# 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 .

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Contributed by: Oleg Kravchenko (January 2015)

Open content licensed under CC BY-NC-SA

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References

[1] W. Hilberg, S. Wolf. "Korrelationsermittlung durch Stapeln von Impulsen," patent DE 19818694 A1, 1999. www.google.com/patents/DE19818694A1?cl=de.

[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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