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,,,