CosFT

CosFT, or Cosinus Transform, refers to the integral transform with kernel $K(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)$;

for function $f$, the CosFT$f$ appears as $g$ defined with

$\displaystyle g(x)=\,$CosFT$f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \cos(xy) \, f(y) \, \mathrm d y$

SinFT and CosFT

CosFT often appears together with SinFT;

the sine transform SinFT of function $f$ appears as $g=\,$SinFT$f$ with rofmula

$\displaystyle g(x)=\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y$

It is assumed that function $f$ decays (or, at least, quickly oscillates) at infinity, in such a way that the integral converges.

Then, SinFT$^2=\,$CosFT$^2=\hat 1$, id est, the identity transform.

Numerical implementation

CosFT can be implemented numerically through the CFT transform at the uniform grid at $N\!+\!1$ nodes; for array $f$, the SFT $g$ is defined with

$g_m=\,$CFT$\displaystyle f_m=$ $\displaystyle \frac{1}{2}\, f_0+\frac{(-1)^m}{2}\, f_{N} + \sum_{n=1}^{N-1} \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n$

The Numerical recipes in C (http://numerical.recipes) suggest the implementation through routines four1 and realft; however, for the serious applications, specification "float" should be replaced to something appropriate, for example, double, or complex double.

At given number $N$ of nodes, the set of the nodes can be denoted with $x_n$ for $n=0 .. N$,

$\displaystyle x_n=\sqrt{\frac{\pi}{N}}~ n$

then, for $f_n=f(x_n)$, at large $N\gg 1$, the transform $~g(x)=\frac{2}{\pi}\int_0^\infty f(y)\,\cos(x\,y)\,\mathrm d y~$ is approximated with

$\displaystyle g(x_m) \approx g_m = \sqrt{\frac{2}{N}}\, \left( \frac{f_0+(-1)^m f_N}{2} + \sum_{n=1}^{N-1} \, \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\right)$

Eigenfunctions

Eigenfunctions $F$ of the CosFT appear as even Oscillator functions.

The simplest of them is just Gaussian;

$F(x)=\exp(-x^2/2)$

This property is used in the C++ test. The thick segmented line in the explicit plots at right show the discrete representation of the Gaussian at the discrete grid with $N\!=\!4$ and $N\!=\!16$.

The thin segmented line shows its SinFT transform, as it is approximated at this grid. The discrete representation and its CFT practically coincide; the deviation is smaller than the thicknesses of the lines.

Keywords

CosFT, FFT, Integral transform, SinFT,,,