Computations on abelian groups.

## Project description

`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
- 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
- The
`LCAFunc`class represents functions from LCAs to complex numbers. - Fundamental methods: evaluation, composition, shift (translation), pullback, pushforward, point-wise operators (i.e. addition).

- The

## 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.

## Project details

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