Harmonic analysis, transforms, and integral equations

Sinc function

The sinc function, also called the cardinal sine (sinus cardinalis), is defined as sin(x)/x with sinc(0) = 1, and is the Fourier transform of a rectangular pulse.

Single and double layer potentials

Single and double layer potentials write solutions of elliptic partial differential equations as surface integrals over a domain's boundary, using a fundamental solution and a density.

The spectral density, or spectral power density, describes how the power or energy of a signal or random process is distributed over frequency, revealing oscillations and noise hidden in the time domain.

Spherical harmonics

Spherical harmonics, also written Ylm, are eigenfunctions of the Laplace-Beltrami operator on the sphere, introduced by Laplace in 1782 as a spherical analog of Fourier series.

Stieltjes transform

The Stieltjes transform of a measure μ on the real line is sμ(z) = ∫ μ(dx)/(x − z), an analytic function that encodes the measure.

Stockwell transform

The Stockwell transform, or S-transform, is a time-frequency analysis method introduced in 1996 that applies a frequency-dependent Gaussian window, used widely in seismology and power systems.

Triangle wave

A triangle wave is a periodic waveform that rises and falls at constant, equal slopes, forming a continuous symmetric zigzag used as the carrier in PWM circuits.

Volterra integral equation

A Volterra integral equation is an integral equation in which the integral's upper limit is the independent variable x, making it always uniquely solvable and equivalent to initial value problems.

Volterra series

The Volterra series is a functional expansion representing the output of nonlinear systems with memory as multi-dimensional convolutions, often described as a Taylor series with memory.

Wavelet packet decomposition

Wavelet packet decomposition is a signal processing method that recursively splits a signal into wavelet subbands at every frequency scale, forming a binary tree of coefficient sets.

Wavelet transform

The wavelet transform, or wavelet transformation, is a linear mathematical transform that decomposes signals into components localized in both time and frequency, unlike the Fourier transform.

Wiener–Hopf method

The Wiener–Hopf method is an analytical technique, created by Norbert Wiener and Eberhard Hopf in 1931, for solving integral equations on a half-line and mixed boundary-value problems in mathematical physics.

Wigner distribution function

The Wigner distribution function, known in signal processing as the Wigner–Ville distribution, is a bilinear transform representing a signal in time and frequency, introduced by Eugene Wigner in 1932.