Discrete Chebyshev transform


In applied mathematics, the discrete Chebyshev transform , 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 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 at the points is given by:
where:
where and otherwise.
Using the definition of,
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
%x=-cos);
f=f;
A=size; N=A;
if exist && A~=1
for i=1:A

a=sqrt*dct;
a=a/sqrt;

end
else
a=sqrt*dct;
a=a/sqrt;
end

The discrete cosine transform is in fact computed using a fast Fourier transform algorithm in MATLAB.
And the inverse transform is given by the MATLAB code:

function f=ifct
%x=-cos)
k=size; N=k;
a=idct*a*sqrt; a;
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. 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 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 and 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