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Topic: Chebyshev polynomials

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  Chebyshev Polynomials
The signal property of Chebyshev polynomials is the trigonometric representation on [-1,1].
Investigate the error for the Chebyshev polynomial approximations in Example 1.
Investigate the error for the Chebyshev polynomial approximations in Example 3.
math.fullerton.edu /mathews/n2003/ChebyshevPolyMod.html   (435 words)

  Chebyshev polynomials - Wikipedia, the free encyclopedia
Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation.
The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.
The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.
en.wikipedia.org /wiki/Chebyshev_polynomials   (948 words)

 Chebyshev filter - Wikipedia, the free encyclopedia
Chebyshev filters have the property that they minimize the error between the idealized filter characteristic and the actual over the range of the filter, but with ripples in the passband.
The group delay is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies.
As is clear from the image, Chebyshev filters are sharper than the Butterworth filter; they are not as sharp as the elliptic one, but they show fewer ripples over the bandwidth.
en.wikipedia.org /wiki/Chebyshev_filter   (1314 words)

 NationMaster - Encyclopedia: Chebyshev polynomials   (Site not responding. Last check: )
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.
In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x)...
In the study of differential equations they arise as the solution to the Chebyshev differential equations Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables.
www.nationmaster.com /encyclopedia/Chebyshev-polynomials   (841 words)

 Chebyshev biography
Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".
Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal.
In the summer of 1846 Chebyshev was examined on his Master's thesis and in the same year published a paper based on that thesis, again in Crelle's journal.
www-history.mcs.st-andrews.ac.uk /Biographies/Chebyshev.html   (3026 words)

 Chebyshev polynomials
The Chebyshev polynomials are named after Pafnuty Chebyshev (Пафнутий Чебышёв) and compose a polynomial sequence.
This article refers to what are commonly known as Chebyshev polynomials of the first kind, which are a solution to the Chebyshev differential equation:
The Chebyshev polynomials can be used in the area of numerical approximation.
www.abacci.com /wikipedia/topic.aspx?cur_title=Chebyshev_polynomials   (217 words)

 Pafnuty Chebyshev Summary
In general, Chebyshev was looking for derivations of the leading results of probability by methods that could not be faulted for rigor, but which were not dependent on mathematical ideas that seemed out of proportion to the depth of the subject.
Chebyshev was able to get a decent approximation for the number of prime numbers less than a fixed number compared to known functions of that fixed number, but he did not prove that there was a limiting value.
Chebyshev is known for his work in the field of probability and statistics.
www.bookrags.com /Pafnuty_Chebyshev   (1783 words)

 Chebyshev polynomials - Biocrawler   (Site not responding. Last check: )
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers.
The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides the best approximation to a continuous function under the maximum norm.
A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [-1,1].
www.biocrawler.com /encyclopedia/Chebyshev_polynomials   (512 words)

 PlanetMath: Chebyshev equation
Chebyshev's equation is the second order linear differential equation
These polynomials are, up to multiplication by a constant, the Chebyshev polynomials.
This is version 3 of Chebyshev equation, born on 2002-11-21, modified 2002-11-21.
planetmath.org /encyclopedia/ChebyshevEquation.html   (167 words)

 Chebyshev Polynomial
Generating Function: The generating function of a Chebyshev Polynomial is:
Recurrence Relation: A Chebyshev Polynomial at one point can be expressed by neighboring Chebyshev Polynomials at the same point.
, are Chebyshev Polynomials of the second kind.
www.efunda.com /math/Chebyshev/index.cfm   (157 words)

 PlanetMath: Chebyshev polynomial
The Chebyshev polynomials of first kind are defined by the simple formula
It is an example of a Trigonometric Polynomial.
This is version 7 of Chebyshev polynomial, born on 2002-02-19, modified 2005-04-28.
planetmath.org /encyclopedia/ChebyshevPolynomial.html   (132 words)

 Chebyshev polynomials
Chebyshev's polynomial of the second kind can be defined in terms of the differential equation
Formulas for Chebyshev polynomials of the second kind, from
A plot of the Chebyshev polynomials of the second kind as functions of
www.unc.edu /~wjt/ChebyshevPolynomials2.htm   (342 words)

