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Inequality in mathematical analysis
In the field of mathematical analysis, an interpolation inequality is an inequality of the form ‖ u 0 ‖ 0 ≤ C ‖ u 1 ‖ 1 α 1 ‖ u 2 ‖ 2 α 2 … ‖ u n ‖ n
Interpolation_inequality
Theorem in mathematical analysis
in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the L
Gagliardo–Nirenberg interpolation inequality
Gagliardo–Nirenberg_interpolation_inequality
Mathematical theory by discovered by Józef Marcinkiewicz
operator norm of T from Lq to Lq,w is at most Nq. Then the following interpolation inequality holds for all r between p and q and all f ∈ Lr: ‖ T f ‖ r ≤ γ N
Marcinkiewicz interpolation theorem
Marcinkiewicz_interpolation_theorem
Fenchel's inequality Friedrichs' inequality Gagliardo–Nirenberg interpolation inequality Gårding's inequality Grothendieck inequality Grunsky's inequalities Hanner's
List_of_inequalities
Mathematical family of interpolation inequalities
Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different
Landau–Kolmogorov_inequality
Canadian-American mathematician (1925–2020)
the Minkowski problem in two-dimensions, the Gagliardo–Nirenberg interpolation inequality, the Newlander-Nirenberg theorem in complex geometry, and the development
Louis_Nirenberg
exponents. Ladyzhenskaya's inequality is one member of a broad class of inequalities known as interpolation inequalities. Let Ω {\displaystyle \Omega
Ladyzhenskaya's_inequality
Root-finding algorithm
perform the next action (to choose) interpolation (when inequality is true) or bisection method (when inequality is not true). Also, if the previous step
Brent's_method
Theorem in mathematics
Robert J. & Schmuckenschläger, Michael (2001). "A Riemannian interpolation inequality à la Borell, Brascamp and Lieb". Invent. Math. 146 (2): 219–257
Borell–Brascamp–Lieb inequality
Borell–Brascamp–Lieb_inequality
Two inequalities in mathematical analysis
mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space
Agmon's_inequality
Mathematical inequality about the convolution of two functions
by Fubini's theorem. Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof. In case p , q >
Young's convolution inequality
Young's_convolution_inequality
Signal (re-)construction algorithm
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real
Whittaker–Shannon interpolation formula
Whittaker–Shannon_interpolation_formula
Theorem on operator interpolation
referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after
Riesz–Thorin_theorem
Inequality between integrals in Lp spaces
the case n = 2 , {\displaystyle n=2,} we obtain the interpolation result Littlewood's inequality—For θ ∈ ( 0 , 1 ) {\displaystyle \theta \in (0,1)} and
Hölder's_inequality
Set of functions between two fixed sets
representation theorem, the Riesz–Thorin theorem, the Gagliardo–Nirenberg interpolation inequality, the Rellich–Kondrachov theorem, the Hardy–Littlewood maximal function
Function_space
(2017). "New Interpolation Inequalities to Euler's R ≥ 2r" (PDF). Forum Geometricorum. 17: 149–156. Lukarevski, Martin: "An inequality for the tanradii
List_of_triangle_inequalities
Mathematical inequality in Sobolev space theory
"L1 Poincare Inequality". Archived from the original on March 3, 2012. Kikuchi, Fumio; Liu, Xuefeng (2007), "Estimation of interpolation error constants
Poincaré_inequality
Vector space in mathematics
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in
Interpolation_space
Vector space of functions in mathematics
complex interpolation is the only way to obtain the H s ( Ω ) {\displaystyle H^{s}(\Omega )} spaces. As a result, the interpolation inequality still holds
Sobolev_space
Problem in complex analysis
\ldots ,z_{n}} in D {\displaystyle \mathbb {D} } , the Nevanlinna–Pick interpolation problem is to find a holomorphic function φ {\displaystyle \varphi }
Nevanlinna–Pick_interpolation
French mathematician (1944–2024)
ISBN 0-387-70913-4 Differential inclusion Gagliardo–Nirenberg interpolation inequality Dalia Karpel (18 April 2002). "Oh my love, comely as Jerusalem"
Haïm_Brezis
Perpendicular line segment from a triangle's side to opposite vertex
2001, p. 118 Andrica, Dorin; Marinescu, Dan Ştefan (2017). "New Interpolation Inequalities to Euler's R ≥ 2r" (PDF). Forum Geometricorum. 17: 149–156. Archived
Altitude_(triangle)
Constants related to interpolation errors
