Chebyshev polynomials explained

Chebyshev polynomials should not be confused with discrete Chebyshev polynomials.

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as

T _{n(x)} and U _{n(x)} . They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind

T _{n} are defined by:$$T\_n(\backslash cos\; \backslash theta)\; =\; \backslash cos(n\backslash theta).$$

Similarly, the Chebyshev polynomials of the second kind

U _{n} are defined by:$$U\_n(\backslash cos\; \backslash theta)\; \backslash sin\; \backslash theta\; =\; \backslash sin\backslash big((n\; +\; 1)\backslash theta\backslash big).$$

That these expressions define polynomials in

\cos\theta may not be obvious at first sight but follows by rewriting \cos(n\theta) and \sin ( (n +1 )\theta ) using de Moivre's formula or by using the angle sum formulas for \cos and \sin repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain T _{2(\cos\theta)} = \cos( 2 \theta) = 2 \cos ^{2\theta-1} and U _{1(\cos\theta)\sin\theta} = \sin( 2 \theta) = 2 \cos\theta\sin\theta , which are respectively a polynomial in \cos\theta and a polynomial in \cos\theta multiplied by \sin\theta . Hence T _{2(x)} = 2 x ^{2} - 1 and U _{1(x)} = 2 x .

An important and convenient property of the is that they are orthogonal with respect to the inner product:$$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_^1\; f(x)\; \backslash ,\; g(x)\; \backslash ,\; \backslash frac,$$and are orthogonal with respect to another, analogous inner product, given below.

The Chebyshev polynomials are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties.^[1]

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;^[2] the roots of, which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

These polynomials were named after Pafnuty Chebyshev.^[3] The letter is used because of the alternative transliterations of the name Chebyshev as French: Tchebycheff, French: Tchebyshev (French) or German: Tschebyschow (German).

Definitions

Recurrence definition

The Chebyshev polynomials of the first kind are obtained from the recurrence relation:The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size

k x k :

The ordinary generating function for is:$$\backslash sum\_^T\_n(x)\backslash ,t^n\; =\; \backslash frac.$$There are several other generating functions for the Chebyshev polynomials; the exponential generating function is:$$\backslash sum\_^T\_n(x)\backslash frac\; =\; \backslash frac\backslash !\backslash left(e^\; +\; e^\backslash right)=\; e^\backslash cosh\backslash left(t\backslash sqrt\backslash right).$$

The generating function relevant for 2-dimensional potential theory and multipole expansion is:$$\backslash sum\backslash limits\_^T\_(x)\backslash ,\backslash frac\; =\; \backslash ln\backslash left(\backslash frac\backslash right).$$

The Chebyshev polynomials of the second kind are defined by the recurrence relation:Notice that the two sets of recurrence relations are identical, except for

T _{1(x)} = x vs. The ordinary generating function for is:$$\backslash sum\_^U\_n(x)\backslash ,t^n\; =\; \backslash frac,$$and the exponential generating function is:$$\backslash sum\_^U\_n(x)\backslash frac\; =\; e^\backslash !\backslash left(\backslash !\backslash cosh\backslash left(t\backslash sqrt\backslash right)\; +\; \backslash frac\; \backslash sinh\backslash left(t\backslash sqrt\backslash right)\backslash !\backslash right).$$

Trigonometric definition

As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying:or, in other words, as the unique polynomials satisfying:$$T\_n(\backslash cos\backslash theta)\; =\; \backslash cos(n\backslash theta)$$for .

The polynomials of the second kind satisfy:$$U\_(\backslash cos\backslash theta)\; \backslash sin\backslash theta\; =\; \backslash sin(n\backslash theta),$$or$$U\_n(\backslash cos\backslash theta)\; =\; \backslash frac,$$which is structurally quite similar to the Dirichlet kernel :$$D\_n(x)\; =\; \backslash frac\; =\; U\_\backslash !\backslash !\backslash left(\backslash cos\; \backslash frac\backslash right).$$(The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)

An equivalent way to state this is via exponentiation of a complex number: given a complex number with absolute value of one:$$z^n\; =\; T\_n(a)\; +\; ib\; U\_(a).$$Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.^[4]

That is an th-degree polynomial in can be seen by observing that is the real part of one side of de Moivre's formula:$$\backslash cos\; n\; \backslash theta\; +\; i\; \backslash sin\; n\; \backslash theta\; =\; (\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)^n.$$The real part of the other side is a polynomial in and, in which all powers of are even and thus replaceable through the identity . By the same reasoning, is the imaginary part of the polynomial, in which all powers of are odd and thus, if one factor of is factored out, the remaining factors can be replaced to create a st-degree polynomial in .

