DCTIV

DCTIV is one of realizations of the Discrete CosTransform. The name is borrowed from Wikipedia , but the dash is omitted from the name DCT-IV in order to use only letters in the identifier and avoid confusion with operation of substruction. For natural number $N$, the $\mathrm{DCTIV}_N$ acts on the array $f$ of length $N$ in the following way\[\displaystyle (\mathrm{DCTIV} ~f )_k = \sqrt{\frac{2}{N}} ~ \sum_{n=0}^{N-1} ~f_n~ \cos \left[\frac{\pi}{N} \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right) \right] \quad \quad k = 0, \dots, N-1.\]

Properties

The square of the DCTIV is identity operator:

$ \mathrm{DCTIV}^2 f=\mathrm{DCTIV}~ \mathrm{DCTIV} f = f$

Approximation of the CosFourier transform

Let $x_n=\sqrt{\frac{\pi}{N}}~ \left(n+\frac{1}{2}\right)~ ~$ , $~ ~ ~ f_n=f(x_n)$

where $f$ is smooth function of real positive argument. One may extend $f$ to the negative values of the argument, assuming that it is even and smooth. Let this $f$ efficiently decay at large values of the argument. Then, the transform can be written as

$ \displaystyle (\mathrm{DCTIV} f)_k= \sqrt{\frac{2}{N}} ~

\sum_{n=0}^{N-1} ~f(x_n) \cos\! \left(\frac{\pi}{N} x_n x_k\right) ~\approx ~

\sqrt{\frac{2}{N}}  \int_0^\infty ~f(x_n) \cos\! \left(x_n x_k\right) ~\mathrm d n $

At large $N$, smoothness and quick decay at infinity is assumed for function $f$. With new variable of integration $y=x_n$, the CosFourier transform $g$ of function $f$ can be approximated at the points $k_n$

$ \displaystyle g(x_k) = \sqrt{\frac{2}{\pi}} ~

\int_0^\infty ~f(y) \cos\! \left(y x_k\right) ~\mathrm d y \approx \displaystyle (\mathrm{DCTIV} f)_k $

Approximation of the Fourier coefficients

Consider the representation of some even continuous function $f$ with the Fouriet series

$ \displaystyle

f(x) = \sum_{n=0}^{\infty} c_n \cos( n x)$

function $f$ is supposed to be symmetric, $f(x)=f(-x)$, and periodic with period $2\pi$. The coefficients can can be expressed through the integrals with function $f$,

$ \displaystyle c_0=\frac{1}{\pi} \int_0^\pi f(x) \mathrm d x$

$ \displaystyle c_m=\frac{2}{\pi} \int_0^\pi f(x) \cos(mx) \mathrm d x~ ~ ~ $, $ ~ ~ ~ m>0$

It seems, the direct representation above does not give a straight way for evaluation of the coefficients $c$; another discretization of the CosFourier operator, it est, DCT, should be used.