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        "author": "Tommy Odland",
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        "description": "=======\nabelian\n=======\n\n.. image:: https://readthedocs.org/projects/abelian/badge/?version=latest\n   :target: http://abelian.readthedocs.io/en/latest/?badge=latest\n   :alt: Documentation Status\n\n``abelian`` is a Python library for computations on elementary locally compact abelian groups (LCAs).\nThe elementary LCAs are the groups R, Z, T = R/Z, Z_n and direct sums of these.\nThe Fourier transformation is defined on these groups.\nWith ``abelian`` it is possible to sample, periodize and perform Fourier\nanalysis on elementary LCAs using homomorphisms between groups.\n\n.. image:: http://tommyodland.com/abelian/intro_figure_75.png\n\nClasses and methods\n^^^^^^^^^^^^^^^^^^^^^\n\nThe most important classes are listed below. The software contains many other functions and methods not listed.\n\n* The ``LCA`` class represents elementary LCAs, i.e. R, Z, T = R/Z, Z_n and direct sums of these groups.\n   * Fundamental methods: identity LCA, direct sums, equality, isomorphic, element projection, Pontryagin dual.\n\n* The ``HomLCA`` class represents homomorphisms between LCAs.\n   * Fundamental methods: identity morphism, zero morphism, equality, composition, evaluation, stacking, element-wise operations, kernel,    cokernel, image, coimage, dual (adjoint) morphism.\n\n* The ``LCAFunc`` class represents functions from LCAs to complex numbers.\n   * Fundamental methods: evaluation, composition, shift (translation), pullback, pushforward, point-wise operators (i.e. addition).\n\nExample\n^^^^^^^^^^^^^^^^^^^^^\n\n.. image:: http://tommyodland.com/abelian/fourier_hexa_33.png\n\nThe following example shows Fourier analysis on a hexagonal lattice.\n\nWe create a Gaussian on R^2 and a homomorphism for sampling.\n\n.. code:: python\n\n    from abelian import LCA, HomLCA, LCAFunc, voronoi\n    from math import exp, pi, sqrt\n    Z = LCA(orders = [0], discrete = [True])\n    R = LCA(orders = [0], discrete = [False])\n\n    # Create the Gaussian function on R^2\n    function = LCAFunc(lambda x: exp(-pi*sum(j**2 for j in x)), domain = R**2)\n\n    # Create an hexagonal sampling homomorphism (lattice on R^2)\n    phi = HomLCA([[1, 1/2], [0, sqrt(3)/2]], source = Z**2, target = R**2)\n    phi = phi * (1/7) # Downcale the hexagon\n    function_sampled = function.pullback(phi)\n\nNext we approximate the two-dimensional integral of the Gaussian.\n\n.. code:: python\n\n    # Approximate the two dimensional integral of the Gaussian\n    scaling_factor = phi.A.det()\n    integral_sum = 0\n    for element in phi.source.elements_by_maxnorm(list(range(20))):\n        integral_sum += function_sampled(element)\n    print(integral_sum * scaling_factor) # 0.999999997457763\n\n\nWe use the FFT to move approximate the Fourier transform of the Gaussian.\n\n.. code:: python\n\n    # Sample, periodize and take DFT of the Gaussian\n    phi_p = HomLCA([[10, 0], [0, 10]], source = Z**2, target = Z**2)\n    periodized = function_sampled.pushforward(phi_p.cokernel())\n    dual_func = periodized.dft()\n\n    # Interpret the output of the DFT on R^2\n    phi_periodize_ann = phi_p.annihilator()\n\n    # Compute a Voronoi transversal function, interpret on R^2\n    sigma = voronoi(phi.dual(), norm_p=2)\n    factor = phi_p.A.det() * scaling_factor\n    total_error = 0\n    for element in dual_func.domain.elements_by_maxnorm():\n        value = dual_func(element)\n        coords_on_R = sigma(phi_periodize_ann(element))\n\n        # The Gaussian is invariant under Fourier transformation, so we can\n        # compare the error using the analytical expression\n        true_val = function(coords_on_R)\n        approximated_val = abs(value)\n        total_error += abs(true_val - approximated_val*factor)\n\n    assert total_error < 10e-15\n\nPlease see `the documentation <http://abelian.readthedocs.io/en/latest/>`_ for more examples and information.\n\n",
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