a function on the sphere by its spherical harmonic series, or a list of such functions; the default representation of a function on the sphere

Syntax

sF = S2FunHarmonic(fhat)

From MTEX’s coefficients.

sF = S2FunHarmonic(fhat, cs, antipodal=True)

In a crystal frame, f(v) = f(-v).

sF = S2FunHarmonic(lambda v: v.z**2, bandwidth=16)

A callable expanded by quadrature.

sF = S2FunHarmonic(psi)

The series of a kernel on the sphere.

sF = S2FunHarmonic(sF, bandwidth=8)

Another function, truncated.

Description

The harmonic series is the form on which convolution, rotation, products and integrals are cheap. The coefficients fhat are MTEX’s, degree by degree, (L + 1)^2 of them for the bandwidth L; a vector ending inside a degree is padded with zeros. A list of functions carries its shape in front of the coefficients.

The first argument may also be:

a callable of vector3d: sampled on a quadrature grid of the bandwidth, by default the preference defaultS2Bandwidth; S2FunHarmonic.quadrature takes values at nodes
an S2Kernel: its series
another S2Fun: its harmonic form, truncated to the bandwidth if given

The function is made invariant under the group of its frame unless symmetrise=False, so a callable that is not invariant becomes its mean over the symmetric directions. A frame with an inversion makes the function antipodal.

Input

`fhat`	array of MTEX’s coefficients, an S2Fun, an S2Kernel or a callable of vector3d
`frame`	the crystalFrame or specimenFrame of the directions

Keywords

keyword	default	description
`real`	`None`	whether the function takes real values only; by default read off the coefficients or the sampled values
`antipodal`	`None`	f(v) = f(-v)
`symmetrise`	`True`	make the coefficients invariant under the group of the frame
`bandwidth`	`None`	the largest degree of the series
`dtype`	`None`	the complex type of the coefficients, complex128 unless given

Output

`sF`	S2FunHarmonic

Properties

`fhat`	spherical harmonic coefficients in MTEX’s public basis
`bandwidth`	largest spherical harmonic degree
`frame`	reference frame carrying the function symmetry
`CS`	alias for the spherical reference frame
`isReal`	whether the function is real-valued
`antipodal`	whether f(v)=f(-v)
`shape`	shape of a list of functions, excluding the coefficient axis

Used on

SO3FunConvolution, S2FunHarmonicRepresentation, S2FunSampling, S2FunVectorValued, SphericalHarmonics

MATLAB’s S2FunHarmonic.S2FunHarmonic

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/S2FunHarmonic.S2FunHarmonic_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.

S2FunHarmonic.S2FunHarmonic