Commuting polynomials definition

Chebyshev polynomials can also be characterized by the following theorem:^[5]

If

F _{n(x)} is a family of monic polynomials with coefficients in a field of characteristic 0 such that \deg F _{n(x)} = n and F _{m(F} _{n(x))} = F _{n(F} _{m(x))} for all m and n , then, up to a simple change of variables, either F _{n(x)} = x ^{n} for all n or F _{n(x)} = 2 ⋅ T _{n(x/2)} for all n .

Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to the Pell equation:$$T\_n(x)^2\; -\; \backslash left(x^2\; -\; 1\backslash right)\; U\_(x)^2\; =\; 1$$in a ring .^[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:$$T\_n(x)\; +\; U\_(x)\backslash ,\backslash sqrt\; =\; \backslash left(x\; +\; \backslash sqrt\backslash right)^n~.$$

Relations between the two kinds of Chebyshev polynomials

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences and with parameters and :It follows that they also satisfy a pair of mutual recurrence equations:^[7]

The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give:$$T\_n(x)\; =\; \backslash frac\; \backslash big(U\_n(x)\; -\; U\_(x)\backslash big).$$

Using this formula iteratively gives the sum formula:while replacing

U _{n(x)} and U _{n-2} (x) using the derivative formula for T _{n(x)} gives the recurrence relationship for the derivative of T _{n} :

$$2\backslash ,T\_n(x)\; =\; \backslash frac\backslash ,\; \backslash frac\backslash ,\; T\_(x)\; -\; \backslash frac\backslash ,\backslash frac\backslash ,\; T\_(x),\; \backslash qquad\; n=2,3,\backslash ldots$$

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

Turán's inequalities for the Chebyshev polynomials are:

The integral relations arewhere integrals are considered as principal value.

Explicit expressions

Different approaches to defining Chebyshev polynomials lead to different explicit expressions. The trigonometric definition gives an explicit formula as follows:From this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold:$$T\_0(\backslash cos\backslash theta)\; =\; \backslash cos(0\backslash theta)\; =\; 1$$and$$T\_1(\backslash cos\backslash theta)\; =\; \backslash cos\backslash theta,$$and that the product-to-sum identity holds:$$2\backslash cos\; n\backslash theta\backslash cos\backslash theta\; =\; \backslash cos\; \backslash lbrack\; (n+1)\backslash theta\; \backslash rbrack\; +\backslash cos\backslash lbrack\; (n-1)\backslash theta\backslash rbrack.$$

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression:$$T\_n(x)\; =\; \backslash dfrac\; \backslash bigg(\backslash Big(x-\backslash sqrt\; \backslash Big)^n\; +\; \backslash Big(x+\backslash sqrt\; \backslash Big)^n\; \backslash bigg)\; \backslash qquad\; \backslash text~\; x\; \backslash in\; \backslash mathbb$$$$T\_n(x)\; =\; \backslash dfrac\; \backslash bigg(\backslash Big(x-\backslash sqrt\; \backslash Big)^n\; +\; \backslash Big(x-\backslash sqrt\; \backslash Big)^\; \backslash bigg)\; \backslash qquad\; \backslash text~\; x\; \backslash in\; \backslash mathbb$$The two are equivalent because

(x + \sqrt{x ^{2} - 1 })(x - \sqrt{x ^{2} - 1 }) = 1 .