 Index to OEIS (Section Ch)
Chebyshev polynomials of the first kind (T- or C- polynomials) (2): A020537 A020538 A020539 A039991 A053120 A001105 A002415 A002492 A005585 A040977 A050486 A008314
Chebyshev polynomials of the first kind (T- or C- polynomials) (3): A053347 A054322 A054323 A054324 A054325 A054326 A054327 A054328 A054329 A054330 A054331 A054332
Chebyshev polynomials of the first kind (T- or C- polynomials) (11): A098249 A098250 A098252 A098253 A098255 A098256 A098258 A098259 A098261 A098262 A098291 A098292 A078070 A004146 A007877 A054493 A011655 A011655 A049683
www.research.att.com /~njas/sequences/Sindx_Ch.html   (724 words)

 P. Wellstead, M. Waller: MODELLING PAPER MACHINE CROSS DIRECTION PROFILES   (Site not responding. Last check: )
Chebyshev polynomials are an example of a basis function set.
The Chebyshev polynomials are useful because they correspond well to the form of bending mode which might be expected of a headbox lip.
For example, the zeroeth polynomial is a constant and its coefficient corresponds to an equal deflection of all actuator bolts.
www2.umist.ac.uk /csc/twod/pubs/art5/625_000.htm   (2072 words)

 Chebyshev polynomial and the Pascal Triangle
He is known for his work in the field of probability and statistics.This article refers to what are commonly known as Chebyshev polynomials of the first kind.
Chebyshev polynomials of the first kind are very important in numerical approximation.
It is obvious that Pascal`s Triangle structure is built in these relations, which certainly indicates the existing connection between the numbers of Pascal`s Triangle and Chebyshev polynomials of the second kind.
milan.milanovic.org /math/english/fibo/fibo6.html   (230 words)

Chebyshev was a prolific mathematician, making contributions to number theory, probability and integration.
In addition to his interest in mathematics, Chebyshev was interested in mechanical systems and their properties.
In 1867 Chebyshev published a paper "On Mean Values" in which he formulated the inequality known today by his name.
tulsagrad.ou.edu /statistics/biographies/Chebyshev.htm   (352 words)

 Chebyshev (print-only)
Brashman was particularly interested in mechanics but his interests were wide ranging and, in addition to courses on mechanical engineering and hydraulics, he taught his students the theory of integration of algebraic functions and the calculus of probability.
Chebyshev's work on prime numbers included the determination of the number of primes not exceeding a given number, published in 1848, and a proof of Bertrand's conjecture.
The closer mutual approximation of the points of view of theory and practice brings most beneficial results, and it is not exclusively the practical side that gains; under its influence the sciences are developing in that this approximation delivers new objects of study or new aspects in subjects long familiar.
www-groups.dcs.st-and.ac.uk /~history/Printonly/Chebyshev.html   (2900 words)

 Citations: The Chebyshev Polynomials - Theodore (ResearchIndex)
The answer is given in terms of the Chebyshev polynomials.
Thus we immediately obtain that t n (x) b Gamma a 2 n T n 2x Gamma a Gamma b b Gamma a is a monic polynomial with the smallest uniform norm on [a; b] among all....
We remark that the Chebyshev constant of a compact set in C is equal to its transfinite diameter and to its logarithmic capacity (cf.
citeseer.ist.psu.edu /context/37828/0   (1046 words)

 Chebyshev Polynomials With Integer Coefficients (ResearchIndex)
We study the asymptotic structure of polynomials with integer coefficients and smallest uniform norms on an interval of the real line.
Introduction Let P n (C) and P n (Z) be the sets of algebraic polynomials of degree at most n, respectively with complex and with integer coefficients.
3 the asymptotic structure of the polynomials of minimal dioph..
citeseer.ist.psu.edu /402403.html   (423 words)

 Chebyshev Polynomials -- Book
Offering a broad treatment of Chebyshev polynomials, this exposition on the subject's state of the art provides a rigorous treatment of theory and an in-depth look at the properties of all four kinds of Chebyshev polynomials and their roles in approximation, series expansions, interpolation, quadrature, and integral equations.
Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods.
A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject's state of the art, Chebyshev Polynomials is just such a treatment.
www.bookspec.com /un/849303559   (311 words)

