Harmonic analysis, transforms, and integral equations

Analytic signal

The analytic signal is a complex-valued function of time whose Fourier transform vanishes at negative frequencies, introduced by Dennis Gabor in 1946 and formed using the Hilbert transform.

Automorphic form

An automorphic form is a function behaving in a controlled way under a discrete group of symmetries, generalizing modular forms and central to number theory and the Langlands program.

Bargmann transform

The Bargmann transform is a unitary integral transform, introduced by Valentine Bargmann in 1961, mapping functions on real space to holomorphic functions on complex space.

Biorthogonal wavelet

A biorthogonal wavelet is a wavelet paired with a dual basis, one used for analysis and one for synthesis, enabling symmetric, linear-phase compactly supported wavelets.

Bounded mean oscillation

Bounded mean oscillation (BMO), also called the John–Nirenberg space, is a function space introduced by Fritz John and Louis Nirenberg in 1961 that strictly contains all bounded functions.

Circular convolution is a mathematical operation that convolves two finite sequences of a fixed period N with wraparound indexing, and the discrete Fourier transform turns it into pointwise multiplication.

Continuous wavelet transform

The continuous wavelet transform (CWT) is a signal analysis method that decomposes signals into wavelets of every scale and position, tracking frequency content over time.

Convolution

Convolution is a mathematical operation that combines two functions to produce a third, expressing how the shape of one modifies the other, defined by an integral.

Convolution theorem

The convolution theorem states that the transform of a convolution of two functions equals the product of their transforms, and underlies FFT-based filtering and deconvolution.

Curvelet transform

The curvelet transform is a multiscale geometric transform that decomposes images into coefficients indexed by scale, orientation, and location, representing curved edges far more efficiently than wavelets.

Daubechies wavelet

A Daubechies wavelet is a compactly supported, orthonormal wavelet from a family constructed by Ingrid Daubechies in 1988, notable for combining orthogonality, compact support, and high smoothness.

Dirichlet conditions

The Dirichlet conditions are sufficient conditions on a periodic function, including absolute integrability over one period, guaranteeing pointwise convergence of its Fourier series, first proved by Dirichlet in 1829.

Discrete Fourier transform

The discrete Fourier transform (DFT) is the form of Fourier analysis computed by computers, converting N samples into complex numbers giving the amplitude and phase of component sinusoids.

Discrete wavelet transform

The discrete wavelet transform (DWT) is a linear transform that decomposes a discrete signal into wavelet coefficients carrying both scale and position information, computed by a fast filter-bank algorithm.

Dyson equation

The Dyson equation is the integral equation of quantum field theory relating the full interacting Green's function to a reference one and the self-energy, introduced by Freeman Dyson in 1949.

Empirical mode decomposition

Empirical mode decomposition (EMD) is an adaptive signal-processing method that decomposes nonlinear, nonstationary time series into intrinsic mode functions; with the Hilbert transform it forms the Hilbert–Huang transform, developed at NASA in 1998.

Exponential integral

The exponential integral, or Ei function, is a special function defined by the integral of e−t/t, introduced by Legendre in 1811 and arising throughout physics and engineering.

Fast Fourier transform

The fast Fourier transform (FFT) is a family of algorithms computing the discrete Fourier transform in about N log N operations instead of N squared.

Fourier analysis

Fourier analysis, also called Fourier theory, is the branch of mathematics that expresses functions as sums of sines and cosines, founded by Joseph Fourier's 1807 work on heat conduction.

Fourier multiplier

A Fourier multiplier is a linear operator acting on a function by multiplying its Fourier transform by a fixed symbol and transforming back, equivalently convolution with a kernel.

Fourier series

A Fourier series, also called a Fourier expansion, represents a periodic function as an infinite sum of sines and cosines at multiples of the fundamental frequency.

Fourier transform

The Fourier transform, also called Fourier transformation, is a mathematical operation that converts a function of time or space into a function of frequency, named for Joseph Fourier and central to signal processing and differential equations.

Fractional Fourier transform

The fractional Fourier transform (FrFT) is a linear integral operator generalizing the Fourier transform, rotating a signal's time-frequency representation through an arbitrary angle, used in signal processing and optics.

Fredholm integral equation

A Fredholm integral equation is an integral equation in which the unknown appears under an integral with fixed limits, in a first kind and a second kind.

Frequency domain

The frequency domain, also known as the Fourier domain or frequency space, is the domain in which a signal is expressed by its frequency content rather than its time content.

Gabor wavelet

A Gabor wavelet is a sinusoidal carrier multiplied by a Gaussian envelope, used for localized time-frequency analysis of signals and for feature extraction from images.

Gibbs phenomenon

The Gibbs phenomenon, also called the Gibbs effect, is the fixed overshoot and undershoot of Fourier series partial sums near a jump discontinuity, persisting at about 8.95 percent of the jump.

Green's function

Green's function is the response of a linear differential equation with its boundary conditions to a point source, named for George Green, who introduced it in 1828.

Haar wavelet

The Haar wavelet is the simplest orthonormal wavelet, a step function equal to +1 and −1 on halves of its support, introduced by Alfréd Haar in 1909.