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75 relations: Abelian group, Addition theorem, Approximation theory, Chebfun, Chebyshev equation, Chebyshev filter, Chebyshev nodes, Chebyshev polynomials, Chebyshev rational functions, Clenshaw algorithm, Clenshaw–Curtis quadrature, Collocation method, Complete metric space, Continuous function, Critical value, Cubic function, Cylindrical multipole moments, De Moivre's formula, Dessin d'enfant, Dickson polynomial, Differential equation, Dirichlet kernel, Discrete Chebyshev transform, Discrete cosine transform, Equioscillation theorem, Even and odd functions, Fibonacci polynomials, Fourier series, Fundamental theorem of algebra, Galerkin method, Gegenbauer polynomials, Generating function, Gibbs phenomenon, Hermite polynomials, Hypergeometric function, Indeterminate form, Inner product space, Intermediate value theorem, Interpolation, Jacobi polynomials, Java applet, Kronecker delta, L'Hôpital's rule, Legendre polynomials, Lissajous curve, Lucas sequence, Markov brothers' inequality, Mathematics, Maxima and minima, Monic polynomial, ..., Numerical analysis, Orthogonal polynomials, Orthogonality, Pafnuty Chebyshev, Pell's equation, Piecewise, Polynomial interpolation, Polynomial sequence, Potential theory, Rational trigonometry, Recurrence relation, Recursion, Romanovski polynomials, Runge's phenomenon, Schröder's equation, Semigroup, Sequence, Smoothness, Sobolev space, Spectral method, Sturm–Liouville theory, Transliteration, Turán's inequalities, Uniform norm, Wigner semicircle distribution. Expand index (25 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Addition theorem
In mathematics, an addition theorem is a formula such as that for the exponential function that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y).
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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.
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Chebfun
Chebfun is a free/open-source software system written in MATLAB for numerical computation with functions of a real variable.
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Chebyshev equation
Chebyshev's equation is the second order linear differential equation (1-x^2) - x + p^2 y.
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Chebyshev filter
Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters.
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Chebyshev nodes
In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind.
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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.
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Chebyshev rational functions
In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal.
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Clenshaw algorithm
In numerical analysis, the Clenshaw algorithm, Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind T^*_n(x).
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Clenshaw–Curtis quadrature
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials.
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Collocation method
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.
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Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Critical value
Critical value may refer to.
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Cubic function
In algebra, a cubic function is a function of the form in which is nonzero.
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Cylindrical multipole moments
Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R. Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential.
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De Moivre's formula
In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) and integer it holds that where is the imaginary unit.
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Dessin d'enfant
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.
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Dickson polynomial
In mathematics, the Dickson polynomials, denoted, form a polynomial sequence introduced by.
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Differential equation
A differential equation is a mathematical equation that relates some function with its derivatives.
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Dirichlet kernel
In mathematical analysis, the Dirichlet kernel is the collection of functions e^.
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Discrete Chebyshev transform
In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind T_n (x) and the discrete Chebyshev transform on the 'extrema' grid of the Chebyshev polynomials of the first kind.
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Discrete cosine transform
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies.
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Equioscillation theorem
The equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm).
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Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.
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Fibonacci polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers.
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Fourier series
In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.
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Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
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Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem.
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Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval with respect to the weight function (1 − x2)α–1/2.
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Generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.
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Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at: and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.
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Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
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Hypergeometric function
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.
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Indeterminate form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form.
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Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
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Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.
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Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.
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Jacobi polynomials
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials.
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Java applet
A Java applet was a small application that is written in the Java programming language, or another programming language that compiles to Java bytecode, and delivered to users in the form of Java bytecode.
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Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.
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L'Hôpital's rule
In mathematics, and more specifically in calculus, L'Hôpital's rule or L'Hospital's rule uses derivatives to help evaluate limits involving indeterminate forms.
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Legendre polynomials
In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre.
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Lissajous curve
In mathematics, a Lissajous curve, also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations which describe complex harmonic motion.
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Lucas sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation where P and Q are fixed integers.
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Markov brothers' inequality
In mathematics, the Markov brothers' inequality is an inequality proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).
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Monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.
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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
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Orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
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Orthogonality
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.
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Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician.
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Pell's equation
Pell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form where n is a given positive nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x.
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Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.
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Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.
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Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3,..., in which each index is equal to the degree of the corresponding polynomial.
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Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
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Rational trigonometry
Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J. Wildberger, currently an associate professor of mathematics at the University of New South Wales.
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Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.
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Recursion
Recursion occurs when a thing is defined in terms of itself or of its type.
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Romanovski polynomials
In mathematics, Romanovski polynomials is an informal term for one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics.
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Runge's phenomenon
In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points.
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Schröder's equation
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function, find the function such that: Schröder's equation is an eigenvalue equation for the composition operator, which sends a function to.
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Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.
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Spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform.
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Sturm–Liouville theory
In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset.
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Transliteration
Transliteration is a type of conversion of a text from one script to another that involves swapping letters (thus trans- + liter-) in predictable ways (such as α → a, д → d, χ → ch, ն → n or æ → e).
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Turán's inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors.
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Uniform norm
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.
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Wigner semicircle distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse): for −R ≤ x ≤ R, and f(x).
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Chebyshev expansion, Chebyshev form, Chebyshev polynomial, Chebyshev polynomial of the first kind, Chebyshev polynomial of the second kind, Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind, Chebyshev polynomials/Proofs, Chebyshev root, Chebyshev roots, Chebyshev series, T-polynomial, Tchebycheff polynomials, Tchebyscheff polynomial, Tchebyscheff polynomials, Tchebyshev polynomials, Tschebyschow polynomial, Tschebyschow polynomials.
References
[1] https://en.wikipedia.org/wiki/Chebyshev_polynomials