"Lebesgue functions and Lebesgue constants in polynomial interpolation", Journal of Inequalities and Applications, 2016 93: 2016:93, doi:10.1186/s13660-016-1030-3
Lebesgue_constant
Canadian-American mathematician, co-developer of Gagliardo–Nirenberg interpolation inequality, Abel Prize winner (2015). Bob Shane, 85, American singer and guitarist
Deaths_in_January_2020
Sufficiency theorem for reconstructing signals from samples
Whittaker–Nyquist–Shannon, and may also be referred to as the cardinal theorem of interpolation. Sampling is a process of converting a signal (for example, a function
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
American mathematician (1953–2026)
pp. 771–831. with L. Caffarelli and L. Nirenberg, "First order interpolation inequalities with weights", Compositio Mathematica n. 53 i. 3, pp. 259–275
Robert_V._Kohn
Formula in matrix theory
matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f (A) of a matrix A as a polynomial in
Sylvester's_formula
Bound on the norm of Fourier coefficients
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered
Hausdorff–Young_inequality
include variational inequalities or complementarities Univariate optimization: Golden section search Successive parabolic interpolation — based on quadratic
List of numerical analysis topics
List_of_numerical_analysis_topics
Mathematical operator in real and harmonic analysis
Hardy–Littlewood maximal inequality in hand, the following strong-type estimate is an immediate consequence of the Marcinkiewicz interpolation theorem: Theorem
Hardy–Littlewood maximal function
Hardy–Littlewood_maximal_function
Polish mathematician (1910–1940)
Marcinkiewicz multiplier theorem, Marcinkiewicz interpolation theorem, and Marcinkiewicz–Zygmund inequality. Following the outbreak of World War II, Marcinkiewicz
Józef_Marcinkiewicz
Compression method for digital images
Navascués, M. A.; Sebastián, M. V. (2006). "Smooth fractal interpolation". Journal of Inequalities and Applications. 2006: 1–20. doi:10.1155/JIA/2006/78734
Fractal_compression
French mathematician (1902–1965)
1946. He died in La Tronche, aged 62. Favard measure Favard inequality Favard interpolation Favard problem COMITE DES AMIS DE JEAN-FAVARD The Lycée Jean
Jean_Favard
Logical error that can often be found in programming
usually caused by the use of non-strict inequality (≤) as the terminating condition where strict inequality (<) should have been used, or vice versa
Off-by-one_error
Property of having a unique mode or maximum value
inequality. Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode. This inequality depends
Unimodality
Function spaces generalizing finite-dimensional p norm spaces
is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality. By contrast, if p > 2 , {\displaystyle
Lp_space
Middle quantile of a data set or probability distribution
calculating the theoretical median of the fitted distribution. Pareto interpolation is an application of this when the population is assumed to have a Pareto
Median
Danish mathematician and statistician (1873–1961)
differences and interpolation. He was professor of actuarial science at the University of Copenhagen from 1923 to 1943. Steffensen's inequality and Steffensen's
Johan_Frederik_Steffensen
Method of estimating the median of a population
Pareto interpolation is a method of estimating the median and other properties of a population that follows a Pareto distribution. It is used in economics
Pareto_interpolation
In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of
Whitney_inequality
Mean in mathematics
semidefinite matrices, and satisfies a similar interpolation formula. Mean Muirhead's inequality Inequality of arithmetic and geometric means E. Heinz (1951)
Heinz_mean
Polish mathematician (1900–1992)
2002) Intégrales singulières (Springer-Verlag, 1971) Trigonometric Interpolation (University of Chicago, 1950) Measure and Integral: An Introduction
Antoni_Zygmund
Optimization technique for solving (mixed) integer linear programs
iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions
Cutting-plane_method
Type of mathematical function
Hardy–Littlewood inequality holds, that is, ∫ f g ≤ ∫ f ∗ g ∗ . {\displaystyle \int fg\leq \int f^{*}g^{*}.} Further, the Pólya–Szegő inequality holds. This
Symmetric decreasing rearrangement
Symmetric_decreasing_rearrangement
French polymath (1749–1827)
probability-generating function of the former. Laplace then shows how, by means of interpolation, these coefficients may be determined from the generating function.
