# DCTIV

DCTIV is one of realizations of the Discrete CosTransform. The name is borrowed from Wikipedia [1], 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.