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The mathematics gives the answer. The Fourier transform of a Dirac function (impulse) transforms to all frequency components present. This is the same as white noise Dirac delta function - Wikipedia, the free encyclopedia
Fourier series - Wikipedia, the free encyclopedia
Check out the first 4 paragraphs. They talk about how the Fourier series was developed, which then led others, like Reimann, to create the more formal integral formulations we see used in today's Fourier Analysis and the Fourier Transform. The math is quite nasty, and complete derivations are probably available in several textbooks.
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