freealg.AlgebraicForm.hilbert

AlgebraicForm.hilbert(x=None, plot=False, latex=False, save=False)

Compute Hilbert transform of the spectral density.

Parameters:

xnumpy.array, default=None

The locations where Hilbert transform is evaluated at. If None, an interval slightly larger than the support interval of the spectral density is used.

plotbool, default=False

If True, density is plotted.

latexbool, default=False

If True, the plot is rendered using LaTeX. This option is relevant only if plot=True.

savebool, default=False

If not False, the plot is saved. If a string is given, it is assumed to the save filename (with the file extension). This option is relevant only if plot=True.

Returns:

hilbnumpy.array

The Hilbert transform on the locations x.

See also

density

stieltjes

Notes

The Hilbert transform of s spectral density is

\[H(x) = \mathcal{H}[\rho](x) = \frac{1}{\pi} \mathrm{p.v.} \int_{\mathbb{R}} \frac{\rho(y)}{x - y}\, \mathrm{d}y.\]

It can be directly computed from the non-tangential limit of the Stieltjes transform by

\[H(x) = -\pi \, \lim_{\epsilon \to 0^{+}} \Re(m(x + i \epsilon)).\]

Examples

>>> # Create a distribution with two bulks
>>> from freealg.distributions import CompoundFreePoisson
>>> cfp = CompoundFreePoisson(t=[2.0,  5.5], w=[0.75, 0.25],
...                           lam=0.1)

>>> # Create AlgebraicForm and fit the distribution
>>> from freealg import AlgebraicForm
>>> af = AlgebraicForm(cfp)
>>> af.fit(deg_m=3, deg_z=1)

>>> # Plot the density of the fitted spectral curve
>>> import numpy
>>> x = numpy.linspace(0, 8, 1000)
>>> hilb = af.hilbert(x, plot=True)