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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Chebyshev polynomials form a special class of orthogonal polynomials especially suited for approximating and polynomial interpolation of other functions. They are widely used in many areas of numerical analysis: uniform approximation, least-squares approximation, numerical solution of ordinary and partial differential equations (the so-called spectral or pseudospectral methods), and so on.

Chebyshev Equations

Pafnuty Chebyshev

Chebyshev differential equations (actually, there are four of them) were discovered in 1859 by the famous Russian mathematician Pafnuty Lvovich Chebyshev (1821--1894). All these equations are used in definitions of singular Sturm--Liouville problems that ask to find bounded (polynomial) solutions on the interval [−1, 1]. Since these differential equations have two regular singular points at x = ±1, Chebyshev equations have two linearly independent solutions, one of them is unbounded at singular points. It is possible to "explicitly" identify polynomial solutions of these equations that now bear Chebyshev name.

Although Chebyshev's equations do not occur often in physical sciences or engineering, solutions of Chebyshev's equation are of importance in numerical analysis such as approximations of solutions to partial differential equations, smoothing of data, etc. Two singular points x = ±1 divide the real axis ℝ into three intervals, (−∞, −1), (−1, 1), (1, ∞), and Chebyshev's equations can be solved in each of them, we consider these equations only within compact interval [−1, 1].

The Chebyshev polynomials are named after Pafnuty Chebyshev, who invented them in 1859 while solving a practical problem for converting horizontal movement of piston in stream engine into rotational one. Of course, contemporary rail road trains have nothing in common with old steam engines first used in 19 century. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebychov, Chebyshov; or Tchebychev, Tchebycheff (French transcriptions); or Tschebyschev, Tschebyschef, Tschebyscheff, Tschebyschow (German transcriptions). He also used two Russian spellings: Чебышёв and Чебышо́в.

There are four known Chebyshev polynomials, denoted by Tn(x), Un(x), Vn(x), and Wn(x). They all are eigenfunctions of the following singular Sturm--Liouville problems:

\begin{align*} \left( 1 - x^2 \right) y'' - x\, y' + \lambda\,y &=0 , \qquad \lambda = n^2 , \quad y= T_n (x) , \tag{T.1} \\ \left( 1 - x^2 \right) y'' - 3x\, y' + \lambda\,y &=0 , \qquad \lambda = n(n+2) , \quad y = U_n (x) , \tag{U.1} \\ \left( 1 - x^2 \right) y'' - \left( 2x -1 \right) y' + \lambda\,y &=0 , \qquad \lambda = n(n+1) , \quad y = V_n (x) , \tag{V.1} \\ \left( 1 - x^2 \right) y'' - \left( 2x +1 \right) y' + \lambda\,y &=0 , \qquad \lambda = n(n+1) , \quad y = W_n (x) . \tag{W.1} \end{align*}
These problems include the corresponding differential equation and a condition that they have a polynomial solution. The latter can be replaced by the condition:
\[ y(-1) < \infty , \qquad y(1) < \infty . \tag{2} \]

Chebyshev Polynomials & Fourier

Each of four kinds of Chebyshev polynomials form a complete orthogonal system in the Hilbert space 𝔏²'([−1,1], w(x) dx), where w(x) is a weight function specified for each kind:

\[ w_1 = \left( 1 - x^2 \right)^{-1/2} , \quad w_2 = \left( 1 - x^2 \right)^{1/2} , \quad w_3 = \sqrt{\frac{1+x}{1-x}} , \quad w_4 = \sqrt{\frac{1-x}{1+x}} , \]
respectively. The substitution x = cos(θ) defines a smooth, strictly decreasing bijection between (0, π) and (-1, 1). Its Jacobian is dx = −sin(θ) dθ and \( \displaystyle \quad \sin \theta = +\sqrt{1-x^2}.\quad \)

So any integral on [-1,1] can be rewritten as an integral on (0,π) with a weight coming from sin(θ), and conversely. This is the basic mechanism behind all the Chebyshev systems: they are trigonometric systems in disguise, transported from θ-space to x-space by x = cos(θ).

