Linked from

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

• 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.

• Change of variablesCompared with: It changes representation through an integral operation rather than a direct variable substitution.

• 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.

• WaveletCompared with: Fourier analysis localizes frequency but not position, unlike wavelet analysis.

• 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.

• DispersionRelated: Dispersion acts differently on the frequency components that make up a pulse.

• Doppler spectroscopyRelated: Spectrometers use dispersive or Fourier methods to resolve wavelength-dependent signals.

• Explicit formulaCompared with: The formula’s zero terms behave like oscillatory frequencies, though its analytic setup is Mellin-based.

• Fourier coefficientCompared with: It extends frequency analysis beyond the discrete coefficients of periodic expansions.

• Fraunhofer diffractionNarrower topic: The far-field pattern is governed by the Fourier transform of the aperture distribution.

• Imaginary unitRelated: Complex exponentials built with i make frequency and phase calculations concise.

• Legendre transformCompared with: It changes representation through oscillatory integration, rather than through convex slope optimization.

• Matter waveRelated: It relates a wave packet's spatial extent to its range of momenta.

• ResidueRelated: Contour integration with residues can evaluate many Fourier transforms and inverse transforms.

• Separation of variablesCompared with: Transforms can reduce differential equations to algebraic or lower-dimensional equations without a product ansatz.

• Fourier opticsRelated: It converts an optical field between position and spatial-frequency representations.

• Gibbs phenomenonNarrower topic: Finite-bandwidth Fourier reconstruction exhibits ringing analogous to the series phenomenon.

• Helmholtz equationRelated: It can convert constant-coefficient Helmholtz problems into algebraic relations in frequency space.

• Integration by partsRelated: Integration by parts transfers derivatives between a function and an exponential kernel, producing frequency factors.

• Small-angle X-ray scatteringRelated: Scattering intensity is related to spatial correlations through Fourier-transform relationships.

• Software-defined radioRelated: It underlies software methods for inspecting spectra and implementing filters.

• Acoustic phoneticsRelated: It underlies many methods for analyzing speech spectra.

• Coherence lengthRelated: Frequency decomposition links a wave’s bandwidth to its coherence distance.

• Fundamental solutionRelated: For constant-coefficient operators, it often converts the fundamental-solution equation into an algebraic relation.

• Mellin transformCompared with: On logarithmic coordinates, the Mellin transform becomes a Fourier transform.

• Scattering theoryRelated: It relates position-space scattering descriptions to momentum-transfer measurements.

• Poisson summation formulaRelated: Transforming the function turns its lattice samples into frequency-space terms.

• Polar formRelated: Its complex-valued outputs are often interpreted by magnitude and phase.

• Terence TaoRelated: Fourier methods are among the main tools in Tao’s harmonic analysis and PDE research.

• AutocorrelationRelated: The Wiener–Khinchin theorem connects autocorrelation with the power spectrum through Fourier transforms.

• Dirac combRelated: The comb transforms into another comb, with spacing set by the reciprocal period.

• Fourier inversion theoremNarrower topic: The theorem reconstructs a function from this transformed representation.

• Integrable functionRelated: For L¹ functions, its defining integral converges absolutely at every frequency.

• Principal symbolRelated: Replacing derivatives by frequency variables explains the symbol’s algebraic form.

• Spectral musicRelated: It provides a way to analyze the component frequencies that spectral composers use as musical material.

• Wavelet transformCompared with: Its global frequency basis contrasts with wavelets' localization in position and scale.

• Bessel's inequalityRelated: Related energy bounds constrain frequency-domain coefficients and transforms.

• Momentum operatorRelated: It converts the position-space derivative into multiplication by momentum in momentum space.

• Jordan's lemmaNarrower topic: Jordan's lemma supports contour methods for integrals containing Fourier oscillations.

• Martin RyleRelated: Aperture synthesis connects measured interferometer data to sky brightness through Fourier relationships.

• Parseval's identityRelated: Its continuous-frequency version relates a function's energy to the energy of its transform.