NationMaster - Encyclopedia: Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev^[1], 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. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted T_n and Chebyshev polynomials of the second kind which are denoted U_n. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Pafnuty Lvovich Chebyshev (Russian: ) (May 16 [O.S. May 4] 1821 â€“ December 8 [O.S. November 26] 1894)[1] was a Russian mathematician. ... 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 which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page â€” a list of pages that otherwise might share the same title. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ... The Lucas numbers are an integer sequence named after the mathematician FranÃ§ois Ã‰douard Anatole Lucas (1842â€“1891), who studied both that sequence and the closely related Fibonacci numbers. ... Transliteration is the practice of transcribing a word or text written in one writing system into another writing system. ...

The Chebyshev polynomials T_n or U_n are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence. For other senses of this word, see sequence (disambiguation). ... 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. ...

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 resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature. In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... In the mathematical subfield of numerical analysis Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... The red curve is the Runge function, the blue curve is a 5th-degree polynomial, while the green curve is a 9th-degree polynomial. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...

In the study of differential equations they arise as the solution to the Chebyshev differential equations Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... Chebyshevs equation is the second order linear differential equation where P is a real constant. ...

$(1-x^2),y'' - x,y' + n^2,y = 0 ,!$

and

$(1-x^2),y'' - 3x,y' + n(n+2),y = 0 ,!$

for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm-Liouville differential equation. In mathematics and its applications, a Sturm-Liouville problem, named after Charles Francois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a second-order linear differential equation of the form (1) often together with specified boundary values of y and dy/dx. ...

1 Definition
2 Explicit formulas
3 Properties
4 Examples
5 As a basis set
- 5.1 Partial sums
- 5.2 Polynomial in Chebyshev form
6 Spread polynomials
7 See also
8 References
9 External links

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

$T_0(x) = 1 ,!$

$T_1(x) = x ,!$

$T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). ,!$

One example of a generating function for T_n is In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

$sum_{n=0}^{infty}T_n(x) t^n = frac{1-tx}{1-2tx+t^2}. ,!$

The Chebyshev polynomials of the second kind are defined by the recurrence relation In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

$U_0(x) = 1 ,!$

$U_1(x) = 2x ,!$

$U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). ,!$

One example of a generating function for U_n is In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

$sum_{n=0}^{infty}U_n(x) t^n = frac{1}{1-2tx+t^2}. ,!$

Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

$T_n(x)=cos(n arccos x)=cosh(n,mathrm{arccosh},x) ,!$

whence:

$T_n(cos(theta))=cos(ntheta) ,!$

for n = 0, 1, 2, 3, ..., while the polynomials of the second kind satisfy:

$U_n(cos(theta)) = frac{sin((n+1)theta)}{sintheta} ,!$

which is structurally quite similar to the Dirichlet kernel. In mathematical analysis, the Dirichlet kernel is the collection of functions It is named after Johann Peter Gustav Lejeune Dirichlet. ...

That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable via the identity cos²(x) + sin²(x) = 1. de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

This identity is extremely useful in conjunction with the recursive generating formula inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle. Evaluating the first two Chebyshev polynomials:

$T_0(x)=cos 0x =1 ,!$

and:

$T_1(cos(x))=cos (x) ,!$

one can straightforwardly determine that:

$cos(2 theta)=2costheta costheta - cos(0 theta) = 2cos^{2},theta - 1 ,!$

$cos(3 theta)=2costheta cos(2theta) - costheta = 4cos^3,theta - 3costheta ,!$

and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting theta equal to zero, for which the cosine is unity), and one sees that 1 = 2 − 1 in the former expression and 1 = 4 − 3 in the latter.

