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Polynomial Atomic Functions for Fourier Analysis

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In computer terminology, an atomic process refers to a sequence of operations that can be executed by the CPU with a single instruction. (It has nothing to do with chemical atoms.) This Demonstration shows Fourier approximations (FA) of the family of polynomial atomic functions (pAF) and their Fourier transforms (spectrum). We consider a set of pAF, such as the functions , , , , , and , each of which is defined as a finite solution of a coordinate functional differential equation (FDE) [1, 2].

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For example, the function is a solution of the FDE of advanced type, , where and the support of is . The FA of is .

Here we use a FA technique to compute all the pAFs considered.

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Contributed by: Oleg Kravchenko (December 2014)
Open content licensed under CC BY-NC-SA

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References

[1] V. F. Kravchenko, H. Perez-Meana, and V. I. Ponomaryov, Adaptive Digital Processing of Multidimensional Signals with Applications, Moscow: Fizmatlit, 2009.

[2] Y. Y. Konovalov, "Iterative Algorithms for Computation Convolutions of Atomic Functions Including New Family ," Proceedings of Days on Diffraction (DD) Conference, 2012, pp. 129–133. doi:10.1109/DD.2012.6402765.

[3] Y. Y. Konovalov and V. F. Kravchenko, "Weight Functions Based on the Convolutions of Atomic Functions," Proceedings of Days on Diffraction (DD) Conference, 2012, pp. 78–82. doi:10.1109/DD.2013.6712807.

[4] 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," Proceedings of MSMW International Kharkov Symposium, 2013, pp. 66–71. doi:10.1109/MSMW.2013.6621989.

[5] "Atomic Functions." (Nov 25, 2014) atomic-functions.ru/en.

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