Chebyshev polynomials

Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted $T_{n}(x)$. They are used as an approximation to a least squares fit, and are a special case of the ultra-spherical polynomial (Gegenbauer polynomial) with $\alpha =0$. Chebyshev polynomial of the first kind, $T_{n}(z)$ can be defined by the contour integral

$T_{n}(z)={\frac {1}{4\pi i}}\oint {\frac {(1-t^{2})t^{-n-1}}{(1-2tz+t^{2})}}{\rm {d}}t$

The first seven Chebyshev polynomials of the first kind are:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_0 (x) \right. =1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_1 (x) \right. =x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_2 (x) \right. =2x^2 -1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_3 (x) \right. =4x^3 - 3x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_4 (x) \right. =8x^4 - 8x^2 +1}

$\left.T_{5}(x)\right.=16x^{5}-20x^{3}+5x$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1}

Orthogonality

The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)^{-1/2}} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-1}^{1} \frac{T_m (x)T_n (x) }{ \sqrt{1-x^2}} \mathrm{d} x= \left\{ \begin{array}{lll} \frac{1}{2}\pi \delta_{(mn)} & ; & m \neq 0, n\neq 0 \\ \pi & ; & m=n=0 \end{array} \right.}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{(mn)}} is the Kronecker delta.

Applications in statistical mechanics

• Computational implementation of integral equations

See also

• Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld