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Theory of getting acceptably close inexact mathematical calculations
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing
Approximation_theory
Something roughly the same as something else
achieved by approximation. Approximation theory is a branch of mathematics, and a quantitative part of functional analysis. Diophantine approximation deals
Approximation
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
German mathematician (1928–2026)
September 2026) was a German mathematician who specialised in analysis (approximation theory, harmonic analysis). He was born in Germany but studied in Montreal
Paul_Butzer
Methods of mathematical approximation
useful approximation for a few terms, but at some point becomes less accurate if even more terms are added. The breakthrough from chaos theory was an
Perturbation_theory
algorithms achieved the best possible approximation ratio. Hardness of approximation theory deals with studying the approximation threshold of such problems. For
Hardness_of_approximation
In approximation theory, a converse to Jackson's theorem
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
Bernstein's theorem (approximation theory)
Bernstein's_theorem_(approximation_theory)
'Best' approximation of a function by a rational function of given order
Diophantine approximation and transcendental number theory, though for sharp results, ad hoc methods—in some sense inspired by the Padé theory—typically
Padé_approximant
Soviet mathematician
differential equations, differential geometry, probability theory, and approximation theory. Bernstein was born into the Jewish family of prominent Ukrainian
Sergei_Bernstein
Rational-number approximation of a real number
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus
Diophantine_approximation
Approximating an arbitrary function with a well-behaved one
classification problem instead. Approximation theory Fitness approximation Kriging Least squares (function approximation) Radial basis function network
Function_approximation
Academic journal
The Journal of Approximation Theory is "devoted to advances in pure and applied approximation theory and related areas." It was founded in 1968. In 2001
Journal of Approximation Theory
Journal_of_Approximation_Theory
Approximation of physical behavior
Bragg–Williams approximation, models on Bethe lattice, Landau theory, Curie-Weiss law for magnetic susceptibility, Flory–Huggins solution theory, and Scheutjens–Fleer
Mean-field_theory
American mathematician
pioneers in the fields of approximation theory and numerical analysis. His 1966 book, An Introduction to Approximation Theory, remains in print and is
Elliott_Ward_Cheney_Jr.
Physical theory with fields invariant under the action of local "gauge" Lie groups
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local
Gauge_theory
Branch of mathematics concerning probability
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations
Probability_theory
Theory of subatomic structure
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called
String_theory
Hungarian mathematician (1913–1996)
discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered
Paul_Erdős
Method to determine the electronic structure of strongly correlated materials
mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent
Dynamical_mean-field_theory
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
Italian mathematician and professor
Research Network on Approximation from 2017 to 2020, and Responsible for the Unione Matematica Italiana Thematic Group on "Approximation Theory and Applications
Stefano_De_Marchi
Italian mathematician (1872–1951)
authored more than 60 papers, mainly in the areas of real analysis, approximation theory and partial differential equations, according to Tricomi (1962).
Carlo_Severini
Study of abstract machines and automata
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical
Automata_theory
Ukrainian mathematician
1958) is a Ukrainian mathematician interested in probability theory and approximation theory, and known for her research on q-Bernstein polynomials, the
Sofiya_Ostrovska
Algorithm to approximate functions
Chebyshev nodes are a common choice for the initial approximation because of their role in the theory of polynomial interpolation. For the initialization
Remez_algorithm
Branch of mathematics
mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives
Mathematical_analysis
Method for estimating new data within known data points
simplest case this leads to least squares approximation. Approximation theory studies how to find the best approximation to a given function by another function
Interpolation
Inequality on approximations of a function by algebraic or trigonometric polynomials
In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials
Jackson's_inequality
Mathematical method that minimizes maximum error
A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function that
Minimax approximation algorithm
Minimax_approximation_algorithm
Mathematical approach to quantum physics
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Concept in several complex variables
{O}}(X)} -convexity, and approximation is by global holomorphic functions on X {\displaystyle X} . This generalizes classical approximation theorems in one complex
Polynomial_convexity
Belgian mathematician and computer scientist
and for the European Mathematical Society. His research concerns approximation theory. Bultheel was born in Zwijndrecht, Belgium on December 14, 1948.