An explicit form of the Chebyshev polynomial in terms of monomials follows from de Moivre's formula:$$T\_n(\backslash cos(\backslash theta))\; =\; \backslash operatorname(\backslash cos\; n\; \backslash theta\; +\; i\; \backslash sin\; n\; \backslash theta)\; =\; \backslash operatorname((\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)^n),$$where denotes the real part of a complex number. Expanding the formula, one gets:$$(\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)^n\; =\; \backslash sum\backslash limits\_^n\; \backslash binom\; i^j\; \backslash sin^j\; \backslash theta\; \backslash cos^\; \backslash theta.$$The real part of the expression is obtained from summands corresponding to even indices. Noting

i ^{2j} = ( -1 ) ^{j} and \sin ^{2j} \theta = ( 1- \cos ^{2} \theta) ^{j} , one gets the explicit formula:$$\backslash cos\; n\; \backslash theta\; =\; \backslash sum\backslash limits\_^\; \backslash binom\; (\backslash cos^2\; \backslash theta\; -\; 1)^j\; \backslash cos^\; \backslash theta,$$which in turn means that:$$T\_n(x)\; =\; \backslash sum\backslash limits\_^\; \backslash binom\; (x^2-1)^j\; x^.$$This can be written as a hypergeometric function:with inverse:^{[8] [9]}

$$x^n\; =\; 2^\backslash mathop^n\_\; \backslash !\backslash !\backslash binom\backslash !\backslash ;T\_j(x),$$where the prime at the summation symbol indicates that the contribution of needs to be halved if it appears.

A related expression for as a sum of monomials with binomial coefficients and powers of two is$$T\_\backslash left(x\backslash right)\; =\; \backslash sum\backslash limits\_^\backslash left(-1\backslash right)^\backslash left(\backslash binom\; +\; \backslash binom\backslash right)\backslash cdot\; 2^\backslash cdot\; x^.$$

Similarly, can be expressed in terms of hypergeometric functions:

Properties

Symmetry

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of . Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of .

Roots and extrema

A Chebyshev polynomial of either kind with degree has different simple roots, called Chebyshev roots, in the interval . The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that:$$\backslash cos\backslash left((2k+1)\backslash frac\backslash right)=0$$one can show that the roots of are:$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\backslash right),\backslash quad\; k=0,\backslash ldots,n-1.$$Similarly, the roots of are:$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\backslash pi\backslash right),\backslash quad\; k=1,\backslash ldots,n.$$The extrema of on the interval are located at:$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\backslash pi\backslash right),\backslash quad\; k=0,\backslash ldots,n.$$

One unique property of the Chebyshev polynomials of the first kind is that on the interval 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:

The extrema of

T _{n(x)} on the interval -1 \leq x \leq 1 where n> 0 are located at n +1 values of x . They are \pm 1 , or \cos\left( (2 \pi k)/(d) \right) where d > 2 , d | 2 n , 0 < k < d/ 2 and (k, d) = 1 , i.e., k and d are relatively prime numbers.

Specifically,^{[10] [11]} when

n is even: T _{n(x)} = 1 if x = \pm 1 , or d > 2 and 2 n/d is even. There are n/ 2 + 1 such values of x .T _{n(x)} = -1 if d > 2 and 2 n/d is odd. There are n/ 2 such values of x .

When

n is odd: T _{n(x)} = 1 if x = 1 , or d > 2 and 2 n/d is even. There are (n +1 )/ 2 such values of x .T _{n(x)} = -1 if x = -1 , or d > 2 and 2 n/d is odd. There are (n +1 )/ 2 such values of x .

This result has been generalized to solutions of

U _{n(x)} \pm 1 = 0 ,^[11] and to V _{n(x)} \pm 1 = 0 and W _{n(x)} \pm 1 = 0 for Chebyshev polynomials of the third and fourth kinds, respectively.^[12]

Differentiation and integration

The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:

The last two formulas can be numerically troublesome due to the division by zero (indeterminate form, specifically) at and . By L'Hôpital's rule:

More generally,$$\backslash left.\backslash frac\; \backslash right|\_\; \backslash !\backslash !\; =\; (\backslash pm\; 1)^\backslash prod\_^\backslash frac~,$$which is of great use in the numerical solution of eigenvalue problems.

