Discrete Chebyshev transform

In applied mathematics, a discrete Chebyshev transform (abbreviated DCT, DChT, or DTT) is an analog of the discrete Fourier transform for a function of a real interval, converting in either direction between function values at a set of Chebyshev nodes and coefficients of a function in Chebyshev polynomial basis. Like the Chebyshev polynomials, it is named after Pafnuty Chebyshev.

The two most common types of discrete Chebyshev transforms use the grid of Chebyshev zeros, the zeros of the Chebyshev polynomials of the first kind ${\displaystyle T_{n}(x)}$ and the grid of Chebyshev extrema, the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind ${\displaystyle U_{n}(x)}$ . Both of these transforms result in coefficients of Chebyshev polynomials of the first kind.

Other discrete Chebyshev transforms involve related grids and coefficients of Chebyshev polynomials of the second, third, or fourth kinds.

Roots grid

The discrete Chebyshev transform of ${\displaystyle {u(x)}}$ at the points ${\displaystyle {x_{n}}}$ is given by:

${\displaystyle a_{m}={\frac {p_{m}}{N}}\sum _{n=0}^{N-1}u(x_{n})T_{m}(x_{n}),}$

where

${\displaystyle x_{n}=-\cos {\frac {{\bigl (}n+{\tfrac {1}{2}}{\bigr )}\pi }{N}},}$

${\displaystyle a_{m}={\frac {p_{m}}{N}}\sum _{n=0}^{N-1}u(x_{n})\cos \left(m\cos ^{-1}(x_{n})\right),}$

with ${\displaystyle p_{m}=1}$ if and only if ${\displaystyle m=0}$ and ${\displaystyle p_{m}=2}$ otherwise.

Using the definition of ${\displaystyle x_{n}}$ ,

${\displaystyle {\begin{aligned}a_{m}&={\frac {p_{m}}{N}}\sum _{n=0}^{N-1}u(x_{n})\cos {\frac {m{\bigl (}N+n+{\tfrac {1}{2}}{\bigr )}\pi }{N}}\\&={\frac {p_{m}}{N}}\sum _{n=0}^{N-1}u(x_{n})(-1)^{m}\cos {\frac {m{\bigl (}n+{\tfrac {1}{2}}{\bigr )}\pi }{N}}.\end{aligned}}}$

The inverse transform is

${\displaystyle u_{n}=\sum _{m=0}^{N-1}a_{m}T_{m}(x_{n})=\sum _{m=0}^{N-1}a_{m}(-1)^{m}\cos {\frac {m{\bigl (}n+{\tfrac {1}{2}}{\bigr )}\pi }{N}}.}$

(This is the standard Chebyshev series evaluated on the roots grid.)

This discrete Chebyshev transform can be computed by manipulating the input arguments to a discrete cosine transform, for example, using the following MATLAB code:

function a=fct(f, l)
% x =-cos(pi/N*((0:N-1)'+1/2));

f = f(end:-1:1,:);
A = size(f); N = A(1); 
if exist('A(3)', 'var') && A(3)~=1
    for i=1:A(3)
        a(:,:,i) = sqrt(2/N) * dct(f(:,:,i));
        a(1,:,i) = a(1,:,i) / sqrt(2);
    end
else
    a = sqrt(2/N) * dct(f(:,:,i));
    a(1,:)=a(1,:) / sqrt(2);
end

MATLAB's built-in dct (discrete cosine transform) function is implemented using the fast Fourier transform.

The inverse transform is given by the MATLAB code:

function f=ifct(a, l)
% x = -cos(pi/N*((0:N-1)'+1/2)) 
k = size(a); N=k(1);

a = idct(sqrt(N/2) * [a(1,:) * sqrt(2); a(2:end,:)]);

end

Extrema grid

This transform uses the grid:

${\displaystyle x_{n}=-\cos {\frac {n\pi }{N}}}$

${\displaystyle T_{n}(x_{m})=\cos \left({\frac {mn\pi }{N}}+n\pi \right)=(-1)^{n}\cos {\frac {mn\pi }{N}}}$

This extrema grid is more widely used.

In this case the transform and its inverse are

${\displaystyle u(x_{n})=u_{n}=\sum _{m=0}^{N}a_{m}T_{m}(x_{n}),}$

${\displaystyle a_{m}={\frac {p_{m}}{N}}{\biggl (}{\tfrac {1}{2}}{\bigl (}u_{0}(-1)^{m}+u_{N}{\bigr )}+\sum _{n=1}^{N-1}u_{n}T_{m}(x_{n}){\biggr )},}$

where ${\displaystyle p_{m}=1}$ if and only if ${\displaystyle m=0}$ or ${\displaystyle m=N}$ and ${\displaystyle p_{m}=2}$ otherwise.

Usage and implementations

The primary uses of the discrete Chebyshev transform are numerical integration, interpolation, and stable numerical differentiation.^[1] An implementation which provides these features is given in the C++ library Boost.^[2]

See also

• Chebyshev polynomials
• Discrete cosine transform
• Discrete Fourier transform
• List of Fourier-related transforms

References

1. Trefethen, Lloyd (2013). Approximation Theory and Approximation Practice.
2. Thompson, Nick; Maddock, John. "Chebyshev Polynomials". boost.org.