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ORF consists of a Java script and a Java application which visualize orthogonal rational functions (ORFs) on the complex unit circle.
If 
is a positive measure of
the complex unit circle, and 
is a sequence of complex numbers inside the unit
disk, then one may consider the spaces
where 
is a polynomial of degree
at most n and
Let these spaces be equiped with the inner product
The ORFs are then a set of orthogonal basis functions for these nested sequence of rational spaces.
If you want to learn more about the mathematics of ORF, you should read Section 2.
ORFs are complex valued functions of a complex variable. Therefore, in the ORF application, one can choose between the 3D representation of the real part, the imaginary part, the modulus or the angle (i.e. the argument) of these ORFs. The degree of the ORFs that can be visualized can be selected between 1 and 30.
At the moment there is only a possibility to choose between the normalized Lebesgue-Borel measure and a second measure which is not absolutely continuous (with respect to the normalized Lebesgue-Borel measure).
However, there is a possibility to choose among different sequences of poles.
One can also choose to see the 3D object from the zenith of the complex plane (angle is 90 degrees) or at a tilted angle (namely 70 degrees).
The pictures were generated by numerical computation with Matlab5 
. The
language Matlab5 has a lot of
useful techniques for visualization. Especially, neccessary
features like truecolor support, phong lighting and multiple
light sources for colored surfaces are used. (We also calculated
the ORFs by using Octave
which is a free clone of Matlab . Octave ist able to
handle most of the m-files which are written for Matlab 4.2. But
Octave is not useful for visualization.) The usage of Matlab5 also gives a
choice between four different colouring types (Spring, Summer,
Autumn, Winter). Also this choice is available to the user.
Moreover, the user has the possibility to start a slideshow where the whole sequence of 30 pictures can be seen. The user can also choose the time interval of the slideshow (that is the time interval between the visualization of the pictures of successive degree). This choice is of course only meaningful when there is a fast internet connection.
ORF is a Java-based interface to view these pictures, which
allows the user to make all the choices mentioned above. All the
choices can be made by clicking the appropriate buttons. The second version
of ORF has all the features of the Java script
ORF v1.2, but it is written as a Java applet and a Java
application. Both of them can be reached through the internet,
but it is also possible to use the latter as a stand-alone
program. Moreover, in ORF v2.0,
a colorbar is added, the position of the sequence 
in the unit disk can be visualized, and
certain panels can be opend. The sound-effects are still
available only in the Java application but we also plan to
implement this effects into the applet. Furthermore, the user
also gets a few interactive help windows which are available in
English and German at the moment.
The feature preloading for animation should be only enabled if you have a fast computer with enough memory. Otherwise this could make your machine crash.
For improvements to be implemented later, we refer to Section 3.
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