Also, we have:$$\backslash frac\backslash ,T\_n(x)\; =\; 2^p\backslash ,n\backslash mathop\_\backslash binom\backslash frac\backslash ,T\_k(x),~\backslash qquad\; p\; \backslash ge\; 1,$$where the prime at the summation symbols means that the term contributed by is to be halved, if it appears.

Concerning integration, the first derivative of the implies that:$$\backslash int\; U\_n\backslash ,\; \backslash mathrmx\; =\; \backslash frac$$and the recurrence relation for the first kind polynomials involving derivatives establishes that for :$$\backslash int\; T\_n\backslash ,\; \backslash mathrmx\; =\; \backslash frac\backslash ,\backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; =\; \backslash frac\; -\; \backslash frac.$$

The last formula can be further manipulated to express the integral of as a function of Chebyshev polynomials of the first kind only:

Furthermore, we have:

Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation:$$T\_m(x)\backslash ,T\_n(x)\; =\; \backslash tfrac\backslash !\backslash left(T\_(x)\; +\; T\_$$

(x)\right)\!,\qquad \forall m,n \ge 0,which is easily proved from the product-to-sum formula for the cosine:$$2\; \backslash cos\; \backslash alpha\; \backslash ,\; \backslash cos\; \backslash beta\; =\; \backslash cos\; (\backslash alpha\; +\; \backslash beta)\; +\; \backslash cos\; (\backslash alpha\; -\; \backslash beta).$$For this results in the already known recurrence formula, just arranged differently, and with it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

The polynomials of the second kind satisfy the similar relation:(with the definition by convention). They also satisfy:$$U\_m(x)\backslash ,U\_n(x)\; =\; \backslash sum\_^n\backslash ,U\_(x)\; =\; \backslash sum\_\backslash underset^\; U\_p(x)~.$$for .For this recurrence reduces to:$$U\_(x)\; =\; U\_2(x)\backslash ,U\_m(x)\; -\; U\_m(x)\; -\; U\_(x)\; =\; U\_m(x)\backslash ,\backslash big(U\_2(x)\; -\; 1\backslash big)\; -\; U\_(x)~,$$which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether starts with 2 or 3.

Composition and divisibility properties

The trigonometric definitions of and imply the composition or nesting properties:For the order of composition may be reversed, making the family of polynomial functions a commutative semigroup under composition.

Since is divisible by if is odd, it follows that is divisible by if is odd. Furthermore, is divisible by, and in the case that is even, divisible by .

Orthogonality

Both and form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight:$$\backslash frac,$$on the interval, i.e. we have:

This can be proven by letting and using the defining identity .

Similarly, the polynomials of the second kind are orthogonal with respect to the weight:$$\backslash sqrt$$on the interval, i.e. we have:

(The measure is, to within a normalizing constant, the Wigner semicircle distribution.)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the Chebyshev differential equations:$$\backslash begin(1\; -\; x^2)T\_n$$ - xT_n' + n^2 T_n &= 0, \\[1ex](1 - x^2)U_n - 3xU_n' + n(n + 2) U_n &= 0,\endwhich are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)

The also satisfy a discrete orthogonality condition:where is any integer greater than, and the are the Chebyshev nodes (see above) of :$$x\_k\; =\; \backslash cos\backslash left(\backslash pi\backslash ,\backslash frac\backslash right)\; \backslash quad\; ~\backslash text~\; k\; =\; 0,\; 1,\; \backslash dots,\; N-1.$$

For the polynomials of the second kind and any integer with the same Chebyshev nodes, there are similar sums:and without the weight function:

For any integer, based on the zeros of :$$y\_k\; =\; \backslash cos\backslash left(\backslash pi\backslash ,\backslash frac\backslash right)\; \backslash quad\; ~\backslash text~\; k=0,\; 1,\; \backslash dots,\; N-1,$$one can get the sum:and again without the weight function:

Minimal -norm

For any given, among the polynomials of degree with leading coefficient 1 (monic polynomials):$$f(x)\; =\; \backslash frac\backslash ,T\_n(x)$$is the one of which the maximal absolute value on the interval is minimal.

