Discrete Chebyshev transform

In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind formula_1 and the discrete Chebyshev transform on the 'extrema' grid of the Chebyshev polynomials of the first kind.
Discrete Chebyshev transform on the roots grid.
The discrete chebyshev transform of u(x) at the points formula_2 is given by:
where:
where formula_6 and formula_7 otherwise.
Using the definition of formula_8,
and its inverse transform:
This can readily be obtained by manipulating the input arguments to a discrete cosine transform.
This can be demonstrated 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
The discrete cosine transform (dct) is in fact computed using a fast Fourier transform algorithm in MATLAB. <br>
And the inverse transform is given by the MATLAB code:<br>
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
Discrete Chebyshev transform on the extrema grid.
This transform uses the grid:
This transform is more difficult to implement by use of a Fast Fourier Transform (FFT). However it is more widely used because it is on the extrema grid which tends to be most useful for boundary value problems. Mostly because it is easier to apply boundary conditions on this grid.
There is a discrete (and in fact fast because it performs the dct by using a fast Fourier transform) available at the MATLAB file exchange that was created by Greg von Winckel. So it is omitted here.
In this case the transform and its inverse are
where formula_6 and formula_7 otherwise.
Usage and implementations.
The primary uses of the discrete Chebyshev transform are numerical integration, interpolation, and stable numerical differentiation.
An implementation which provides these features is given in the C++ library Boost