a function on the circle by its Fourier coefficients fhat, k = -L..L, over e^{-ikx} as MATLAB’s

Syntax

sF = S1FunHarmonic([0.25, 0, 1, 0, 0.25])

From the coefficients, k = -2..2.

sF = S1FunHarmonic(lambda x: np.sin(2*x), bandwidth=16)

A callable by quadrature.

sF = S1FunHarmonic(psi, bandwidth=32)

The Fourier series of an S1 kernel.

sF = S1FunHarmonic(sF, bandwidth=8)

Another function, cut to a bandwidth.

Description

The coefficients run from the degree -L to L, so a coefficient vector has odd length 2 L + 1. A callable is sampled on the equispaced quadrature grid of the bandwidth, by default the preference maxS1Bandwidth; an S1 kernel, an S1FunHandle or another S1FunHarmonic gives its own coefficients, cut or padded to the bandwidth when one is given. A real function keeps c_{-k} = conj(c_k), which is checked from the coefficients unless real says it.

Input

fhat array of 2 L + 1 coefficients, a callable of the angle, an S1Kernel or an S1Fun

Keywords

keyword	default	description
`real`		whether the function takes real values only; by default read off the coefficients or the sampled values
`bandwidth`		the largest degree L of the series
`dtype`		the complex type of the coefficients, complex128 unless given

Output

`sF`	S1FunHarmonic

Properties

`fhat`	complex coefficients ordered from Fourier mode -L through L
`bandwidth`	largest Fourier mode L
`isReal`	whether the function is constrained to have real values
`antipodal`	whether f(x+pi)=f(x), read from its coefficients
`even, odd`	boolean symmetry tests under x -> -x

Members

abs	the pointwise absolute value, a Fourier series of a smoothing bandwidth
abs	the pointwise absolute value, a Fourier series of a smoothing bandwidth
antipodal	whether f(x+pi)=f(x), tested by checking that every odd Fourier mode vanishes
approximate	the Fourier series of a function on the circle by quadrature
bandwidth	the bandwidth L, the largest frequency of the coefficients
cat	cat(a, b, …) is type(a).cat(a, b, …): the list of the operands, MATLAB’s [a, b], the carried attributes joined
conj	the complex conjugate function
conj	the complex conjugate function
conv	smoothing by a kernel is one factor per mode
conv	smoothing by a kernel is one factor per mode
diff	differentiate the circular function with respect to its angle in radians
dot	the inner product with g for the normalised measure, by the coefficients
eval	evaluate the Fourier series at angles in radians
even	whether f(x)=f(-x), tested from the Fourier coefficients
exp	the pointwise exponential, a Fourier series of a smoothing bandwidth, or at the bandwidth given
grad	the derivative as a function, or its values at the angles x
interpolate	the Fourier series through noisy samples at angles on the circle, fitted by regularised least squares
isclose	whether the coefficients of g agree within atol + rtol times their norm
log	the pointwise logarithm, a Fourier series of a smoothing bandwidth, or at the bandwidth given
max	global, local and pointwise maxima of a function on the circle
max	global, local and pointwise maxima of a function on the circle
mean	the mean over the circle, the coefficient c_0
mean	the mean over the circle, the coefficient c_0
min	global, local and pointwise minima of a function on the circle
min	global, local and pointwise minima of a function on the circle
minus
mrdivide
mtimes
mtimes
norm	the L2 norm for the normalised measure, the Euclidean norm of the coefficients
norm	the L2 norm for the normalised measure, the Euclidean norm of the coefficients
odd	whether f(x)=-f(-x), tested from the Fourier coefficients
plot	plot of a function on the circle, its values as the radius of a polar line
plotSpektra	the Chebyshev coefficients of a kernel, or the harmonic power of a function, per degree
plus
plus
power
power
quadrature	the Fourier series of a function on the circle by quadrature
rdivide
rdivide
rotate	shift the circular function by an angle in radians
S1Fun	a function on the circle by its Fourier coefficients fhat, k = -L..L, over e^{-ikx} as MATLAB’s
S1FunHarmonic	a function on the circle by its Fourier coefficients fhat, k = -L..L, over e^{-ikx} as MATLAB’s
smooth	the convolution with a de la Vallee Poussin kernel, 2.9 degree unless a kernel or a halfwidth is given
sqrt	the pointwise square root, a Fourier series of a smoothing bandwidth, or at the bandwidth given
sqrt	the pointwise square root, a Fourier series of a smoothing bandwidth, or at the bandwidth given
sum	the integral over the circle, 2 pi times the mean, MATLAB’s sum
sum	the integral over the circle, 2 pi times the mean, MATLAB’s sum
times
times
truncate	return the same circular function represented through Fourier mode L
uminus
uminus

Used on

S1FunHarmonics

MATLAB’s S1FunHarmonic_index

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/S1FunHarmonic_index_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.

The class S1FunHarmonic