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Chebyshev Polynomials: Simple Definition

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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

Properties

For n ≥ 2 (Smith, 2011):

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:
rodrigues 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

References

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.

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