An immediate corollary is the composition identity (or the "nesting property")

$T_n(T_m(x)) = T_{ncdot m}(x).,!$

Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to the Pell equation Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

$T_i^2 - (x^2-1) U_{i-1}^2 = 1 ,!$

in a ring R[x].^[2] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

$T_i + U_{i-1} sqrt{x^2-1} = (x + sqrt{x^2-1})^i. ,!$

Relation between Chebyshev polynomials of the first and second kind

The Chebyshev polynomials of the first and second kind are closely related by the following equations

$frac{d}{dx} , T_n(x) = n U_{n-1}(x) mbox{ , } n=1,ldots$

$T_n(x) = frac{1}{2} (U_n(x) - , U_{n-2}(x)).$

$T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x),$

$T_n(x) = U_n(x) - x , U_{n-1}(x).$

The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations

$2 T_n(x) = frac{1}{n+1}; frac{d}{dx} T_{n+1}(x) - frac{1}{n-1}; frac{d}{dx} T_{n-1}(x) mbox{ , }quad n=1,ldots$

This relationship is used in the Chebyshev spectral method of solving differential equations.

Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:

$T_0(x) = 1,!$

$U_{-1}(x) = 0,!$

$T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x),$

$U_n(x) = xU_{n-1}(x) + T_n(x),$

These can be derived from the trigonometric formulae; for example, if $scriptstyle x = cosvartheta$ , then

$begin{align} T_{n+1}(x) &= T_{n+1}(cos(vartheta)) &= cos((n + 1)vartheta) &= cos(nvartheta)cos(vartheta) - sin(nvartheta)sin(vartheta) &= T_n(cos(vartheta))cos(vartheta) - U_{n-1}(cos(vartheta))sin^2(vartheta) &= xT_n(x) - (1 - x^2)U_{n-1}(x). end{align}$

Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our U_n (the polynomial of degree n) with U_n+1 instead.

Explicit formulas

Different approaches to defining Chebyshev polynomials lead to different explicit formulas such as:

$T_n(x) = begin{cases} cos(narccos(x)), & x in [-1,1] cosh(n , mathrm{arccosh}(x)), & x ge 1 (-1)^n cosh(n , mathrm{arccosh}(-x)), & x le -1 end{cases} ,!$

$T_n(x)=frac{(x+sqrt{x^2-1})^n+(x-sqrt{x^2-1})^n}{2} = sum_{k=0}^{lfloor n/2rfloor} binom{n}{2k} (x^2-1)^k x^{n-2k}$

$U_n(x)=frac{(x+sqrt{x^2-1})^{n+1}-(x-sqrt{x^2-1})^{n+1}}{2sqrt{x^2-1}} = sum_{k=0}^{lfloor n/2rfloor} binom{n+1}{2k+1} (x^2-1)^k x^{n-2k}$

$T_n(x) = 1+n^2 {(x-1)} prod_{k=1}^{n-1} left( { 1+{{{x-1}}over 2 sin^2left({k pi over n}right)}}right)$ (due to M. Hovdan)

Properties

Orthogonality

Both the T_n and the U_n form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner...

$frac{1}{sqrt{1-x^2}}, ,!$

on the interval [−1,1], i.e. we have:

$int_{-1}^1 T_n(x)T_m(x),frac{dx}{sqrt{1-x^2}}=left{ begin{matrix} 0 &: nne m~~~~~ pi &: n=m=0 pi/2 &: n=mne 0 end{matrix} right. ,!$

This can be proven by letting x= cos(θ) and using the identity T_n (cos(θ))=cos(nθ). Similarly, the polynomials of the second kind are orthogonal with respect to the weight

$sqrt{1-x^2} ,!$

on the interval [−1,1], i.e. we have:

$int_{-1}^1 U_n(x)U_m(x)sqrt{1-x^2},dx = begin{cases} 0 &: nne m, pi/2 &: n=m. end{cases} ,!$

(Note that the weight $sqrt{1-x^2} ,!$ is, to within a normalizing constant, the density of the Wigner semicircle distribution). The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [âˆ’R, R] 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...

Minimal ∞-norm

For any given $1 le n$ , among the polynomials of degree $n$ with leading coefficient 1,

$f(x) = frac1{2^{n-1}}T_n(x)$

is the one of which the maximal absolute value on the interval $[ - 1,1]$ is minimal.

This maximal absolute value is

$frac1{2^{n-1}}$

and $| f (x) |$ reaches this maximum exactly $n + 1$ times: in $- 1$ and $1$ and the other $n - 1$ extremal points of $f$ $I n s e r t f o r m u l a h e r e$ $I n s e r t f o r m u l a h e r e$

Differentiation and integration

The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it's easy to show that:

$frac{d T_n}{d x} = n U_{n - 1},$

$frac{d U_n}{d x} = frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1},$

$frac{d^2 T_n}{d x^2} = n frac{n T_n - x U_{n - 1}}{x^2 - 1} = n frac{(n + 1)T_n - U_n}{x^2 - 1}.,$

The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown (see proof) that: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. ...

