abelian

abelian is a Python library for computations on elementary locally compact abelian groups (LCAs). The elementary LCAs are the groups R, Z, T = R/Z, Z_n and direct sums of these. The Fourier transformation is defined on these groups. With abelian it is possible to sample, periodize and perform Fourier analysis on elementary LCAs using homomorphisms between groups.

Classes and methods

The most important classes are listed below. The software contains many other functions and methods not listed.

• The LCA class represents elementary LCAs, i.e. R, Z, T = R/Z, Z_n and direct sums of these groups.

  • Fundamental methods: identity LCA, direct sums, equality, isomorphic, element projection, Pontryagin dual.

• The HomLCA class represents homomorphisms between LCAs.

  • Fundamental methods: identity morphism, zero morphism, equality, composition, evaluation, stacking, element-wise operations, kernel, cokernel, image, coimage, dual (adjoint) morphism.

• The LCAFunc class represents functions from LCAs to complex numbers.

  • Fundamental methods: evaluation, composition, shift (translation), pullback, pushforward, point-wise operators (i.e. addition).

Example

The following example shows Fourier analysis on a hexagonal lattice.

We create a Gaussian on R^2 and a homomorphism for sampling.

from abelian import LCA, HomLCA, LCAFunc, voronoi
from math import exp, pi, sqrt
Z = LCA(orders = [0], discrete = [True])
R = LCA(orders = [0], discrete = [False])

# Create the Gaussian function on R^2
function = LCAFunc(lambda x: exp(-pi*sum(j**2 for j in x)), domain = R**2)

# Create an hexagonal sampling homomorphism (lattice on R^2)
phi = HomLCA([[1, 1/2], [0, sqrt(3)/2]], source = Z**2, target = R**2)
phi = phi * (1/7) # Downcale the hexagon
function_sampled = function.pullback(phi)

Next we approximate the two-dimensional integral of the Gaussian.

# Approximate the two dimensional integral of the Gaussian
scaling_factor = phi.A.det()
integral_sum = 0
for element in phi.source.elements_by_maxnorm(list(range(20))):
    integral_sum += function_sampled(element)
print(integral_sum * scaling_factor) # 0.999999997457763

We use the FFT to move approximate the Fourier transform of the Gaussian.

# Sample, periodize and take DFT of the Gaussian
phi_p = HomLCA([[10, 0], [0, 10]], source = Z**2, target = Z**2)
periodized = function_sampled.pushforward(phi_p.cokernel())
dual_func = periodized.dft()

# Interpret the output of the DFT on R^2
phi_periodize_ann = phi_p.annihilator()

# Compute a Voronoi transversal function, interpret on R^2
sigma = voronoi(phi.dual(), norm_p=2)
factor = phi_p.A.det() * scaling_factor
total_error = 0
for element in dual_func.domain.elements_by_maxnorm():
    value = dual_func(element)
    coords_on_R = sigma(phi_periodize_ann(element))

    # The Gaussian is invariant under Fourier transformation, so we can
    # compare the error using the analytical expression
    true_val = function(coords_on_R)
    approximated_val = abs(value)
    total_error += abs(true_val - approximated_val*factor)

assert total_error < 10e-15

Please see the documentation for more examples and information.