse function of f(x), where (f(x))−1 specifically denotes a pointwise reciprocal. Where f is bijective specifying an output codomain of every
−1
Approximation in mathematics
narrower with decreasing ε {\displaystyle \varepsilon } , the approximations converge to the outer solution pointwise, but not uniformly, almost everywhere.
Method of matched asymptotic expansions
Method_of_matched_asymptotic_expansions
Property of functions which is weaker than continuity
\liminf \int f_{n}} where lim inf {\displaystyle \liminf } denotes the (pointwise) limit inferior. What this means, in full generality, is that if ( X
Semi-continuity
Space of bounded sequences
fulfills the conditions of being localizable and therefore semifinite). Pointwise multiplication gives them the structure of a Banach algebra, and in fact
L-infinity
Locally convex topology on function spaces
can be viewed as more natural, too, since it is simply the topology of pointwise convergence. The SOT topology also provides the framework for the measurable
Strong_operator_topology
Type of Kac–Moody algebras
-valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak
Affine_Lie_algebra
Indicator function of rational numbers
Blumberg theorem. The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: ∀ x ∈ R , 1 Q
Dirichlet_function
Concept in mathematics
complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory
Complex_reflection_group
Maximal proper filter
natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering.
Ultrafilter
Theorem in probability theory
τ {\displaystyle X_{\tau }} is defined as the almost surely existing pointwise limit of X = ( X t ) t ∈ N 0 {\displaystyle X=(X_{t})_{t\in \mathbb {N}
Optional_stopping_theorem
Theorems concerning stochastic processes
bounded below, so by the martingale convergence theorem it converges pointwise almost surely to a random variable Y {\displaystyle Y} . But if Y n >
Doob's martingale convergence theorems
Doob's_martingale_convergence_theorems
Statement in complex analysis
result in complex differential geometry that estimates the (squared) pointwise norm | ∂ f | 2 {\displaystyle |\partial f|^{2}} of a holomorphic map f
Schwarz_lemma
Mathematical theory
{\displaystyle \Omega } which, when fixed, destroys all symmetry, i.e. its pointwise stabilizer is trivial. A base is irredundant if each element further reduces
Cameron–Fon-Der-Flaass IBIS theorem
Cameron–Fon-Der-Flaass_IBIS_theorem
Multiplication algorithm
{\displaystyle {\widehat {C}}_{i}={\widehat {A}}_{i}{\widehat {B}}_{i}} (pointwise product), and compute the inverse transform C {\displaystyle C} of the
Schönhage–Strassen_algorithm
Algebras arising in harmonic analysis
bounded continuous complex-valued functions on G {\displaystyle G} with pointwise multiplication. We call A ( G ) {\displaystyle A(G)} the Fourier algebra
Fourier_algebra
Way to write a number as a product of other numbers
partitions of finite sequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions has been ongoing since
Multiplicative_partition
Branch of mathematics
field of scalars F, viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector
Linear_algebra
Algebraic structure in linear algebra
form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function ( f + g ) {\displaystyle
Vector_space
Non-parametric statistic used to estimate the survival function
hazards test. Other statistics that may be of use with this estimator are pointwise confidence intervals, the Hall-Wellner band and the equal-precision band
Kaplan–Meier_estimator
Topics referred to by the same term
Look up pmi in Wiktionary, the free dictionary. PMI may stand for: Pointwise mutual information, in statistics Privilege Management Infrastructure in
PMI
Romanian-American mathematician (1935–2025)
with the following properties: (I) H is compact (for the topology of pointwise convergence); (II) H is convex; (III) H satisfies the "separation property"
Alexandra_Bellow
Type of mathematical space
numbers x. The coarsest such topology, sometimes called the topology of pointwise convergence, is the product topology. With this topology, K is a compact
Compact_space
group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every
Algebraic_topology_(object)
Mathematical group of loops in a Lie group
maps from the circle S1 to a Lie group G, with multiplication defined pointwise. When G is a compact Lie group, LG is a basic example of an infinite-dimensional
Loop_group
Mathematical use of "for all" and "there exists"
pointwise continuity, whose definitions differ only by an exchange in the positions of two quantifiers. A function f from R to R is called Pointwise continuous
Quantifier_(logic)
Mathematical statement of uniqueness, except for an equivalent structure
employs the equivalence relation R between functions, defined by fRg if the pointwise difference f−g is a constant function, and means that the solution and
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