Let w(x) be a positive weight on (-1,1), and consider the weighted space ℌx = 𝔏²([−1,1], w). We want to relate it to a space of functions of θ. Under x = cosθ, we have

\[ \int_{-1}^1 \,|f(x)|^2 \,w(x)\,{\text d}x = \int_0^{\pi } \,|f(\cos \theta )|^2 \,w(\cos \theta )\, \sin \theta \,{\text d}\theta . \]
Hence, if we define x = cosθ, then the map
\[ (U\,f)(\theta ):=f(\cos \theta ) \]
is an isometry from ℌx into ℌθ:
\[ \| U\,f\| _{\theta}^2 = \int_0^{\pi} \, |f(\cos \theta )|^2 \,\mu (\theta )\,{\text d}\theta = \int _{-1}^1 \,|f(x)|^2 \,w(x)\,{text d}x =\| f\| _{x}^2 . \]
Thus, the Hilbert space 𝔏²([-1,1], w(x) dx) is unitarily equivalent to 𝔏²([0,π], w(cosθ) sinθ dθ). Because x ↦ cos−1x is a bijection between (-1,1) and (0,π), this isometry is onto: every g ∈ ℌθ can be written as g(θ) = f(cosθ) for a unique f ∈ ℌx. Thus, U is unitary:
\[ U\ : \ ℌ_x \rightarrow ℌ_{\theta }\quad \mathrm{is\ a\ unitary\ isomorphism.} \]
This makes the precise sense in which “working in θ” is not just a trick, but an exact equivalence of Hilbert spaces.

Suppose { ϕₙ } is an orthogonal system in ℌθ. Define

\[ \psi _n(x):=\phi _n(\arccos x),\qquad x\in (-1,1). \]
Then ψₙ = U−1ϕₙ, and orthogonality is preserved:
\[ \langle \psi _m ,\psi _n\rangle_{x} = \langle U\psi_m ,U\psi_n \rangle_{\theta} = \langle \phi_m , \phi_n \rangle_{\theta} . \]
So { ψₙ } is orthogonal in ℌx. More importantly, completeness is preserved. If the linear span of { ϕₙ } is dense in ℌθ, then for any f ∈ ℌx,
\[ U\,f\in ℌ_{\theta} \]
can be approximated in norm by finite linear combinations of ϕₙ. Applying U−1, we see that f can be approximated in norm by finite linear combinations of ψₙ. Symbolically:
\[ \overline{\mathrm{span}\{ \phi _n\} } = ℌ_{\theta }\quad \Longleftrightarrow \quad \overline{\mathrm{span}\{ \psi _n\} } = ℌ_x . \]
Hemce, completeness is not something to be‑proved in x-space; it is inherited from the trigonometric system via the unitary map.

For Chebyshev systems, the weight w(x) and corresponding Chebyshev polynomials upon trigonometric transformation x = cos(θ) are as follows:

\begin{align*} w_1 (\cos\theta ) &= \frac{1}{\sin\theta} \quad \Longrightarrow \quad T_n (\cos\theta ) = \cos\left( n\,\theta \right) , \\ w_2 (\cos\theta ) &= \sin\theta \quad \Longrightarrow \quad U_n (\cos\theta ) = \frac{\sin\left( (n+1)\,\theta \right)}{\sin\theta} , \\ w_3 (\cos\theta ) &= \frac{1 + \cos\theta}{\sin\theta} \quad \Longrightarrow \quad V_n (\cos\theta ) = \frac{\cos\left( \frac{2n+1}{2}\,\theta \right)}{\cos\left( \frac{\theta}{2} \right)} , \\ w_4 (\cos\theta ) &= \frac{1 - \cos\theta}{\sin\theta} \quad \Longrightarrow \quad W_n (\cos\theta ) =\frac{\sin\left( \frac{2n+1}{2}\,\theta \right)}{\sin\left( \frac{\theta}{2} \right)} c \end{align*} The role of x = cosθ in completeness proofs is not cosmetic: it is the bridge that turns polynomial questions on [-1,1] into trigonometric questions on (0,π), where the structure is already fully understood. Since Chebyshev polynomials Tₙ(x) and Uₙ(x) are expressed through cosine and sime functions upon substitution x = cosθ, we conclude that these polynomials form complete sets in the x-space, 𝔏²([−1,1], w).

Chebyshev III and IV sit symmetrically:

\[ V_n (x) = T_n(x)-(1-x)\, U_{n-1}(x),\qquad W_n (x) = T_n(x)+(1+x)\, U_{n-1}(x) . \]
Therefore, these two “exotic” Chebyshev families in θ-space are just regular Fourier trigonometric polynomials.

