The Fourier Transform
In the chapter on Fourier series we showed that every continuous periodic function can be written as a sum of simple waves. A similar result holds for aperiodic functions on R, provided that they are square integrable. In the periodic case the possible waves were cos(2πkx) and sin(2πkx) where k has to be an an integer, which means that the possible “wave lengths” are 1, 1/2, 1/3,.... In the aperiodic case there is no restriction on the wavelengths, so every positive real number can occur. Consequently, the sum in the case of Fourier series will have to be replaced by an integral over ℝ, thus giving the Fourier transform.
KeywordsFourier Series Convergence Theorem Dominate Convergence Theorem Simple Wave Theta Series
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