Pierre-Simon_Laplace
Mathematical framework
functional connections (TFC) is a mathematical framework for functional interpolation. It provides a method for deriving a functional—a function that operates
Theory of functional connections
Theory_of_functional_connections
Statistical method of dividing data into equal-sized intervals for analysis
linear interpolation between data points, and differ only in how the index h used to choose the point along the piecewise linear interpolation curve,
Quantile
American mathematician
Calderón-Zygmund, fonctions para-accrétives et interpolation. [Calderón-Zygmund operators, para-accretive functions and interpolation] Rev. Mat. Iberoamericana 1 (1985)
Stephen_Semmes
first kind Abel–Goncharov interpolation Abel–Plana formula Abel function Abel's integral equation Abel's identity Abel's inequality Abel's irreducibility
List of things named after Niels Henrik Abel
List_of_things_named_after_Niels_Henrik_Abel
Argentine mathematician
singular integral operators to partial differential equations, from interpolation theory to Cauchy integrals on Lipschitz curves, from ergodic theory
Alberto_Calderón
Mathematical algorithm for eliminating variables from a system of linear inequalities
mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier
Fourier–Motzkin_elimination
Geometry of the location of polynomial roots
the polynomial on a basis related to Lagrange interpolation to define discs centered at the interpolation points, each containing a root of the polynomial;
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Function space
(E_{n})^{1/p}\|_{\ell ^{q}(\mathbb {Z} )}\leq A_{p,q}C} . Interpolation space Hardy–Littlewood inequality Grafakos, Loukas (2008), Classical Fourier analysis
Lorentz_space
Pair of polynomial sequences
are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon
Chebyshev_polynomials
Open problem on 3x+1 and x/2 functions
{1}{2}}-\cos(\pi z)\right)\sin(\pi z)+h(z)\sin ^{2}(\pi z)\end{aligned}}} is an interpolation of the Collatz map to the complex plane. The reason for adding the extra
Collatz_conjecture
Optimizing objective functions that have constrained variables
= 1 , … , n Equality constraints h j ( x ) ≥ d j for j = 1 , … , m Inequality constraints {\displaystyle {\begin{array}{rcll}\min &~&f(\mathbf {x} )&\\\mathrm
Constrained_optimization
competed for the prize with his Discourse on the Origin and Basis of Inequality Among Men, but did not win the prize that year. The Académie des Sciences
Académie des Sciences, Arts et Belles-Lettres de Dijon
Académie_des_Sciences,_Arts_et_Belles-Lettres_de_Dijon
Ancient Hindu text
10.131 are transitional verses. Olivelle notes instances of likely interpolation and insertions in the notes to this section, in both the presumed vulgate
Manusmriti
Optimization algorithm
{\displaystyle f} is continuously differentiable, we may prove that: This inequality implies that the amount by which we can be sure the function f {\displaystyle
Gradient_descent
Hungarian mathematician
introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's inequality. He also introduced
Marcel_Riesz
Theorem in mathematics
differentiable functions. A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the
Mean_value_theorem
Overview of and topical guide to geometry
multiplication Vector addition Zero vector Complex plane Imaginary axis Linear interpolation One-to-one Orthogonal Polar coordinate system Pole Real axis Secant
Outline_of_geometry
Relationship of various quantum subsystems
Computation and Quantum Information" "Quantum Entropy and Its Use" Trace Inequalities and Quantum Entropy: An Introductory Course We use the following notation
Strong subadditivity of quantum entropy
Strong_subadditivity_of_quantum_entropy
2024 song by Beyoncé
within the United States in the context of economic, racial and social inequality. Upon the album's release, the song debuted in the top 40 of the charts
Ya_Ya_(Beyoncé_song)
Optimization problem in computer science
expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where
Nearest_neighbor_search
Optimization algorithm
{\displaystyle \sigma } are Lagrange multipliers. If the problem does not have inequality constraints (that is, m I = 0 {\displaystyle m_{I}=0} ), the first-order
Sequential quadratic programming
Sequential_quadratic_programming
Divergent sum of positive unit fractions
function provides a continuous interpolation of the factorials, the digamma function provides a continuous interpolation of the harmonic numbers, in the
Harmonic_series_(mathematics)
Branch of number theory
analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers
P-adic_analysis
Mathematics award
dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis." 2026 Philadelphia, US Yu Deng University
Fields_Medal
Order-preserving mathematical function
functions and predicates are monotonic and Boolean. Monotone cubic interpolation Pseudo-monotone operator Spearman's rank correlation coefficient - measure
Monotonic_function
Iterative method for minimizing convex functions
over the vector x {\displaystyle x} (containing n variables); Convex inequality constraints of the form f i ( x ) ⩽ 0 {\displaystyle f_{i}(x)\leqslant
Ellipsoid_method
Function in discrete mathematics
\mathbf {X} } and Y {\displaystyle \mathbf {Y} } . The trigonometric interpolation polynomial p ( t ) = { 1 N [ X 0 + X 1 e i 2 π t + ⋯ + X N 2 − 1 e i
Discrete_Fourier_transform
Punctuation mark
