CosFourier

CosFourier is linear operator acting on continuous functions defined at the non–negative values of the artument. Function $F$ is converted to function $\mathrm {CosFourier} ~F$ in such a way, that

$\!\!\!\!\!\!\!\!\!\!\ (1)~~~(\mathrm {CosFourier} ~F)(x)=\displaystyle {\sqrt {\frac {2}{\pi }}}~\int _{0}^{\infty }\cos(xy)~F(y)~\mathrm {d} y$

Incerse operator

The CosFourier is self-inverse operator; its square is identity operator.

Eigenfunctions of CosFourier

Eigenfunctions of the Fourier Operator with eigenvalue unity are also eigenfunctions of the CosFourier. Such functions can be called Self-Fourier. Below are three examples of the self-Fourier functions:

$a_{0}(x)=\exp(-x^{2}/2)~$

$a_{1}(x)=\exp(-x^{2}/2)~(x^{4}-3x^{2})$

$a_{2}(x)=\exp(-x^{2}/2)~(x^{8}-14x^{6}+35x^{4})$

These functions are good for testing of the numerical implementations of the FourierOperator.

Relation to the FourierOperator

The Fourier operator acts on a function $F$ in the following way:

$(\mathrm {Fourier} ~F)(x)={\sqrt {\frac {1}{2\pi }}}\int _{-\infty }^{\infty }\exp(\mathrm {i} xy)~f(y)~\mathrm {d} y$

For a continuous even function $F$, the Fourier operator give the same result as CosFourier.

Numerical implementation

In principle, the CosFourier coud be implemented directly through the numerical implementation of the Discrete Fourier transform, extending the function to the negative values of the argument. However, there exist more efficient implementations.

For the numerical implementation of CosFourier, the choice of equidistant nodes is important. In particular, there exist the following DCT, id est, the followin discrete analogies of the CosFourier are available: DCTI, DCTII, DCtIII,

Keywords

Linear operator, Fourier operator, DCT, DCTI, DCTII, DCTIII,

References

This article is adopted from http://tori.ils.uec.ac.jp/TORI/index.php/CosFourier