Fourier Transform

A decomposition of a signal into constituent frequencies, mapping functions from time domain to frequency domain.

• transforms a function f(t) into F(w) by integrating against complex exponentials
• inverse Fourier transform reconstructs the original signal from its spectrum
• FFT (Fast Fourier Transform):: the O(n log n) algorithm enabling practical computation
• convolution theorem:: convolution in time domain equals multiplication in frequency domain
• Foundation of signal processing, audio analysis, image compression, and wave physics
• spectral analysis reveals hidden periodicities in data, connecting to statistics
• Central to quantum mechanics (momentum-position duality) and differential equations (solving PDEs)
• Related: calculus, linear algebra, information theory, optimization