The Fourier transform is a linear operator in harmonic analysis that maps a time-domain function to its frequency-domain representation by decomposing signals into complex exponentials based on the principle of orthogonality and center-of-mass analysis. Theoretically, this transformation relies on integrating the product of a signal $g(t)$ against rotating unit vectors defined by Euler's formula over an infinite interval ($-\infty$ to $\infty$) or finite bounds, resulting in a complex-valued output where magnitude indicates frequency strength and phase encodes temporal alignment. This concept establishes the duality between time and frequency variables within functional analysis, allowing for the identification of periodic components in continuous signals through weighted superposition of basis functions without requiring physical rotation mechanisms.
The Fourier transform is a linear operator in harmonic analysis that maps a time-domain function to its frequency-domain representation by decomposing signals into complex exponentials based on the p…