This maximal absolute value is:$$\backslash frac1$$and reaches this maximum exactly times at:$$x\; =\; \backslash cos\; \backslash frac\backslash quad\backslash text0\; \backslash le\; k\; \backslash le\; n.$$

Remark

By the equioscillation theorem, among all the polynomials of degree, the polynomial minimizes on if and only if there are points such that .

Of course, the null polynomial on the interval can be approximated by itself and minimizes the -norm.

Above, however, reaches its maximum only times because we are searching for the best polynomial of degree (therefore the theorem evoked previously cannot be used).

Chebyshev polynomials as special cases of more general polynomial families

C ^{( λ )} _{n} (x) , which themselves are a special case of the Jacobi polynomials P ^{(\alpha,\beta)} _{n} (x) :

Chebyshev polynomials are also a special case of Dickson polynomials:$$D\_n(2x\backslash alpha,\backslash alpha^2)=\; 2\backslash alpha^T\_n(x)\; \backslash ,$$ $$E\_n(2x\backslash alpha,\backslash alpha^2)=\; \backslash alpha^U\_n(x).\; \backslash ,$$In particular, when

\alpha = \tfrac{ 1 }{ 2 } , they are related by D _{n(x,\tfrac{1}{4})} = 2 ^{1-n} T _{n(x)} and E _{n(x,\tfrac{1}{4})} = 2 ^{-n} U _{n(x)} .

Other properties

The curves given by, or equivalently, by the parametric equations,, are a special case of Lissajous curves with frequency ratio equal to .

Similar to the formula:$$T\_n(\backslash cos\backslash theta)\; =\; \backslash cos(n\backslash theta),$$we have the analogous formula:$$T\_(\backslash sin\backslash theta)\; =\; (-1)^n\; \backslash sin\backslash left(\backslash left(2n+1\backslash right)\backslash theta\backslash right).$$

For :$$T\_n\backslash !\backslash left(\backslash frac\backslash right)\; =\; \backslash frac$$and:$$x^n\; =\; T\_n\backslash !\; \backslash left(\backslash frac\backslash right)+\; \backslash frac\backslash \; U\_\backslash !\backslash left(\backslash frac\backslash right),$$which follows from the fact that this holds by definition for .

There are relations between Legendre polynomials and Chebyshev polynomials

\sum ^{n} _{k =0} P _{k} \left(x\right)T _{n-k} \left(x\right) = \left(n +1 \right)P _{n} \left(x\right) \sum ^{n} _{k =0} P _{k} \left(x\right)P _{n-k} \left(x\right) = U _{n} \left(x\right)

These identities can be proven using generating functions and discrete convolution

Examples

First kind

The first few Chebyshev polynomials of the first kind are

Second kind

The first few Chebyshev polynomials of the second kind are

As a basis set

In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on, be expressed via the expansion:^[13] $$f(x)\; =\; \backslash sum\_^\backslash infty\; a\_n\; T\_n(x).$$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.

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.^[13] These attributes include:

• The Chebyshev polynomials form a complete orthogonal system.
• The Chebyshev series converges to 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 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,^[13] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Example 1

Consider the Chebyshev expansion of . One can express:$$\backslash log(1+x)\; =\; \backslash sum\_^\backslash infty\; a\_n\; T\_n(x)~.$$

One can find the coefficients either through the application of an inner product or by the discrete orthogonality condition. For the inner product:$$\backslash int\_^\backslash ,\backslash frac\backslash ,\backslash mathrmx\; =\; \backslash sum\_^a\_n\backslash int\_^\backslash frac\backslash ,\backslash mathrmx,$$which gives:

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for approximate coefficients:$$a\_n\; \backslash approx\; \backslash frac\backslash ,\backslash sum\_^T\_n(x\_k)\backslash ,\backslash log(1+x\_k),$$where is the Kronecker delta function and the are the Gauss–Chebyshev zeros of :$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\backslash right)\; .$$For any, these approximate coefficients provide an exact approximation to the function at with a controlled error between those points. The exact coefficients are obtained with, thus representing the function exactly at all points in . The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients very efficiently through the discrete cosine transform:$$a\_n\; \backslash approx\; \backslash frac\backslash sum\_^\backslash cos\backslash left(\backslash frac\backslash right)\backslash log(1+x\_k).$$

Example 2

To provide another example:

Partial sums

The partial sums of:$$f(x)\; =\; \backslash sum\_^\backslash infty\; a\_n\; T\_n(x)$$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 are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation.