 Systat Software Inc. - TableCurve 2D - HTML Help
These equations are evaluated using the recurrence relation, which is quite efficient, although in general you should expect a Chebyshev polynomial or rational evaluation to require about twice the time required for a standard polynomial or rational of the same coefficient count.
Because Chebyshev models are too complex to code in-line, each of the code generation languages uses a separate function to map the range and perform the evaluation.
It is often desirable, though not always wise, to convert a Chebyshev polynomial or rational to a standard polynomial or rational representation.
www.systat.com /products/TableCurve2D/help/?sec=1085   (406 words)

 Normalized Chebyshev Filter Polynomials
Computes the numerator and denominator polynomials for a normalized Chebyshev filter with n poles, where n is an integer greater than 0.
The output value of numout is set to the double-precision column vector corresponding to the numerator polynomial of the Chebyshev filter (numout is equal to the scalar 1 because normalized Chebyshev filters have no zeros).
The output value of denout is set to the double-precision column vector corresponding to the denominator polynomial of the Chebyshev filter.
www.omatrix.com /manual/fncheb.htm   (109 words)

 GNU Scientific Library -- Reference Manual - Chebyshev Approximations   (Site not responding. Last check: )
A Chebyshev approximation is a truncation of the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}.
The computation of the Chebyshev approximation is an O(n^2) process, and requires n function evaluations.
For smooth functions the Chebyshev approximation converges extremely rapidly and errors would not be visible.
gnuwww.epfl.ch /software/gsl/manual/gsl-ref_28.html   (468 words)

 Polynomials From Pascal's Triangle
There are many interesting things about polynomials whose coefficients are taken from slices of Pascal's triangle.
However, if N is a prime, it IS possible to factor the polynomial P_N[-x^2], which is the polynomial formed by substituting -x^2 in place of x.
Of course, if we construct a polynomial whose coefficients are all the numbers on the Nth horizontal row of Pascal's triangle, then the roots are all -1's, because the polynomial is just (1+x)^N. However, if you take every OTHER number from a horizontal row, you get an interesting result.
www.mathpages.com /home/kmath304.htm   (507 words)

 Ephemerides of the Largest Asteroids: README
Converting the ephemerides from ASCII to Chebyshev polynomials is recommended because the Chebyshev polynomials are much more compact than the tabular data and positions and velocities can quickly be interpolated from them for any arbitrary date.
The Chebyshev polynomial representation has the advantage that the position and velocity for any date within the valid time period of the ephemeris can be interpolated easily.
The disadvantage is that the values produced by the Chebyshev polynomials are not exactly the same as that of the tabulated ephemeris; however, the difference between the tabulated and the Chebyshev representations can be made arbitrarily small.
aa.usno.navy.mil /hilton/ephemerides/USNOAE98/README.HTM   (1960 words)

 Orthogonal Polynomials
Series of Chebyshev polynomials are often used in making numerical approximations to functions.
The name "Chebyshev" is a transliteration from the Cyrillic alphabet; several other spellings, such as "Tschebyscheff", are sometimes used.
Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
documents.wolfram.com /v4/MainBook/3.2.9.html   (240 words)

 Generalized Chebyshev polynomials
In this paper we study polynomials that are orthogonal with respect to a weight function which is zero on a set of positive measure.
These were initially introduced by Akhiezer as a generalization of the Chebyshev polynomials where the interval of orthogonality is [-1,α]∪[β,1].
The simpler case, where g = 1, is extensively dealt with and the reduction to the Chebyshev polynomials in the limiting situation, α→β, where the two intervals merge into one, is demonstrated.
stacks.iop.org /0305-4470/35/4651   (345 words)

 Chebyshev Polynomials   (Site not responding. Last check: )
By default, all of the Chebyshev polynomials up to the given degree are plotted, but the user can choose to plot only the Chebyshev polynomial of highest degree.
The equi-oscillation property of the Chebyshev polynomials is evident: the extreme function values are all equidistant from the horizontal axis and alternate in sign.
The Chebyshev points correspond to equally-spaced points on a circle, but their values on the horizontal axis are not equally spaced.
www.cse.uiuc.edu /eot/modules/interpolation/chbshvp   (148 words)

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