Fouriertransformed

Fourier-transformed refers to the representation obtained when a function or signal is converted from its original domain, typically time, into the frequency domain by means of the Fourier transform. This representation highlights the frequency components that make up the signal and is widely used in science and engineering for analysis, filtering, and problem solving.

For a function f(t) in L1(R) or L2(R), the Fourier transform F(ξ) is defined by F(ξ) = ∫_{-∞}^{∞} f(t) e^{-i 2π ξ t} dt. The inverse transform recovers f from F via f(t) = ∫_{-∞}^{∞} F(ξ) e^{i 2π ξ t} dξ, under appropriate conditions. Conventions may vary (for example using ω instead of 2πξ). Generalized versions extend the transform to broader classes of functions and to distributions.

Key properties include linearity, time-shifting and frequency-shifting, modulation, and the convolution theorem: the transform of a convolution corresponds to the product of transforms. The magnitude spectrum |F(ξ)| indicates the strength of frequency components, while the phase spectrum encodes timing information. Real-valued signals yield conjugate-symmetric spectra, reflecting their symmetry in time.

Discrete variants exist for finite data. The discrete Fourier transform (DFT) applies to finite sequences, and its efficient computation is made possible by the fast Fourier transform (FFT). Applications span spectral analysis, digital filtering, solving differential equations, imaging, optics, and quantum mechanics. Classic examples include the Gaussian function, which is (up to scale) its own Fourier transform, and complex exponentials, which transform to impulses at their respective frequencies.