With the theory of trigonometric function system in Advanced Mathematics, it is proved that any periodic function defined in the real number field with a period of 2π and satisfying Dirichlet condition can be expanded into Fourier series, and any function with a period of 2l can be expanded through telescopic transformation. (Department of Mathematics, Tongji University M: Advanced Mathematics 6th Edition (Volume II). Beijing: Higher Education Press, 2007)
If it is not a periodic function, we can turn the above conclusion into infinity, that is, the period of the function is infinite, and then express the Fourier series by exponent and replace the sum in the series by integral. Finally, the expression of Fourier transform is obtained naturally. (m, Yao Zhengduan, Liang Jiabao, etc. : mathematical and physical methods. Beijing: Science Press, 20 10)
This is not the equality you think, but the actual equality. Series and integral can completely eliminate the gap between real function and "approximately equal".