Chebyshev Polynomials of the First Kind

Chebyshev differential equation of the first kind

\[ \left( 1 - x^2 \right) y'' - x\, y' + n^2 y =0 , \qquad n=0,1,2,3,\ldots , \tag{T.3} \]
can be explicitly solved by substitution
\[ x = \cos \theta \qquad \Longrightarrow \qquad \frac{{\text d}x}{{\text d} \theta} = - \sin \theta . \]
Then in new variable, Chebyshev's equation (T.3) becomes
\[ \frac{{\text d}^2 y}{{\text d} \theta^2} + n^2 y = 0 \]
because
\[ y' = \frac{{\text d}y}{{\text d}x} = - \frac{1}{\sin \theta}\, \frac{{\text d}y}{{\text d}\theta} , \qquad y'' = \frac{{\text d}^2 y}{{\text d} x^2} = \frac{1}{\sin \theta} \, \frac{{\text d}}{{\text d}\theta} \left( \frac{1}{\sin \theta}\, \frac{{\text d}y}{{\text d}\theta} \right) . \]
Although the differential equation \( y_{\theta \theta} + n^2 y = 0 \) has two linearly independent solutions y1 = cos(nθ) and y2 = sin(nθ), only former provides a polynomial solution to the Chebyshev equation (T.1). This polynomial solution is known as the Chebyshev polynomial of the first kind, denoted by
\[ T_n (x) = \cos \left( n\,\arccos x \right) , \qquad n = 0,1,2,3,\ldots ; \tag{T.4} \]
and determined by Rodrigues' formula
\[ T_n (x) = \frac{(-1)^n 2^n n!}{(2n)!} \,\sqrt{1-x^2} \,\frac{{\text d}^n}{{\text d}x^n} \left( 1 - x^2 \right)^{n-1/2} , \qquad n=0,1,2,\ldots . \tag{T.5} \]
The name "Rodrigues formula" was introduced by a German mathematician Eduard Heine (1821--1881) in 1878, after Charles Hermite pointed out in 1865 that a French banker, mathematician, and social reformer Rodrigues (1795--1851) was the first to discover it. Chebyshev polynomials can be defined for arbitrary real argument:
\begin{align*} T_n (x) &= \frac{1}{2} \left[ \left( x + \sqrt{x^2 -1} \right)^n + \left( x - \sqrt{x^2 -1} \right)^n \right] \qquad \mbox{for} \quad |x| \ge 1, \\ &= \begin{cases} \cos \left( n\,\mbox{arccos} x \right) , & \quad \mbox{if $|x| \le 1$}, \\ \cosh \left( n\,\mbox{arccosh} x \right) , & \quad \mbox{if $x\ge 1$}, \\ (-1)^n \cosh \left( n\,\mbox{arccosh} (-x) \right) , & \quad \mbox{if $x\le -1$}; \end{cases} \\ &= x^n \sum_{k=0}^{\lfloor n/2 \rfloor} \,\binom{n}{2k} \left( 1- x^{-2} \right)^k . \end{align*}

Another option to define the Chebyshev polynomials directly through explicit formula:

CT[n_, x_] := Module[{k, tmp, tab}, If[n == 0, Return[1]];
If[n == 1, Return[x]];
tmp = (2*x)^n/2;
tab = Table[
tmp = -tmp/4/k*(n - 2*k + 2)*(n - 2*k + 1)/x^2/(n - k), {k, 1, Floor[n/2]}];
(2*x)^n/2 + Apply[Plus, tab]]

Chebyshev Polynomials of the Second Kind

Chebyshev differential equation of the second kind
\[ \left( 1 - x^2 \right) y'' - 3x\, y' + n\left( n+2 \right) y =0 , \qquad n=0,1,2,3,\ldots , \tag{U.3} \]
also can be solved by substitution \( x= \cos t \) or \( x= \cosh t \) depending whether |x| is less than 1 or greater than 1. Chebyshev's polynomials of the second kind satisfy the identity:
\[ U_n \left( \cos \theta \right) = \frac{\sin (n+1)\theta}{\sin \theta} , \qquad \theta \in [0, \pi ], \quad n=0,1,2,\ldots . \tag{U.4} \]
They can be determined by Rodrigues' formula
\[ U_n (x) = \frac{(-1)^n 2^n (n+1)!}{(2n+1)!} \,\frac{1}{\sqrt{1-x^2}} \,\frac{{\text d}^n}{{\text d}x^n} \left( 1 - x^2 \right)^{n+1/2} , \qquad n=0,1,2,\ldots . \tag{U.5} \]
In general, we have
\begin{align*} U_n (x) &= \frac{1}{2\,\sqrt{x^2 -1}} \left[ \left( x + \sqrt{x^2 -1} \right)^{n+1} - \left( x - \sqrt{x^2 -1} \right)^{n+1} \right] = \frac{\sinh \left( (n+1)\,\mbox{arccosh} \,x \right)}{\sinh \left( \mbox{arccosh} \,x \right)} \qquad \mbox{for} \quad |x| \ge 1, \\ &= \frac{\sin \left( (n+1)\,\mbox{arccos} \,x \right)}{\sin \left( \mbox{arccos} \,x \right)} = \frac{\sin \left( (n+1)\,\mbox{arccos} \,x \right)}{\sqrt{1-x^2}} \quad \mbox{for } |x| \le 1; \\ &= x^n \sum_{k=0}^{\lfloor n/2 \rfloor} \,\binom{n+1}{2k+1} \left( 1- x^{-2} \right)^k . \end{align*}