As an interpolant, the coefficients of the st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto^[14] 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:$$x\_k\; =\; -\backslash cos\backslash left(\backslash frac\backslash right);\; \backslash qquad\; k\; =\; 0,\; 1,\; \backslash dots,\; N\; -\; 1.$$

Polynomial in Chebyshev form

An arbitrary polynomial of degree can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial is of the form:$$p(x)\; =\; \backslash sum\_^N\; a\_n\; T\_n(x).$$

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

Families of polynomials related to Chebyshev polynomials

Polynomials denoted

C _{n(x)} and S _{n(x)} closely related to Chebyshev polynomials are sometimes used. They are defined by:$$C\_n(x)\; =\; 2T\_n\backslash left(\backslash frac\backslash right),\backslash qquad\; S\_n(x)\; =\; U\_n\backslash left(\backslash frac\backslash right)$$and satisfy:$$C\_n(x)\; =\; S\_n(x)\; -\; S\_(x).$$A. F. Horadam called the polynomials C _{n(x)} Vieta–Lucas polynomials and denoted them v _{n(x)} . He called the polynomials S _{n(x)} Vieta–Fibonacci polynomials and denoted them Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII.^[15] The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of i and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials and of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by:^[16] The Chebyshev polynomials of the third kind are defined as:$$V\_n(x)=\backslash frac=\backslash sqrt\backslash fracT\_\backslash left(\backslash sqrt\backslash frac\backslash right)$$and the Chebyshev polynomials of the fourth kind are defined as:$$W\_n(x)=\backslash frac=U\_\backslash left(\backslash sqrt\backslash frac\backslash right),$$where

\theta = \arccos x . In the airfoil literature V _{n(x)} and W _{n(x)} are denoted t _{n(x)} and u _{n(x)} . The polynomial families T _{n(x)} , U _{n(x)} , V _{n(x)} , and W _{n(x)} are orthogonal with respect to the weights:$$\backslash left(1-x^2\backslash right)^,\backslash quad\backslash left(1-x^2\backslash right)^,\backslash quad(1-x)^(1+x)^,\backslash quad(1+x)^(1-x)^$$and are proportional to Jacobi polynomials P ^{(\alpha,\beta)} _{n} (x) with:$$(\backslash alpha,\backslash beta)=\backslash left(-\backslash frac,-\backslash frac\backslash right),\backslash quad(\backslash alpha,\backslash beta)=\backslash left(\backslash frac,\backslash frac\backslash right),\backslash quad(\backslash alpha,\backslash beta)=\backslash left(-\backslash frac,\backslash frac\backslash right),\backslash quad(\backslash alpha,\backslash beta)=\backslash left(\backslash frac,-\backslash frac\backslash right).$$

All four families satisfy the recurrence

p _{n(x)=2xp} _{n-1} (x) - p _{n-2} (x) with p _{0(x)} = 1 , where p _{n} = T _{n} , U _{n} , V _{n} , or W _{n} , but they differ according to whether p _{1(x)} equals x , 2 x , 2 x -1 , or

Even order modified Chebyshev polynomials

Some applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications. Even order Chebyshev filter designs using equally terminated passive networks are an example of this.^[17] However, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect. Such modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials. Even order modified Chebyshev polynomials may be created from the Chebyshev nodes in the same manner as standard Chebyshev polynomials.$$P\_N\; =\; \backslash prod\_^N(x-C\_i)$$

where

P _{N} is an N-th order Chebyshev polynomialC _{i} is the i-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the even order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials.$$Pe\_N\; =\; \backslash prod\_^N(x-Ce\_i)$$

where

P e _{N} is an N-th order even order modified Chebyshev polynomialCe _{i} is the i-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the example above is

X ^{4-X} ^{2+.125} , which by inspection contains no roots of zero. Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of X ^{4-.828427X} ^{2} , which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

