Harmonic Analysis & Time-Frequency Functions

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Abstractions about functions and operators in harmonic analysis, including time-frequency atoms and transforms such as the Gabor atom, Mathieu wavelet and Zak transform, boundary and potential-theoretic operators like harmonic measure and the Neumann-Poincare operator, and maximal-function tools such as the Hardy-Littlewood maximal function.

12 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Fourier analysis — The representation and study of functions or signals through sinusoidal or character components indexed by frequency.
• Gabor atom — A time–frequency atom formed by translating, modulating and sometimes scaling a localized window function.
• Hardy–Littlewood maximal function — The pointwise supremum of local absolute-value averages of a function over all balls or cubes containing the evaluation point.
• Harmonic measure — A boundary probability measure giving the likelihood that Brownian motion started inside a domain first exits through each boundary subset, equivalently representing solutions of the Dirichlet problem.
• Hermitian function — A complex-valued function satisfying conjugate symmetry f(-x)=conjugate(f(x)), equivalently having an even real part and an odd imaginary part.
• Mathieu wavelet — A wavelet family constructed from periodic Mathieu functions and their associated filter coefficients.
• Maximal function — A harmonic-analysis operator assigning each point the supremum of local averages of a function over neighborhoods containing or centered there.
• Neumann–Poincaré operator — A boundary integral operator built from the normal derivative of the Laplace fundamental solution and used to reduce harmonic boundary-value problems to Fredholm integral equations.
• Pluriharmonic function — A function on a complex manifold whose restriction to every complex line is harmonic, locally the real part of a holomorphic function in the real-valued case.
• Progressive function — An L2 signal whose Fourier transform is supported only on nonnegative frequencies, equivalently a boundary function in the upper-half-plane Hardy space under the stated convention.
• Subharmonic function — An upper-semicontinuous function whose value at each point is no greater than the average over every sufficiently small surrounding sphere or ball.
• Zak transform — A quasi-periodic time–frequency transform that maps a function on the real line to a function on a two-dimensional fundamental cell indexed by position and frequency phase.