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Optimization
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Approximation
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that wh...
Approximation theory - Wikipedia
Theorems in approximation theory
Approximation theory - Wikipedia
Theorems in approximation theory
Szász–Mirakjan–Kantorovich operator
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polyn...
Carleman's condition
In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure μ satisfies Carleman's condition, th...
Universal differential equation
A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line t...
Bramble–Hilbert lemma
Bramble–Hilbert lemma - Wikipedia
Journal of Approximation Theory
The Journal of Approximation Theory is "devoted to advances in pure and applied approximation theory and related areas."
Journal of Approximation Theory - Wikipedia
Hilbert matrix
In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractionsFor example, this is the 5 × 5 Hilbert matrix:The Hilbert matrix c...
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usu...
Chebyshev polynomials - Wikipedia
Favard operator
In functional analysis, a branch of mathematics, the Favard operators are defined by:where , , and . They are named after Jean Favard.
A common generalization is:where is a positive sequence that...
Szász–Mirakyan operator
Bernstein's inequality (mathematical analysis)
In mathematical analysis, Bernstein's inequality is named after Sergei Natanovich Bernstein. The inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial ti...
Unisolvent functions
In mathematics, a collection of n functions f1, f2, ..., fn is unisolvent on domain Ω if the vectorsare linearly independent for any choice of n distinct points x1, x2 ... xn in Ω. Equivalently, the c...
Erdős–Turán inequality
In mathematics, the Erdős–Turán inequality bounds the distance between a probability measure on the circle and the Lebesgue measure, in terms of Fourier coefficients. It was proved by Paul Erdős and P...
Baskakov operator
In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators. They are defined bywhere ( can be ...
Remez algorithm
The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations...
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of...
Fekete problem
In mathematics, the Fekete problem is, given a natural number N and a real s ≥ 0, to find the points x1,...,xN on the 2-sphere for which the s-energy, defined byfor s > 0...
Euler–Maclaurin formula
Unisolvent point set
Unisolvent point set - Wikipedia
Lebesgue's lemma
For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number lemmaIn mathematics, Lebesgue's lemma is an important statement in approximation theory. It provides a bound f...
Bernstein's theorem (approximation theory)
In approximation theory, Bernstein's theorem is a converse to Jackson's theorem. The first results of this type were proved by Sergei Bernstein in 1912.For approximation by trigonometric polynomials,...
Constructive Approximation
Constructive Approximation is "an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation ...
Least squares (function approximation)
In mathematics, the idea of least squares can be applied to approximating a given function by a weighted sum of other functions. The best approximation can be defined as that which minimises the diffe...
Favard's theorem
In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomia...
Semi-infinite programming
In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a fin...
Jackson's inequality
In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or m...
Müntz–Szász theorem
The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884–1952) in 1916. Roughly speaking, the theorem shows to what extent the Weierstrass...
Discrete Chebyshev polynomials
In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev (1864) and red...