originated in the C shell and the string generation mechanism is a simple interpolation that can occur anywhere in a command line and takes no account of existing
Bracket
Soviet and Ukrainian mathematician
and with S. G. Krein and E. M. Semenov contributed to the theory of interpolation of linear operators. He solved Banach's problem of norming subspaces
Yuri_Petunin
Indian mathematician
Singapore, 1993, p. 312. R.P. Agarwal and P.J.Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Academic Publishers, Dordrecht
Ravi_Agarwal
Probability distribution
targets Pareto efficiency – Weakly optimal allocation of resources Pareto interpolation – Method of estimating the median of a population Power law probability
Pareto_distribution
Romanian-American mathematician (1903–1990)
won a Lester R. Ford Award. Schoenberg, I. J. (1973), Cardinal Spline Interpolation, Society for Industrial and Applied Mathematics Schoenberg, I. J. (1982)
Isaac_Jacob_Schoenberg
Prediction of digital video quality
been introduced and studied in the context of radial basis function interpolation. SSIMULACRA and SSIMULACRA2 are variants of SSIM developed by Cloudinary
Structural similarity index measure
Structural_similarity_index_measure
Mathematical theorem in the study of analysis
both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass
Stone–Weierstrass_theorem
Polynomial sequence
Erdélyi et al. 1955, p. 207. Szegő 1975. Indritz, Jack (1961), "An inequality for Hermite polynomials", Proceedings of the American Mathematical Society
Hermite_polynomials
MR 0196341. Natanson, I. P. (1965). Constructive function theory. Vol. III. Interpolation and approximation quadratures. New York: Ungar Publishing Co. MR 0196342
Constructive_function_theory
Theorem in complex analysis
space and plays an important role in complex interpolation theory. It can be used to prove Hölder's inequality for measurable functions ∫ | g h | ≤ ( ∫ |
Hadamard_three-lines_theorem
Swedish mathematician (born 1944)
Analysis, Interpolation Theory, PDE:s and Homogenization Theory, Convexity Theory, Hardy-Type Inequalities, Continuous and Matricial Inequalities, Functional
Lars-Erik_Persson
Inequalities for inexact line search
(also known as the Armijo-Wolfe conditions in some books) are a set of inequalities for performing inexact line search, especially in quasi-Newton methods
Wolfe_conditions
Method of solving linear programming problems
M method introduces surplus and artificial variables to convert all inequalities into that form and there by extends the simplex in higher dimensions
Big_M_method
3rd century calculation of π by Liu Hui
<3.1415927} . That was the famous Zu Chongzhi π inequality. Zu Chongzhi then used the interpolation formula by He Chengtian (何承天, 370–447) and obtained
Liu_Hui's_π_algorithm
Fundamental trigonometric functions
trigonometric polynomial's ample applications may be acquired in its interpolation, and its extension of a periodic function known as the Fourier series
Sine_and_cosine
Method to solve optimization problems
of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set
Linear_programming
Algorithm for linear programming
him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at that time he didn't
Simplex_algorithm
American mathematician (1916–2001)
coding theorem Shannon-Weaver model of communication Whittaker–Shannon interpolation formula Atmar, Wirt (2001). "A Profoundly Repeated Pattern". Bulletin
Claude_Shannon
half" reports have been split into three year-long reports through interpolation. Errazzouki, Samia (2022). "The People vs. the Palace: Power and Politics
Human_rights_in_Morocco
u^{\prime \prime }\right\vert .} This inequality also follows from the well-known error estimate for linear interpolation by choosing v {\displaystyle \textstyle
Bramble–Hilbert_lemma
Matrix operation generalizing exponentiation of scalar numbers
roots of P are simple, and the "interpolation" characterization indicates that St is given by the Lagrange interpolation formula, so it is the Lagrange−Sylvester
Matrix_exponential
Technique for the generative modeling of a continuous probability distribution
various architectural improvements. For example, they proposed log-space interpolation during backward sampling. Instead of sampling from x t − 1 ∼ N ( μ ~
Diffusion_model
English polymath (1642–1727)
regarded as "the single most significant contributor to finite difference interpolation", with many formulas created by Newton. He was the first to state Bézout's
Isaac_Newton
Extension of the factorial function
combinatorics. The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve y = f ( x ) {\displaystyle y=f(x)}
Gamma_function
Empirical observation in economics that as income rises, less is spent on food
with the regression line and applying the adequate coefficients of interpolation, would fit the equation perfectly. Unfortunately, the first known publication
Engel's_law
1895 dystopian science fiction novella by H. G. Wells
Machine is interpreted in modern times as a commentary on the increasing inequality and class divisions of Wells's era, which he projects as giving rise to
The_Time_Machine
theorem (mathematical logic) Craig's theorem (mathematical logic) Craig's interpolation theorem (mathematical logic) Cut-elimination theorem (proof theory)
List_of_theorems
original paper, G.H. Hardy and J.E. Littlewood explained their maximal inequality in the language of cricket averages. Given a function f defined on Rn
Maximal_function
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