Adhemar_Bultheel
Mathematical inequality
disk. It was proven by Sergei Bernstein while he was working on approximation theory. Let max | z | = 1 | f ( z ) | {\displaystyle \max _{|z|=1}|f(z)|}
Bernstein's theorem (polynomials)
Bernstein's_theorem_(polynomials)
In approximation theory, a Haar space or Chebyshev space is a finite-dimensional subspace V {\displaystyle V} of C ( X , K ) {\displaystyle {\mathcal {C}}(X
Haar_space
Scientific study of digital information
Sun; Verdú, Sergio (May 1993). "Approximation theory of output statistics". IEEE Transactions on Information Theory. 39 (3): 752–772. Bibcode:1993ITIT
Information_theory
mathematical theories. Almgren–Pitts min-max theory Approximation theory Arakelov theory Asymptotic theory Automata theory Bass–Serre theory Bifurcation theory Braid
List_of_mathematical_theories
American mathematician
mathematician, specializing in complex analysis, approximation theory, numerical analysis, and potential theory. Saff received in 1964 his bachelor's degree
Edward_B._Saff
Concept in number theory
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers α
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Armenian mathematician
mathematician, who made major contributions to the Approximation theory. The modern Complex Approximation Theory is based on Mergelyan's classical work. Corresponding
Sergey_Mergelyan
Solution method for linear differential equations
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially
WKB_approximation
universal approximation properties of two-layer neural networks. It has applications in approximation theory and statistical learning theory. It is named
Barron_space
Multivariate functions can be written using univariate functions and summing
In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous
Kolmogorov–Arnold representation theorem
Kolmogorov–Arnold_representation_theorem
Approximation for factorials
mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate
Stirling's_approximation
Computational quantum mechanical modelling method to investigate electronic structure
calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation
Density_functional_theory
{\displaystyle \arg \min _{y\in M}d(x,y)} are also called elements of best approximation. This term comes from constrained optimization: we want to find an element
Metric_projection
Mathematical method
approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions. The best approximation
Least-squares function approximation
Least-squares_function_approximation
Square matrix where a[i,j]=1/(i+j-1)
introduced the Hilbert matrix to study the following question in approximation theory: "Assume that I = [a, b], is a real interval. Is it then possible
Hilbert_matrix
modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines. The modulus
Modulus_of_smoothness
Element of a basis for a function space
functions are useful in numerical methods. In numerical analysis and approximation theory, basis functions are also called blending functions, particularly
Basis_function
Description of limiting behavior of a function
statistics, however. Non-asymptotic bounds are provided by methods of approximation theory. Examples of applications are the following. In applied mathematics
Asymptotic_analysis
Branch of applied probability theory
Decision theory or the theory of rational choice is a branch of probability, economics, and analytic philosophy that uses expected utility and probability
Decision_theory
Egyptian mathematician
editorial boards of several journals including Constructive Approximation, Journal of Approximation Theory, Journal of Physics A, and The Ramanujan Journal. He
Mourad_Ismail
polynomials are listed under Polynomial interpolation Approximation theory Orders of approximation Lebesgue's lemma Curve fitting Vector field reconstruction
List of numerical analysis topics
List_of_numerical_analysis_topics
Polynomials used in approximation theory
polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the origin
Zolotarev_polynomials
Mathematical theorem
Harmonic approximation. Cambridge: Cambridge University Press. p. 39. ISBN 9780521497992. Arakeljan, N. U. (1968). "Uniform and tangential approximations by
Arakelyan's_theorem
Supposition or system of ideas intended to explain something
Mathematics: Approximation theory — Arakelov theory — Asymptotic theory — Bifurcation theory — Catastrophe theory — Category theory — Chaos theory — Choquet
Theory
Study of the properties of codes and their fitness
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography
Coding_theory
Scattering theory
Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total
Born_approximation
Describes the range of energies of an electron within the solid
energies so that there are no band gaps at higher energies. Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting
Electronic_band_structure
Field of mathematical analysis
constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related
Constructive_function_theory
German mathematician (1910–1990)
and G Walz, In memoriam : the work of Lothar Collatz in approximation theory, J. Approx. Theory 67 (2) (1991), 119–128. G Meinardus and G Nürnberger, In
Lothar_Collatz
Expressions for approximation accuracy
quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is in terms of the number of parameters
Order_of_approximation
continuous function can be approximated by polynomials. Korovkin approximation theory provides a way to establish the convergence of a sequence of positive
Korovkin_approximation
Function in mathematical analysis
American Mathematical Society. ISBN 978-0-8218-6963-5. Steffens, K.-G. (2006). The History of Approximation Theory. Boston: Birkhäuser. ISBN 0-8176-4353-2.
