Concept
An extension of Fourier analysis to groups and general spaces, with decomposition and convergence depending on the domain and function space.
Understand it in one breath
"A mirror between space and frequency." Under suitable conditions, a music signal can be analysed by the amplitude and phase of its frequency components and recovered by an inverse transform. JPEG and MP3 are not one-step schemes that merely discard high or “inaudible” frequencies: they combine transforms with perceptual models, quantization, and coding, and lossy settings can create visible or audible artifacts.
At a glance
Domain | Object | "Basis functions" |
|---|---|---|
ℝ (time signals) | Fourier transform | e^(2πiξx) |
Tⁿ (periodic functions) | Fourier series | e^(2πin·x) |
ℤⁿ (lattice) | Discrete Fourier transform | e^(2πik/n) |
Compact group G | Representation theory | Irreducible representations |
Sphere S² | Spherical harmonics | Yₗᵐ(θ, φ) |
Need localization on ℝ | Wavelet transform | ψ(2ʲx − k) |
The single idea "a function on a space = a decomposition into a suitable basis" extends from Fourier analysis to modern representation theory and quantum field theory.
Key formula
Modern applications
JPEG and MP3 compression, MRI reconstruction, 5G signal processing, and quantum field theory.
Beyond MathVoyage
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