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An introduction to chebyshev polynomials, their definitions, properties, and applications. The text also covers the legendre differential equation, orthogonality of chebyshev polynomials, and their zeros and extrema. Peter blomgren's lecture notes offer examples and comparisons between different interpolation methods.
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Chebyshev Polynomials Least Squares, redux
Lecture Notes # — Approximation Theory — Chebyshev Polynomials & Least Squares, redux
Peter Blomgren, 〈[email protected]〉
Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182- http://terminus.sdsu.edu/
Fall 2009
Chebyshev Polynomials Least Squares, redux
Outline
(^1) Chebyshev Polynomials Orthogonal Polynomials Chebyshev Polynomials, Intro & Definitions Properties
2 Least Squares, redux Examples More than one variable? — No problem!
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & DefinitionsProperties
The Legendre Polynomials Background
The Legendre polynomials are solutions to the Legendre Differential Equation (which arises in numerous problems exhibiting spherical symmetry)
(1 − x^2 )
d^2 y dx^2
− 2 x
dy dx
or equivalently
d dx
(1 − x^2 )
dy dx
Applications: Celestial Mechanics (Legendre’s original applica- tion), Electrodynamics, etc...
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & DefinitionsProperties
Other Orthogonal Polynomials Background
“Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. [... They] provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization.”
“The roots of orthogonal polynomials possess many rather surprising and useful properties.”
(http://mathworld.wolfram.com/OrthogonalPolynomials.html)
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & DefinitionsProperties
More Orthogonal Polynomials Background
Polynomials Interval w(x) Chebyshev (1st) [− 1 , 1 ] 1 /
√ 1 − x^2 Chebyshev (2nd) [− 1 , 1]
√ 1 − x^2 Gegenbauer [− 1 , 1] (1 − x^2 )α−^1 /^2 Hermite∗^ (−∞, ∞) e−x^2 Jacobi (− 1 , 1) (1 − x)α(1 + x)β Legendre [− 1 , 1] 1 Laguerre [0, ∞) e−x Laguerre (assoc) [0, ∞) xk^ e−x
Today we’ll take a closer look at Chebyshev polynomials of the first kind.
∗ (^) These are the Hermite orthogonal polynomials, not to be confused with the Hermite interpolating polynomials...
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
Chebyshev Polynomials: The Sales Pitch.
Tn(z) =
2 πi
1 − t^2
t−(n+1) (1 − 2 tz + t^2 )
dt
Chebyshev Polynomials are used to minimize approximation error. We will use them to solve the following problems:
[1] Find an optimal placement of the interpolating points {x 0 , x 1 ,... , xn} to minimize the error in Lagrange interpola- tion.
[2] Find a means of reducing the degree of an approximating poly- nomial with minimal loss of accuracy.
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
Chebyshev Polynomials: Definitions.
The Chebyshev polynomials {Tn(x)} are orthogonal on the interval (− 1 , 1) with respect to the weight function w (x) = 1/
1 − x^2 , i.e.
〈Ti (x), Tj (x)〉w (x) ≡
∫ (^1)
− 1
Ti (x)Tj (x)∗^ w (x)dx = αi δi,j.
We could use the Gram-Schmidt orthogonalization process to find them, but it is easier to give the definition and then check the properties...
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
Chebyshev Polynomials: Definitions.
The Chebyshev polynomials {Tn(x)} are orthogonal on the interval (− 1 , 1) with respect to the weight function w (x) = 1/
1 − x^2 , i.e.
〈Ti (x), Tj (x)〉w (x) ≡
∫ (^1)
− 1
Ti (x)Tj (x)∗^ w (x)dx = αi δi,j.
We could use the Gram-Schmidt orthogonalization process to find them, but it is easier to give the definition and then check the properties...
Definition (Chebyshev Polynomials) For x ∈ [− 1 , 1], define
Tn(x) = cos(n arccos x), ∀n ≥ 0.
Note: T 0 (x) = cos(0) = 1, T 1 (x) = x.
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
Chebyshev Polynomials, Tn(x), n ≥ 2.
We introduce the notation θ = arccos x, and get
Tn(θ(x)) ≡ Tn(θ) = cos(nθ), where θ ∈ [0, π].
We can find a recurrence relation, using these observations:
Tn+1(θ) = cos((n + 1)θ) = cos(nθ) cos(θ) − sin(nθ) sin(θ) Tn− 1 (θ) = cos((n − 1)θ) = cos(nθ) cos(θ) + sin(nθ) sin(θ) Tn+ 1 (θ) + Tn− 1 (θ) = 2 cos(nθ) cos(θ).
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
Chebyshev Polynomials, Tn(x), n ≥ 2.
We introduce the notation θ = arccos x, and get
Tn(θ(x)) ≡ Tn(θ) = cos(nθ), where θ ∈ [0, π].
We can find a recurrence relation, using these observations:
Tn+1(θ) = cos((n + 1)θ) = cos(nθ) cos(θ) − sin(nθ) sin(θ) Tn− 1 (θ) = cos((n − 1)θ) = cos(nθ) cos(θ) + sin(nθ) sin(θ) Tn+ 1 (θ) + Tn− 1 (θ) = 2 cos(nθ) cos(θ).
Returning to the original variable x, we have
Tn+1(x) = 2x cos(n arccos x) − Tn− 1 (x),
or Tn+ 1 (x) = 2xTn(x) − Tn− 1 (x).
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
The Chebyshev Polynomials
-1 -1 -0.5 0 0.5 1
-0.
0
1 T1(x) T2(x)
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
The Chebyshev Polynomials
-1 -1 -0.5 0 0.5 1
-0.
0
1 T1(x) T2(x) T3(x)
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
The Chebyshev Polynomials
-1 -1 -0.5 0 0.5 1
-0.
0
1 T1(x) T2(x) T3(x) T4(x) T5(x)
Chebyshev Polynomials Least Squares, redux Chebyshev Polynomials, Intro & Definitions Properties
Orthogonality of the Chebyshev Polynomials, I
− 1
Tn(x)Tm(x) √ 1 − x^2
dx =
− 1
cos(n arccos x) cos(m arccos x)
dx √ 1 − x^2