Modulus_of_continuity
Roots of the Chebyshev polynomials of the first kind
the greatest to the smallest. The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial
Chebyshev_nodes
Concept in mathematical analysis
{\displaystyle 2\pi } is the n {\displaystyle n} th-degree Fourier series approximation to f {\displaystyle f} , i.e., we have ( D n ∗ f ) ( x ) = ∫ − π π f
Dirichlet_kernel
Property in the Law of Total Probability and Its Application in Communication Theory," in IEEE Communications Letters, doi: 10.1109/LCOMM.2024.3447352. S. S
Carleman's_condition
Swiss mathematician
invented the Swiss cheese set and made significant contributions to approximation theory. She was born, lived and died in Bern, Switzerland. Alice attended
Alice_Roth
Concept in mathematics
1007/978-3-642-65711-5_3. ISBN 978-3-642-65713-9. Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7 Riesz
Trigonometric_polynomial
American mathematician (1928–2022)
areas of mathematics, including matrix analysis, complex analysis, approximation theory, and scientific computation. He was the author of the classic textbook
Richard_S._Varga
Generalization of basis splines (B-splines) to multiple variables
In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines
Box_spline
Method in approximation theory
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured
Radial basis function interpolation
Radial_basis_function_interpolation
Sequence of operations for a task
approximate While many algorithms reach an exact solution, approximation algorithms seek an approximation that is close to the true solution. Such algorithms
Algorithm
Failure of convergence in interpolation
Applications Conference on Mathematics in Defence Cheney, Ward; Light, Will (2000), A Course in Approximation Theory, Brooks/Cole, p. 19, ISBN 0-534-36224-9
Runge's_phenomenon
Unrelated vertices in graphs
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a
Independent set (graph theory)
Independent_set_(graph_theory)
main fields of research were orthogonal polynomials, approximation theory and potential theory. In 1947 he proved a generalization of Egorov's theorem:
Pavel_Korovkin
Application of mathematical methods to other fields
principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods
Applied_mathematics
field of approximation theory for obtaining upper estimates on the errors of best approximation. Denote the value of the best uniform approximation of a function
Whitney_inequality
Area of mathematics
R. (1986). Computational Mathematics: An Introduction to Numerical Approximation. John Wiley and Sons. ISBN 978-0-470-20260-9. Gentle, J. E. (2007).
Computational_mathematics
Class of algorithms that find approximate solutions to optimization problems
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems
Approximation_algorithm
In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints
Semi-infinite_programming
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Type of discrete orthogonal polynomials
polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were
Discrete Chebyshev polynomials
Discrete_Chebyshev_polynomials
Mathematical approximation of a function
– best approximation by a rational function Puiseux series – power series with rational exponents Approximation theory Function approximation Banner 2007
Taylor_series
Method for describing the electronic structure of molecules using quantum mechanics
combinations of atomic orbitals (LCAO). These approximations are made by applying the density functional theory (DFT) or Hartree–Fock (HF) models to the Schrödinger
Molecular_orbital_theory
Pair of polynomial sequences
Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of Tn(x), which are
Chebyshev_polynomials
Soviet mathematician
mathematics such as complex analysis, approximation theory and numerical analysis. He also worked on elasticity theory, which is used in applied math and
Georgii_Polozii
Method in cosmology and astrophysics
The approximation provides a relatively simple way to model the growth of structure in the early universe, bridging the gap between linear theory and
Zeldovich_approximation
Armenian and Soviet mathematician (1936–2023)
in approximation theory and complex analysis. He was known for Arakelian's approximation theorem. Also, on the basis of his approximation theory results
Norair_Arakelian
important statement in approximation theory. It provides a bound for the projection error, controlling the error of approximation by a linear subspace based
Lebesgue's_lemma
Solution of a simplified form of an equation
Newton polygon method. Newton developed this method to find an explicit approximation for an algebraic function. Newton expressed the function as proportional
Method_of_dominant_balance
are for science, engineering, finance, economics and logistics. Approximation theory part of analysis that studies how well functions can be approximated
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Basic result of approximation theory
The Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász in 1916. Roughly speaking, the theorem
Müntz–Szász_theorem
Method in physics
{\displaystyle \nu _{n}} is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength, E n
Debye_model
Property of an irrational number
In number theory, specifically in Diophantine approximation theory, the Markov constant M ( α ) {\displaystyle M(\alpha )} of an irrational number α {\displaystyle
Markov_constant
Bagby, T.; Gauthier, P. M. (1992). "Uniform approximation by global harmonic functions". Approximations by solutions of partial differential equations
Walsh–Lebesgue_theorem
Erdős–Turán–Koksma inequality. Erdős, P.; Turán, P. (1948). "On a problem in the theory of uniform distribution. I." (PDF). Proceedings of the Koninklijke Nederlandse
Erdős–Turán_inequality
British school master and mathematician (1786–1837)
functional equations, number theory and approximation theory, but also on optics. His contribution to approximation theory is honoured in the designation
William_George_Horner
Theorem in complex analysis
Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951. Let
Mergelyan's_theorem
Study of discrete mathematical structures
beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields. Algebraic structures occur as
Discrete_mathematics
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