Harmonic Analysis

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12 domain-specific abstractions whose origin domain is Harmonic Analysis.

• Fourier algebra — The commutative Banach algebra of coefficient functions of the left regular representation of a locally compact group, under pointwise multiplication.
• Fourier analysis — The representation and study of functions or signals through sinusoidal or character components indexed by frequency.
• Fourier transform on finite groups — A harmonic transform mapping a function on a finite group to matrices indexed by irreducible representations, generalizing the scalar discrete Fourier transform beyond abelian groups.
• 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.
• Idempotent measure — A probability measure on a topological group that is unchanged by convolution with itself.
• Maximal function — A harmonic-analysis operator assigning each point the supremum of local averages of a function over neighborhoods containing or centered there.
• 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.
• Multiresolution Analysis — A dilation-linked nested sequence of approximation spaces whose successive orthogonal complements isolate wavelet detail across scales.
• 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.
• Riesz potential — Apply the convolution kernel proportional to |x|^{α−n} to realize a fractional inverse power of the Laplacian on Euclidean space.
• 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.