Fractal Fourier Coefficient
Adil M. Ahmed1 and Nadia M. G. Al-Saidi2
1Department
of Mathematics, Ibn al-Haytham College, University of Baghdad
2Applied Sciences Department-Applied Mathematics-University of Technology
E-mail: nadiamg08@gmail.com
Abstract
Fourier analysis is the abstract science of frequency. Fourier series is important aspect in
science and engineering. It is allow us to model periodic signals in term of distinct
harmonic component. Frequencies can occur over the analysis of symmetries in geometry
or harmonic analysis. In many applications the signal is not always smooth, whereas
many kinds of noise are seen as rapid, random changes in amplitude or gaps from point to
point within the signal. Many attempts are proposed towered reducing this noise. In this
paper a new approach for estimating Fourier coefficients defined on a subset Y of [0,1]
which is fractal, and using pixel covering method is proposed. This can be performed by
counting the number of point in the mesh Im that needed to cover the fractal set YIm. It is
defined as fractal Fourier series 𝑓(𝑥) = ∑𝑘∈𝐼𝑚 𝑓̂(𝑘)𝑒 2𝜋𝑖⟨𝑡|𝑘⟩ 𝑑𝑡, where ̂𝑓(𝑘) =
∫𝑡∈𝑌 𝑓(𝑡)𝑒 −2𝜋𝑖⟨𝑡|𝑘⟩ 𝑑𝑡, 𝑓𝑜𝑟 𝑡𝑌, which is called fractal Fourier coefficient. The work in
this abstracting set Y upon the hole set [0,1]m, will help to improve the accuracy and the
complexity of the calculation. Also, it will help to expand the employment of this
science of frequency that plays an important role in many applications. The proposed
method is performed first by eliminating the exponential in 𝑓̂(𝑥) to enable us estimate in
a discrete space, then determine the coefficient according to covering the set Y by the
non-escaped pixels of the set [0,1]m.
Key words: Fractal, Fourier series, Fractal Fourier Coefficient (FFC), attractor.