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Harmonic Transforms & Wave Expansions

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Abstractions about harmonic functions, time-frequency atoms, integral transforms, special functions, plane-wave expansions, and pseudospectral methods.

9 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.

  • Gabor atom — A time–frequency atom formed by translating, modulating and sometimes scaling a localized window function.
  • Mehler–Fock transform — An integral transform using conical Legendre functions as its kernel, with a weighted inverse transform on the half-line under suitable analytic conditions.
  • Meijer G-function — Define a highly general special function by a Mellin–Barnes contour integral whose gamma-factor parameters subsume many hypergeometric and classical functions.
  • Plane wave expansion method — A frequency-domain eigenmethod that expands periodic material parameters and electromagnetic fields in reciprocal-lattice plane waves to compute photonic-crystal band structures.
  • Positive harmonic function — Characterize a nonnegative harmonic function on the unit disc as the Poisson integral of a unique finite positive boundary measure, with normalization at the origin fixing the measure's total mass.
  • Pseudospectral time-domain method — A wave-simulation method that computes spatial derivatives through global spectral transforms while advancing the field on a discrete time grid.
  • Sombrero function — The radial two-dimensional analogue of sinc, commonly defined as 2J1(πρ)/(πρ), and arising as the Fourier transform of a circular aperture.
  • Wigner–Weyl transform — An invertible correspondence between phase-space functions and quantum operators that underlies the quasiprobability formulation of quantum mechanics.
  • 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.