Transforms

Collections of classes that allow to compute the Fourier transform of a radially symmetric function and Hankel transforms.

• RadiallySymmetricFourierTransform: Compute the Fourier transform of a radially symmetric function using the correspondence with the Hankel transform

• HankelTransformBaddourChouinard: Compute the Hankel transform using Baddour and Chouinard discrete Hankel transform

• HankelTransformOgata: Compute the Hankel transform using Ogata quadrature

For more details, we refer to [HGBLachiezeR22].

class structure_factor.transforms.RadiallySymmetricFourierTransform(dimension)

Bases: object

Compute the Fourier transform of a radially symmetric function using the correspondence with the Hankel transform.

__init__(dimension)

Initialize the \(d\)-dimensional Fourier transform.

Parameters

dimension (int) – Dimension of the ambient space.

transform(f, k, method, **params)

Evaluate the Fourier transform of the radially symmetric function \(f\) at \(k\) using the correspondence with the Hankel transform.

Parameters

• f (callable) – Function to transform.

• k (scalar or numpy.ndarray) – Point or vector of points where the Fourier transform is to be evaluated.

• method (str) –

  Name of the method used to compute the underlying Hankel transform.

  • "Ogata" transform()

  • "BaddourChouinard" transform()

Keyword Arguments

params (dict) –

• If method="BaddourChouinard" (see [BC15]):

  • r_max (float): Threshold radius characterizing the space-limited feature of the function f, i.e., \(f(r)=0\) for r > r_max.

  • nb_points (int, optional): Number of quadrature nodes. Defaults to 300.

  • see also transform()

• If method="Ogata" (see [Oga05]):

  • r_max (float, optional): Maximum radius on which the input function \(f\) to be Hankel transformed was evaluated before the interpolation. Parameter used to conclude a lower bound on \(k\) on which \(f\) to be Hankel transformed. Defaults to None.

  • step_size (float, optional): Step size of the discretization scheme. Defaults to 0.01.

  • nb_points (int, optional): Number of quadrature nodes. Defaults to 300.

  • see also transform()

Returns

• k: Point(s) where the Fourier transform is to be evaluated.

• F_k: Fourier transform of f at k.

Return type

tuple (numpy.ndarray, numpy.ndarray)

Definition

The Hankel transform \(\mathcal{H}_{\nu}\) of order \(\nu\) of \(f\) is defined by

\[\mathcal{H}_{\nu -1}(f)(k) = \int_0^\infty f(r) J_{\nu}(kr)r \mathrm{d}k,\]

where \(J_{\nu}\) is the Bessel function of first kind.

The \(d\)-dimensional Fourier transform \(\mathcal{F}\) of the radially symmetric function \(f\) at \(k\) could be defined using the Hankel transform of \(x \rightarrow x^{d/2 -1}f(x)\) of order \(d/2 -1\) as follows,

\[k^{d/2-1} \mathcal{F}[f](k) = (2 \pi)^{d/2} \int_{0}^{+\infty} r^{d/2-1} f(r) J_{d/2-1}(kr) r \mathrm{d}r = (2 \pi)^{d/2} \mathcal{H}_{d/2-1}[\cdot^{d/2-1} f(\cdot)](k).\]

class structure_factor.transforms.HankelTransform(order)

Bases: object

Compute the Hankel transform of order \(\nu\).

See also

• RadiallySymmetricFourierTransform

• HankelTransformBaddourChouinard

• HankelTransformOgata

__init__(order)

Initialize the Hankel transform with prescribed order.

Parameters

order (int, optional) – Order of the Hankel transform.

class structure_factor.transforms.HankelTransformBaddourChouinard(order=0)

Bases: structure_factor.transforms.HankelTransform

Compute the Hankel transform, using the method of [BC15] considering that the input function is space-limited, i.e., \(f(r)=0\) for \(r>r_{max}\).

See also

• MatLab code of Baddour Chouinard

• Pyhank Python package

__init__(order=0)

Initialize the Hankel transform with prescribed order.

Parameters

order (int, optional) – Order of the Hankel transform.

compute_transformation_parameters(r_max, nb_points)

Compute the parameters involved in the evaluation of the corresponding Hankel-type transform using the discretization scheme of [BC15].

The following object’s attributes are defined

• bessel_zeros

• r_max

• transformation_matrix

Parameters

• r_max (float) – Threshold radius. Considering that the input function \(f\) to be Hankel transformed is space-limited, then r_max satisfies \(f(r)=0\) for r > r_max.

• nb_points (int) – Number of quadrature nodes.

transform(f, k=None, **interpolation_params)

Compute the Hankel transform of f at k.

Parameters

• f (callable) – Function to be Hankel transformed.

• k (numpy.ndarray, optional) –

  Points of evaluation of the Hankel transform. Defaults to None.

  • If k is None (default), then k = self.bessel_zeros[:-1] / self.r_max derived from compute_transformation_parameters().

  • If k is provided, the Hankel transform is first computed at the above k values (case k is None), then interpolated using scipy.interpolate.interp1d() with interpolation_params and finally evaluated at the provided k values.

Keyword Arguments

interpolation_params (dict) – Keyword arguments of scipy.interpolate.interp1d().

Returns

k and the evaluations of the Hankel transform of f at k.

Return type

tuple (scalar or numpy.ndarray, scalar or numpy.ndarray)

class structure_factor.transforms.HankelTransformOgata(order=0)

Bases: structure_factor.transforms.HankelTransform

Compute the Hankel transform using Ogata quadrature [Oga05], (Section 5).

See also

• Hankel Python package

__init__(order=0)

Initialize the Hankel transform with prescribed order.

Parameters

order (int, optional) – Order of the Hankel transform. Defaults to 0.

compute_transformation_parameters(r_max=None, nb_points=300, step_size=0.01)

Compute the quadrature nodes and weights used by [Oga05] (Equation (5.2)), to evaluate the corresponding Hankel-type transform.

Parameters

• r_max (float, optional) – Maximum radius on which the input function \(f\), to be Hankel transformed, was evaluated before the interpolation. Parameter used to conclude a lower bound on \(k\) on which \(f\) to be Hankel transformed. Defaults to None.

• step_size (float, optional) – Step size of the discretization scheme. Defaults to 0.01.

• nb_points (int, optional) – Number of quadrature nodes. Defaults to 300.

Returns

Quadrature nodes and weights.

Return type

tuple (numpy.ndarray, np.ndarray)

transform(f, k)

Compute the Hankel transform of f evaluated at k, following the work of [Oga05] (Section 5).

Parameters

• f (callable) – Function to be Hankel transformed.

• k (numpy.ndarray, optional) – Points of evaluation of the Hankel transform (1d array). Defaults to None.

Returns

• k: Points of evaluation of the Hankel transform.

• H_k: Evaluations of the Hankel transform of f on k.

Return type

tuple(numpy.ndarray, np.ndarray)

Important

Please call HankelTransformOgata.compute_transformation_parameters() to define quadrature attributes nodes and weights, before applying HankelTransformOgata.compute_transform().