Chebyshev polynomials crop up in many areas of calculus, including numerical integration, orthogonal polynomials and spectral methods for partial differential equations. They can also be used for curve fitting (finding a function that models a curve), interpolation and in multiple other areas of numerical analysis.
The general formula for a Chebyshev polynomial, for an integer n ≥ 0, is:
Tn(x) = cos(n cos-1x) ; -1 ≤ x ≤ 1
For n ≥ 2 (Smith, 2011):
- Tn(x) is an n th-order polynomial in x.
- When n is an even integer, Tn(x) is an even function.
- When n is an odd integer, Tn(x) is an odd function.
- Tn(x) has n zeros in the open interval (-1, 1).
- Tn(x) has n + 1 extrema in the closed interval [-1, 1].
Chebyshev Polynomials of the First Kind
Some authors refer to Chebyshev polynomials as just the Chebyshev polynomial of the first kind (Tn)—a polynomial in x of degree n, defined by the relation (Mason & Handscomb, 2002):
Tn(x) = cos nθ when x = cosθ.
The following table (Culham, 2020) lists the first 12 Chebyshev Polynomials of the first kind, obtained from Rodrigue’s formula:
|T0(x) = 1|
|T1(x) = x|
|T2(x) = 2x2 – 1|
|T3(x) = 4x3 – 3x|
|T4(x) = 8x4 – 8x2 + 1|
|T5(x) = 16x5 – 20x3 + 5x|
|T6(x) = 32x6 – 48x4 + 18x2 – 1|
|T7(x) = 64x7 – 112x5 + 56x3 – 7x|
|T8(x) = 128x8 – 256x6 + 160x4 – 32x2 + 1|
|T9(x) = 256x9 – 576x7 + 432x5 – 120x3 + 9x|
|T10(x) = 512x10 – 1280x8 + 1120x6 – 400x4 + 50x2 – 1|
|T11(x) = 1024x11 – 2816x9 + 2616x7 – 1232x5 + 220x3 – 11x|
Culham, J. (2020). Chebyshev Polynomials. Retrieved August 22, 2020 from: mhtl.uwaterloo.ca/courses/me755/web_chap6.pdf
Mason, J. & Handscomb, S. (2002). Chebyshev Polynomials. CRC Press.
Smith, J.O. Spectral Audio Signal Processing, http://ccrma.stanford.edu/~jos/sasp/, online book, 2011 edition, accessed August 23, 2020.
Stephanie Glen. "Chebyshev Polynomials: Simple Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/chebyshev-polynomials/
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