 

Chebyshev Polynomials of the Third Kind

Chebyshev differential equation of the third kind
\[ \left( 1 - x^2 \right) y'' - \left( 2x -1 \right) y' + n\left( n+1 \right) y =0 , \qquad n=0,1,2,3,\ldots , \tag{V.3} \]
has a polynomial solution
\[ V_{n} (x) = \frac{\cos \left( \frac{2n+1}{2}\,\theta \right)}{\cos \left( \frac{\theta}{2} \right)} = \sum_{k = \lceil n/2 \rceil}^n \binom{k}{n-k} 2^{2k-n-1} (-1)^{n-k} x^{2k-n-1} \left( 2x -2 + \frac{n}{k} \right) , \quad n\ge 2, \qquad x= \cos\theta , \qquad n=0,1,2,\ldots . \tag{V.4} \]
V[n_] = FullSimplify[((x + Sqrt[x^2 - 1])^(n + 1) + (x - Sqrt[x^2 - 1])^(n) )/(Sqrt[x^2 - 1] + x + 1)]
\[ V_{n} (x) = \frac{\left( x + \sqrt{x^2 -1} \right)^{n+1} + \left( x - \sqrt{x^2 -1}\right)^{n}}{x+1+\sqrt{x^2 -1}} , \qquad n=0,1,2,\ldots . \]
They can be determined by Rodrigues' formula
\[ V_n (x) = \frac{(-1)^n 2^n n!}{(2n)!} \,\sqrt{\frac{1-x}{1+x^2}} \,\frac{{\text d}^n}{{\text d}x^n} \left[ \left( 1 - x^2 \right)^{n} \sqrt{\frac{1+x}{1-x}} \right] , \qquad n=0,1,2,\ldots . \tag{V.5} \]

 

Chebyshev Polynomials of the Fourth Kind

Chebyshev differential equation of the fourth kind
\[ \left( 1 - x^2 \right) y'' - \left( 2x +1 \right) y' + n\left( n+1 \right) y =0 , \qquad n=0,1,2,3,\ldots , \tag{W.3} \]
has a polynomial solution
\[ W_n (x) = \frac{\sin \left( \frac{2n+1}{2}\,\theta \right)}{\sin \left( \frac{\theta}{2} \right)} = \sum_{k = \lceil n/2 \rceil}^n \binom{k}{n-k} 2^{2k-n-1} (-1)^{n-k} x^{2k-n-1} \left( 2x +2 - \frac{n}{k} \right) , \quad n\ge 2, , \qquad x= \cos\theta , \qquad n=0,1,2,\ldots . \tag{W.4} \]
W[n_] = FullSimplify[((x + Sqrt[x^2 - 1])^(n + 1) - (x - Sqrt[x^2 - 1])^(n) )/(Sqrt[x^2 - 1] + x - 1)]
\[ W_{n} (x) = \frac{\left( x + \sqrt{x^2 -1} \right)^{n+1} - \left( x - \sqrt{x^2 -1}\right)^{n}}{x-1+\sqrt{x^2 -1}} , \qquad n=0,1,2,\ldots . \]
They can be determined by Rodrigues' formula
\[ W_n (x) = \frac{(-1)^n 2^n n!}{(2n)!} \,\sqrt{\frac{1+x}{1-x^2}} \,\frac{{\text d}^n}{{\text d}x^n} \left[ \left( 1 - x^2 \right)^{n} \sqrt{\frac{1-x}{1+x}} \right] , \qquad n=0,1,2,\ldots . \tag{W.5} \]
The leading coefficient of polynomial Wn(x) is 2n.

 

 

  1. Davis, P., Interpolation and Approximation, ‎ Dover Publications; 2014, ISBN-13 ‏ : ‎ 978-0486624952
  2. Chebyshev, P., L. "Sur le développement des fonctions à une seule variable." Bull. Ph.-Math., Acad. Imp. Sc. St. Pétersbourg 1, 193-200, 1859.
  3. Gasull, A., Li, C., Torregrosa, J., A new Chebyshev family with applications to Abel equations, Journal of Differential Equations, 2012, Volume 252, Issue 2, 15 January 2012, Pages 1635-1641; https://doi.org/10.1016/j.jde.2011.06.010
  4. Mason, J.C. and Handscomb, D.C., Chebyshev Polynomials, (Chapman and Hall/CRC, Boca Raton, 2003).
  5. Rivlin, T.J., Chebyshev Polynomials, Wiley, 1974.
  6. Rivlin, T.J., Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Wiley, 2nd edition, 1990. ISBN-13 ‏ : ‎ 978-0471628965

 

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