$frac{d^2 T_n}{d x^2} Bigg|_{x = 1} !! = frac{n^4 - n^2}{3},$

$frac{d^2 T_n}{d x^2} Bigg|_{x = -1} !! = (-1)^n frac{n^4 - n^2}{3};$

indeed, the following, more general formula holds:

$frac{d^p T_n}{d x^p} Bigg|_{x = pm 1} !! = (pm 1)^{n+p}prod_{k=0}^{p-1}frac{n^2-k^2}{2k+1}.$

This latter result is of great use in the numerical solution of eigenvalue problems.

Concerning integration, the first derivative of the T_n implies that

$int U_n, dx = frac{T_{n + 1}}{n + 1},$

and the recurrence relation for the first kind polynomials involving derivatives establishes that

$int T_n, dx = frac{1}{2} left(frac{T_{n + 1}}{n + 1} - frac{T_{n - 1}}{n - 1}right) = frac{n T_{n + 1}}{n^2 - 1} - frac{x T_n}{n - 1}.,$

Roots and extrema

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1,1]. The roots are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that In the mathematical subfield of numerical analysis Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind. ...

$cosleft(frac{pi}{2},(2k+1)right)=0$

one can easily prove that the roots of T_n are

$x_k = cosleft(frac{pi}{2},frac{2k-1}{n}right) mbox{ , } k=1,ldots,n.$

Similarly, the roots of U_n are

$x_k = cosleft(frac{k}{n+1}piright) mbox{ , } k=1,ldots,n.$

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by: Local and global maxima and minima for cos(3Ï€x)/x, 0. ... In differential topology, a critical value of a differentiable map between differentiable manifolds is the image of a critical point. ... In mathematics, a dessin denfant (French for a childs drawing) is a connected graph with a cyclic ordering of edges at each vertex, and each vertex being colored black or white and with no edge having endpoints of the same color. ...

$T_n(1) = 1,$

$T_n(-1) = (-1)^n,$

$U_n(1) = n + 1,$

$U_n(-1) = (n + 1)(-1)^n,$

Other properties

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials. In mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of orthogonal polynomials. ... In mathematics, Jacobi polynomials are a class of orthogonal polynomials. ...

For every nonnegative integer n, T_n(x) and U_n(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. ...

The leading coefficient of T_n is 2^n − 1 if 1 ≤ n, but 1 if 0 = n.

T_n are a special case of Lissajous curves with frequency ratio to equal to n. Lissajous figure on an oscilloscope- the shape of the ABC logo Lissajous figure in three dimensions In mathematics, a Lissajous curve (Lissajous figure or Bowditch curve) is the graph of the system of parametric equations which describes complex harmonic motion. ...

Examples

This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat T0, and T1, T2, T3, T4 and T5.

This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat T₀, and T₁, T₂, T₃, T₄ and T₅.

The first few Chebyshev polynomials of the first kind are Image File history File links Chebyshev. ...

$T_0(x) = 1 ,$

$T_1(x) = x ,$

$T_2(x) = 2x^2 - 1 ,$

$T_3(x) = 4x^3 - 3x ,$

$T_4(x) = 8x^4 - 8x^2 + 1 ,$

$T_5(x) = 16x^5 - 20x^3 + 5x ,$

$T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1 ,$

$T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x ,$

$T_8(x) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 ,$

$T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. ,$

This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat U0, and U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n + 1 and Un(−1) = (n + 1)(−1)n.

This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat U₀, and U₁, U₂, U₃, U₄ and U₅. Although not visible in the image, U_n(1) = n + 1 and U_n(−1) = (n + 1)(−1)ⁿ.

The first few Chebyshev polynomials of the second kind are Image File history File links Chebyshev2. ...

