Harmonic Analysis skills for AI agents

Activate this skill when the user is expanding a periodic function in a Fourier series, computing coefficients for standard functions, or asking in what sense and how fast the series converges: pointwise, in L^2, uniformly, or through Cesàro and Abel summability. Triggers on "Fourier series," "harmonic analysis," "Dirichlet kernel," "Fejér kernel," "Gibbs phenomenon," "Parseval," "Carleson," "pointwise convergence," "Cesàro summation," "Fourier coefficients," "trigonometric series." Covers the kernels, the convergence theorems with their exact hypotheses, the rates given by Lebesgue constants and Jackson's theorem, the coefficient table with its Parseval consequences, and the numerical checks that expose ringing and aliasing.

Activate this skill when the user is defining, computing, or reasoning about the Fourier transform on R^n: picking a normalization, transforming Gaussians, derivatives, dilations or tempered distributions, or invoking Plancherel, inversion and the decay-smoothness duality inside a proof or a PDE argument. Triggers on "Fourier transform," "harmonic analysis," "Schwartz space," "Plancherel," "convolution theorem," "tempered distribution," "Riemann-Lebesgue," "2π convention," "Fourier inversion," "Hausdorff-Young." Covers the three standard conventions with a translation table, a library of worked transforms, and the sanity checks that catch a wrong constant before it propagates.

Activate this skill when the user is doing Fourier analysis on a group other than R^n: characters and the dual of a locally compact abelian group, Haar measure, the DFT on Z/nZ, Fourier series on the torus, Walsh-Hadamard analysis on the hypercube, Peter-Weyl for compact groups, representation-theoretic Fourier transforms on finite nonabelian groups, or Poisson summation and its consequences. Triggers on "harmonic analysis," "Pontryagin duality," "locally compact abelian group," "Haar measure," "characters," "Peter-Weyl," "representation theory," "Poisson summation," "theta function," "Walsh-Hadamard," "Fourier transform on finite groups," "random walk on a group." Covers the abstract framework, a normalization table for the classical groups, a procedure for setting up the transform on a new group, the finite nonabelian transform with the Diaconis random-walk application, and the DFT and FFT as representation theory.

Activate this skill when the user needs to decompose a function into dyadic frequency pieces, bound a Fourier multiplier on L^p, define or compare Sobolev, Besov and Triebel-Lizorkin spaces, apply Bernstein inequalities, or estimate products and nonlinear terms with paraproducts in a PDE argument. Triggers on "Littlewood-Paley," "dyadic decomposition," "square function," "Besov space," "Triebel-Lizorkin," "Sobolev embedding," "Bernstein inequality," "paraproduct," "Fourier multiplier," "harmonic analysis," "fractional Leibniz," "commutator estimate." Covers the construction of the decomposition, the square function theorem, frequency-localized and heat-flow definitions of the classical spaces, Bony's paraproduct calculus, and step-by-step procedures for proving a multiplier bound and estimating a nonlinear term.

Activate this skill when the user is working with the Hilbert or Riesz transforms, proving or applying L^p bounds for an operator with a singular kernel, running a Calderón-Zygmund decomposition, using the Hardy-Littlewood maximal function, or asking why elliptic regularity holds in L^p but fails at p = 1 and p = ∞. Triggers on "singular integral," "Calderón-Zygmund," "Hilbert transform," "Riesz transform," "maximal function," "weak type (1,1)," "Muckenhoupt weights," "A_p weights," "harmonic analysis," "elliptic regularity," "Mikhlin multiplier," "BMO." Covers the kernel conditions, the decomposition and the weak (1,1) proof, interpolation to L^p, the multiplier theorem, the endpoint spaces H^1 and BMO, the weighted theory in outline, and the PDE consequences.

Activate this skill when the user is estimating a power spectrum from sampled data and needs to choose between the periodogram, Welch averaging and multitaper methods, pick a window, understand leakage and the resolution-variance trade-off, detect sinusoidal lines against noise, or interpret a plotted spectrum without fooling themselves. Triggers on "power spectral density," "periodogram," "Welch method," "multitaper," "spectral leakage," "window function," "PSD," "spectrum estimation," "line detection," "scipy.signal.welch," "harmonic analysis." Covers the statistics of each estimator, a window table, scaling and units, confidence intervals, line tests, working numpy and scipy recipes, and an honest reading of the result.

Activate this skill when the user is expanding a function on the sphere, solving Laplace's equation in spherical coordinates, using the addition theorem or Funk-Hecke formula, choosing normalizations and phase conventions, or computing spherical harmonic transforms for geodesy, graphics lighting or cosmology. Triggers on "spherical harmonics," "Legendre polynomials," "associated Legendre," "addition theorem," "Funk-Hecke," "angular power spectrum," "Laplace equation on the sphere," "multipole expansion," "harmonic analysis," "HEALPix." Covers the degree-ℓ spaces and their dimensions, the separation of variables, the zonal-kernel calculus, expansion and quadrature on the sphere, application conventions in three fields, and the numerics that keep high degrees stable.

Activate this skill when the user is reasoning about how concentrated a function and its Fourier transform can simultaneously be, reconstructing a band-limited signal from samples, diagnosing aliasing, choosing a sampling rate or a window for time-frequency analysis, or citing Heisenberg, Hardy, Donoho-Stark, Shannon-Nyquist or Paley-Wiener correctly. Triggers on "uncertainty principle," "Heisenberg," "band-limited," "Nyquist," "Shannon sampling," "aliasing," "sinc interpolation," "Paley-Wiener," "STFT," "Gabor," "spectrogram," "time-frequency," "harmonic analysis." Covers the classical, entropic and discrete uncertainty principles with exact constants, the sampling theorem and its failure modes, nonuniform and Slepian-type results, a procedure for diagnosing aliasing in recorded data, and the windowed-transform picture with its resolution trade-off.

Activate this skill when the user is building, choosing, or debugging a wavelet decomposition: setting up a multiresolution analysis, deriving scaling and wavelet filters, running the fast wavelet transform, thresholding coefficients for denoising or compression, or dealing with artifacts at signal boundaries. Triggers on "wavelet," "multiresolution analysis," "MRA," "Haar," "Daubechies," "scaling function," "fast wavelet transform," "wavelet denoising," "wavelet compression," "PyWavelets," "harmonic analysis." Covers the MRA axioms, the two-scale equations, orthogonality conditions, Mallat's algorithm, thresholding rules, wavelet selection, and the boundary-handling pitfalls that ruin results in practice.