See also

• Chebyshev filter
• Chebyshev cube root
• Dickson polynomials
• Legendre polynomials
• Laguerre polynomials
• Hermite polynomials
• Minimal polynomial of 2cos(2pi/n)
• Romanovski polynomials
• Chebyshev rational functions
• Approximation theory
• Function approximation
• The Chebfun system
• Discrete Chebyshev transform
• Markov brothers' inequality
• Clenshaw algorithm

Sources

• • Dette . Holger . 1995 . A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials . Proceedings of the Edinburgh Mathematical Society . 38 . 2 . 343–355 . math/9406222 . 10.1017/S001309150001912X. 16703489 .
• David . Elliott . 1964 . The evaluation and estimation of the coefficients in the Chebyshev Series expansion of a function . Math. Comp. . 18 . 86 . 274–284 . 0166903 . 10.1090/S0025-5718-1964-0166903-7 . free.
• Eremenko . A. . Lempert . L. . 1994 . An Extremal Problem For Polynomials . 10.1090/S0002-9939-1994-1207536-1 . 1207536 . . 122 . 1 . 191–193 . free .
• M. A. . Hernandez . Chebyshev's approximation algorithms and applications . 2001 . Computers & Mathematics with Applications . 41 . 3–4 . 433–445 . 10.1016/s0898-1221(00)00286-8 . free .
• Book: J. C. . Mason . Rational Approximation and Interpolation . 1984 . Some properties and applications of Chebyshev polynomial and rational approximation . Lecture Notes in Mathematics . 1105 . 27–48 . 10.1007/BFb0072398 . free . 978-3-540-13899-0.
• Book: Mason . J. C. . Handscomb . D.C. . 2002 . [{{Google books|8FHf0P3to0UC|Chebyshev Polynomials|plainurl=yes}} Chebyshev Polynomials ]. Chapman and Hall/CRC . 978-1-4200-3611-4 . 10.1201/9781420036114.
• Richard J. . Mathar . 2006 . Chebyshev series expansion of inverse polynomials . Journal of Computational and Applied Mathematics . 196 . 2 . 596–607 . 10.1016/j.cam.2005.10.013 . free . math/0403344 . 2006JCoAM.196..596M. 16476052 .
  • Web site: Remes . Eugene . On an Extremal Property of Chebyshev Polynomials .
• Herbert E. . Salzer . 1976 . Converting interpolation series into Chebyshev series by recurrence formulas . Mathematics of Computation . 30. 134 . 295–302 . 10.1090/S0025-5718-1976-0395159-3 . free . 0395159.
• R.E. . Scraton . 1969 . The Solution of integral equations in Chebyshev series . Mathematics of Computation . 23 . 108 . 837–844 . 0260224 . 10.1090/S0025-5718-1969-0260224-4 . free.
• Lyle B. . Smith . 1966 . Computation of Chebyshev series coefficients . Algorithm 277 . Comm. ACM . 9 . 2 . 86–87 . 10.1145/365170.365195. 8876563 . free .

External links

• • • Web site: John H. . Mathews . 2003 . Module for Chebyshev polynomials . Course notes for Math 340 Numerical Analysis & Math 440 Advanced Numerical Analysis . California State University . Fullerton, CA . Department of Mathematics . dead . 2020-08-17 . dmy-all . https://web.archive.org/web/20070529221407/http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html . 2007-05-29.
• Web site: Chebyshev interpolation: An interactive tour . Mathematical Association of America (MAA) . none. – includes illustrative Java applet.
• Web site: Numerical computing with functions . The Chebfun Project .
• Web site: Is there an intuitive explanation for an extremal property of Chebyshev polynomials? . Math Overflow . Question 25534 .
• Web site: Chebyshev polynomial evaluation and the Chebyshev transform . Boost . Math .

