A Continuous Analog of the 1D Thue-Morse Sequence

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This Demonstration shows a connection [1, 2] between the Thue–Morse (Prouhet–Thue–Morse) sequence and the infinitely differentiable or atomic functions. By definition, an atomic function is a finite solution of a functional differential equation (FDE) of advanced type, such as , where and is a linear differential operator with constant coefficients. The simplest example of such an FDE is , with . Its solution is called the (or ) function with support .


This function was found independently by Vladimir L. Rvachev, Vladimir A. Rvachev, and Wolfgang Hilberg in 1971. That was the beginning of research into the theory of atomic functions theory. Later, Wolfgang Hilberg found an extremely bizarre connection between the function and the Thue–Morse sequence [4–6], which allows the calculation of the function (red curve in the plot) by iterations of the substitution system that generates the Thue–Morse sequence (filled rectangular pulses in the plot). Such a connection helps in the study of both atomic function theory (a branch of functions) and Thue–Morse-like sequences. It takes advantage of numerical methods, digital signal processing, and many other applications of atomic functions and Thue–Morse sequences [2].


Contributed by: Oleg Kravchenko (January 2015)
Open content licensed under CC BY-NC-SA




[1] W. Hilberg, S. Wolf. "Korrelationsermittlung durch Stapeln von Impulsen," patent DE 19818694 A1, 1999. 

[2] W. Hilberg, V. F. Kravchenko, O. V. Kravchenko, and Y. Y. Konovalov, "Atomic Functions and Generalized Thue–Morse Sequence in Digital Signal and Image Processing," International Kharkov Symposium on Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves (MSMW) 2013, Kharkiv, Ukraine: IEEE 2013. doi:10.1109/MSMW.2013.6621989.

[3] Technische Universität Darmstadt. "Wolfgang Hilberg." (Jan 12, 2015) www.tu-darmstadt.de/universitaet/selbstverstaendnis/profil_geschichte/persoenlichkeiten/thema_perso_k6.en.jsp.

[4] Wikipedia. "Wolfgang Hilberg." (Jan 12, 2015)de.wikipedia.org/wiki/Wolfgang_Hilberg.

[5] W. Hilberg. "Prof. em. Dr. Ing. Wolfgang Hilberg." (Jan 12, 2015) www.hilberg-wolfgang.de.

[6] Atomic Functions. (Jan 12, 2015) atomic-functions.ru/en.

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