Linked from

The 135 pages that link to Fourier transform, each with the reason it gives.

• Magnetic resonance imagingRelated: MRI reconstruction transforms measured spatial-frequency data into an image.

• Complex numberRelated: Its complex exponentials encode both the amplitude and phase of frequency components.

• RadarRelated: Radar processing uses frequency analysis to estimate Doppler shifts and separate targets.

• Nuclear magnetic resonance spectroscopyRelated: NMR instruments transform recorded free-induction decays into frequency-domain spectra.

• X-ray diffractionRelated: The diffraction pattern is related to the Fourier transform of the sample's electron density.

• GradientRelated: Spatial differentiation becomes multiplication by frequency in the Fourier domain.

• Lebesgue integralRelated: Lebesgue integrability provides a standard condition for defining the transform.

• Heat equationRelated: It converts the constant-coefficient heat equation into independently decaying frequency modes.

• Nuclear magnetic resonanceRelated: It converts measured time-domain NMR signals into frequency-domain spectra.

• Poisson's equationRelated: It converts constant-coefficient Poisson equations into algebraic relations in frequency space.

• Measure theoryRelated: Measure-theoretic integration defines Fourier transforms for broad classes of functions and measures.

• Amplitude modulationRelated: It reveals the carrier and sidebands that make up an AM spectrum.

• CrystallographyRelated: Crystal electron-density maps are reconstructed from diffraction data using Fourier methods.

• Euler's formulaRelated: Its complex exponential kernels encode the sine and cosine oscillations being measured.

• Angular frequencyRelated: Its angular-frequency form describes signal components in radians per unit time.

• Complex exponentialRelated: Its complex exponential kernel encodes each frequency as a rotating phase.

• Improper integralRelated: Many Fourier transforms are improper integrals whose convergence depends on decay or cancellation.

• Vibrational spectroscopyRelated: It converts interferometer measurements into the frequency-domain spectra used in modern instruments.

• Diffraction gratingRelated: Periodic grating structures can be analyzed through their spatial-frequency components.

• Frequency responseRelated: It connects a system’s time-domain behavior with its behavior across frequencies.

• Signal processingRelated: It exposes the frequency components that many processing methods analyze or modify.

• Aperture synthesisRelated: Image reconstruction converts sampled Fourier components into a brightness distribution.

• Dirac delta functionRelated: The delta transforms into a constant, linking point localization to all frequencies.

• Entire functionRelated: Compact support in one domain can produce entire extensions in the transform variable.

• Heisenberg uncertainty principleRelated: Position and momentum wavefunctions are Fourier-related, producing a spread tradeoff.

• Spatial frequencyRelated: It decomposes an image into spatial frequencies and their orientations.

• Linear mapRelated: On suitable function spaces, it acts linearly, carrying sums and scalar multiples to corresponding outputs.

• Linear operatorRelated: It changes representation while preserving linear combinations.

• Lp spaceRelated: Lp bounds describe when Fourier transforms extend beyond especially well-behaved functions.

• QuasicrystalRelated: It connects quasiperiodic atomic arrangements to the discrete diffraction peaks they produce.

• Radio interferometryRelated: Image reconstruction converts sampled Fourier information into a sky-brightness map.

• AliasingRelated: It provides the frequency-domain view used to analyze sampling and aliasing.

• Fubini's theoremRelated: Interchanging integrals is often needed to derive transform identities and inversion formulas.

• Canonical commutation relationRelated: Position and momentum wavefunctions are related by a Fourier transform, reflecting their conjugate roles.

• Cryo-electron microscopyRelated: Fourier methods underpin image alignment, filtering, and three-dimensional reconstruction.

• Dominated convergence theoremRelated: Convergence of Fourier integrands often uses domination to pass limits through the transform.

• HertzRelated: Fourier analysis identifies which frequencies, measured in hertz, make up a signal.

• Wave functionRelated: It relates position-space and momentum-space wave-function representations.

• WavenumberRelated: Its spatial-frequency variable is often expressed as wavenumber.

• Plane waveRelated: It decomposes many fields into plane-wave components.

• Cauchy principal valueRelated: Principal values arise in Fourier transforms of nonintegrable oscillatory kernels.

• Distribution (mathematics)Related: Tempered distributions extend Fourier analysis beyond integrable functions.

• Fourier-transform infrared spectroscopyRelated: It converts the interferogram’s path-difference signal into intensity across infrared frequencies.

• Gaussian functionRelated: The Fourier transform maps a Gaussian to another Gaussian.

• Kernel (linear algebra)Related: Restricted or sampled Fourier measurements can have kernels containing unrecoverable signals.

• Riesz representation theoremRelated: The theorem underlies identifications between Hilbert spaces of functions and their continuous duals.

• Carrier waveRelated: It reveals the carrier and sidebands as components of a modulated signal’s spectrum.

• Complex conjugateRelated: For real signals, conjugate symmetry relates Fourier components at opposite frequencies.

• Complex multiplicationRelated: Its frequency components are built from complex-valued products and sums.

• Contour integrationRelated: Contour deformation evaluates transforms and inverse transforms in many cases.