Notes and References

1. Book: Rivlin, Theodore J. . 1974 . The Chebyshev Polynomials . 1st . Chapter  2, Extremal properties . 56–123 . Pure and Applied Mathematics . Wiley-Interscience [John Wiley & Sons] . New York-London-Sydney . 978-047172470-4.
2. Solution of systems of linear equations by minimized iterations . 1952 . 33 . Journal of Research of the National Bureau of Standards . 10.6028/jres.049.006 . 49 . 1 . Lanczos . C. . free .
3. Chebyshev polynomials were first presented in Chebyshev . P. L. . 1854 . Théorie des mécanismes connus sous le nom de parallélogrammes . fr . Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg . 7 . 539–586 .
4. Schaeffer . A. C. . 1941 . Inequalities of A. Markoff and S. Bernstein for polynomials and related functions . Bulletin of the American Mathematical Society . 47 . 8 . 565–579 . 10.1090/S0002-9904-1941-07510-5 . 0002-9904. free .
5. J. F. . Ritt . Joseph Ritt . 10.1090/S0002-9947-1922-1501189-9 . Prime and Composite Polynomials . Trans. Amer. Math. Soc. . 1922. 23 . 51–66 . free.
6. Jeroen . Demeyer . Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields . https://web.archive.org/web/20070702185523/https://cage.ugent.be/~jdemeyer/phd.pdf . 2007-07-02 . Ph.D. . 2007 . 70.
7. Book: Bateman . Harry . Harry Bateman . Bateman Manuscript Project . Erdélyi . Arthur . Arthur Erdélyi . Higher Transcendental Functions . . 1953 . McGraw-Hill . 1st . II . New York . , eq. (3), (4) . 53-5555 . Reprint: 1981. Melbourne, FL: Krieger. .
8. W. J. . Cody . A survey of practical rational and polynomial approximation of functions . 1970 . SIAM Review . 12 . 3 . 400–423 . 10.1137/1012082.
9. R. J. . Mathar . 2006 . math/0403344 . Chebyshev series expansion of inverse polynomials . J. Comput. Appl. Math. . 196 . 2 . 596–607 . 10.1016/j.cam.2005.10.013 . 2006JCoAM.196..596M. 16476052 .
10. Y. Z. . Gürtaş . Chebyshev Polynomials and the minimal polynomial of \cos ( 2 \pi/n) . 2017 . American Mathematical Monthly . 124 . 1 . 74–78 . 10.4169/amer.math.monthly.124.1.74. 125797961 .
11. D. A. . Wolfram . Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of \cos ( 2 \pi /d ) . 2022 . American Mathematical Monthly . 129 . 2 . 172–176 . 10.1080/00029890.2022.2005391. 245808448 .
12. D. A. . Wolfram . Factoring Chebyshev polynomials with minimal polynomials of \cos ( 2 \pi /d) . 2022 . Bulletin of the Australian Mathematical Society . 10.1017/S0004972722000235. 2106.14585 .
13. Book: Boyd, John P.. Chebyshev and Fourier Spectral Methods. 0-486-41183-4. second. 2001. Dover. 2009-03-19. https://web.archive.org/web/20100331183829/http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf. 2010-03-31. dead.
14. Web site: Chebyshev Interpolation: An Interactive Tour . 2016-06-02 . https://web.archive.org/web/20170318214311/http://www.scottsarra.org/chebyApprox/chebyshevApprox.html . 2017-03-18 . dead .
15. Book: Viète, François. Francisci Vietae Opera mathematica : in unum volumen congesta ac recognita / opera atque studio Francisci a Schooten. 1646. Bibliothèque nationale de France.
16. $$T\_n^*(x)\; =\; T\_n(2x-1),\backslash qquad\; U\_n^*(x)\; =\; U\_n(2x-1).$$

    When the argument of the Chebyshev polynomial satisfies the argument of the shifted Chebyshev polynomial satisfies . Similarly, one can define shifted polynomials for generic intervals .

    Around 1990 the terms "third-kind" and "fourth-kind" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name airfoil polynomials. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to Walter Gautschi, "in consultation with colleagues in the field of orthogonal polynomials."

17. Book: Saal, Rudolf . Handbook of Filter Design . Allgemeine Elektricitais-Gesellschaft . January 1979 . 3-87087-070-2 . 1st . Munich, Germany . 25, 26, 56–61, 116, 117 . English, German.