The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second, for fractal interpolation function, odd prolongation is done and Fourier sine series formula of fractal interpolation function is proved. Final, Fourier series expansion of fractal interpolation function on the closed interval [Formula: see text] is proved. The result shows that complex fractal interpolation function can be represented by Fourier sine series and Fourier cosine series, so relatively simple Fourier series can be used to represent relatively complicated fractal interpolation function.

中文翻译：

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分形插值函数的傅里叶级数表示

本研究的目的是展示复杂且不规则的分形插值函数如何用傅里叶级数表示。首先，在闭区间[0,1]上，对由仿射变换构成的迭代函数系统生成的分形插值函数进行偶数延拓，证明了分形插值函数的傅里叶余弦级数表示。其次，对分形插值函数进行奇延长，证明了分形插值函数的傅里叶正弦级数公式。最后，证明了闭区间上分形插值函数的傅里叶级数展开[公式：见正文]。结果表明，复分形插值函数可以用傅里叶正弦级数和傅里叶余弦级数表示，