Module functional_analysis 

Functional analysis tools for physics simulation.

Provides Hilbert and Banach spaces, operator spectrum, Sobolev spaces, functional derivatives (Gâteaux/Fréchet), and variational problems (Euler-Lagrange, Lagrange multiplier). Also includes L² inner products, Gram–Schmidt orthogonalization, Fourier/Chebyshev/Legendre expansions, Sobolev norms, and operator norm estimation via power iteration.

Structs

BanachSpace

A discrete Banach space with Lᵖ norm.

FunctionSpace

A finite-dimensional function space spanned by a list of basis functions.

FunctionalDerivative

Gâteaux and Fréchet derivatives of functionals on function spaces.

HilbertSpace

A discrete Hilbert space over a uniform grid with spacing dx.

OperatorSpectrum

Spectral analysis of compact operators represented as finite matrices.

SobolevSpace

A discrete Sobolev space H^k on a uniform grid.

VariationalProblem

Variational problem solver: Euler-Lagrange equations and constrained minimization with Lagrange multipliers.

Functions

chebyshev_expansion

Compute the Chebyshev expansion coefficients cₙ for a function sampled on [-1, 1] using the discrete cosine approach.

fourier_series_coeffs

Compute the Fourier series coefficients (aₙ, bₙ) for n = 0, 1, …, n_terms-1.

gram_schmidt_orthogonalize

Orthogonalize a set of sampled basis vectors using the modified Gram–Schmidt process in the L² inner product.

l2_inner_product

Compute the discrete L² inner product ⟨f, g⟩ = dx · Σ f[i]·g[i].

l2_norm

Compute the discrete L² norm ‖f‖ = √(⟨f, f⟩).

legendre_expansion

Compute the Legendre expansion coefficients for f sampled on [-1, 1].

operator_norm_estimate

Estimate the operator (spectral) norm of a matrix via power iteration.

sobolev_norm

Compute an approximate H^s Sobolev norm ‖f‖_{H^s}.

wavelet_haar_inverse

Compute the inverse Haar wavelet transform.

wavelet_haar_transform

Compute the full-length in-place Haar wavelet transform.