$U_0(x) = 1 ,$

$U_1(x) = 2x ,$

$U_2(x) = 4x^2 - 1 ,$

$U_3(x) = 8x^3 - 4x ,$

$U_4(x) = 16x^4 - 12x^2 + 1 ,$

$U_5(x) = 32x^5 - 32x^3 + 6x ,$

$U_6(x) = 64x^6 - 80x^4 + 24x^2 - 1 ,$

$U_7(x) = 128x^7 - 192x^5 + 80x^3 - 8x ,$

$U_8(x) = 256x^8 - 448 x^6 + 240 x^4 - 40 x^2 + 1 ,$

$U_9(x) = 512x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x. ,$

As a basis set

The non-smooth function (top)

y = - x 3 H ( - x)

, where

H

is the Heaviside step function, and (bottom) the 5^th partial sum of its Chebyshev expansion. The 7^th sum is indistinguishable from the original function at the resolution of the graph.

In the appropriate Sobolev space, the set of Chebyshev polynomials form a complete basis set, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:^[3] The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite Lp norm, for given p â‰¥ 1. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

$f(x) = sum_{n = 0}^infty a_n T_n(x).$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients a_n can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc that apply to Fourier series have a Chebyshev counterpart.^[3] These attributes include: The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

The Chebyshev polynomials form a complete orthogonal system.

The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases — as long as there are a finite number of discontinuities in f(x) and its derivatives.

At a discontinuity, the series will converge to the average of the right and left limits.

The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method^[3], often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem). In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In applied mathematics, Spectral methods are algorithms to solve certain kinds of partial differential equations numerically using some sort of Fast Fourier Transform. ...

Partial sums

The partial sums of

$f(x) = sum_{n = 0}^infty a_n T_n(x)$

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients a_n are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation. In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... In applied mathematics, Spectral methods are algorithms to solve certain kinds of partial differential equations numerically using some sort of Fast Fourier Transform. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a collocation method is a method for the numerical solution of ordinary differential equation and partial differential equations and integral equations. ... For other uses, see Interpolation (disambiguation). ...

As an interpolant, the N coefficients of the (N − 1)^th partial sum are usually obtained on the Chebyshev-Gauss-Lobatto^[4] points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by: The red curve is the Runge function, the blue curve is a 5th-degree polynomial, while the green curve is a 9th-degree polynomial. ...

$x_i = -cosleft(frac{i pi}{N - 1}right) ; qquad i = 0, 1, dots, N - 1.$

Polynomial in Chebyshev form

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial p(x) is of the form

$p(x) = sum_{n=0}^{N} a_n T_n(x)$

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm. In the mathematical subfield of numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) is a recursive method to evaluate polynomials in Chebyshev form. ...

Spread 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. In mathematics, the nth-degree spread polynomial Sn, for n = 0, 1, 2, ..., may be characterized by the trigonometric identity Although that is probably the simplest way to explain what spread polynomials are to those versed in well-known topics in mathematics, spread polynomials were introduced by Norman Wildberger for... Divine Proportions: Rational Trigonometry to Universal Geometry is a book by Dr. Norman Wildberger of The University of New South Wales, presenting the authors reformulation of trigonometry. ...

References

Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 22”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .

^ Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom parallelogrammes," Mémoires des Savants étrangers présentes à l'Academie de Saint-Pétersbourg, vol. 7, pages 539-586.
^ Jeroen Demeyer Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields, Ph.D. theses (2007), p.70.
^ ^a ^b ^c Chebyshev and Fourier Spectral Methods by John P. Boyd.
^ Chebyshev Interpolation: An Interactive Tour

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

External links

Eric W. Weisstein, Chebyshev Polynomial of the First Kind at MathWorld.
Module for Chebyshev Polynomials by John H. Mathews
Chebyshev Interpolation: An Interactive Tour, includes illustrative Java applet.
chebfun project, representing functions by automatic Chebyshev polynomial interpolation in MATLAB.

Categories: Polynomials | Special hypergeometric functions | Orthogonal polynomials | Numerical analysis | Approximation theory

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... A Java applet is an applet delivered in the form of Java bytecode. ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ...

Results from FactBites:

Chebyshev Polynomials (435 words)
	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.

Chebyshev polynomials - Wikipedia, the free encyclopedia (948 words)
	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.

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