e3x.nn.functions.trigonometric.reciprocal_fourier

e3x.nn.functions.trigonometric.reciprocal_fourier(x, num, kind='shifted', use_reciprocal_weighting=False)

Reciprocal Fourier basis functions.

Computes the basis functions (see basic_fourier and reciprocal_mapping)

\[\mathrm{reciprocal\_fourier}_k(x) = \mathrm{fourier}_k(1-\mathrm{reciprocal\_mapping}(x))\]

where \(k=0 \dots K-1\) with \(K\) = num.

Plot for \(K = 5\) (kind = 'shifted', use_reciprocal_weighting = False):

Plot for \(K = 5\) (kind = 'shifted', use_reciprocal_weighting = True):

Plot for \(K = 5\) (kind = 'damped', use_reciprocal_weighting = False):

Plot for \(K = 5\) (kind = 'damped', use_reciprocal_weighting = True):

Plot for \(K = 5\) (kind = 'cuspless', use_reciprocal_weighting = False):

Plot for \(K = 5\) (kind = 'cuspless', use_reciprocal_weighting = True):

Parameters:

• x (<class 'Float[Array, '...']'>) – Input array.

• num (int) – Number of basis functions \(K\).

• kind ({'shifted', 'damped', 'cuspless'}, default: 'shifted') – Which kind of reciprocal mapping is used.

• use_reciprocal_weighting (bool, default: False) – If True, the functions are weighted by the value of the reciprocal mapping.

Return type:

<class 'Float[Array, '... num']'>

Returns:

Value of all basis functions for all values in x. The output shape follows the input, with an additional dimension of size num appended.