hofa.norm#
This module contains the Numpy implementation for computing Gowers norms
on functions defined on finite abelian groups represented by numpy.ndarray.
Conventions#
The following conventions are used for functions in this file:
Any finite abelian group \(Z\) is isomorphic to \(\mathbb{Z}/n_1\mathbb{Z} \times \mathbb{Z}/n_2\mathbb{Z} \times ... \times \mathbb{Z}/n_k\mathbb{Z}\). Therefore, a function \(f: Z \to \mathbb{C}\) is represented as a
numpy.ndarraytensor with shape(n_1, n_2, ..., n_k).Sometimes we abreviate
numpybynp.
Functions#
Module Contents#
- hofa.norm.u_pow(f: numpy.ndarray, k: int)#
The \(2^k\) power of the Gowers \(U^k\) norm of
f.- Parameters:
f (np.ndarray) – A function \(f: Z \to \mathbb{C}\) defined on a finite abelian group \(Z\).
k (int) – An integer representing the order of the Gowers norm.
- Returns:
The \(2^k\) power of the \(U^k\) norm of
f.- Return type:
float
- hofa.norm.u(f: numpy.ndarray, k: int)#
The Gowers \(U^k\) norm of
f.- Parameters:
f (np.ndarray) – A function \(f: Z \to \mathbb{C}\) defined on a finite abelian group \(Z\).
k (int) – An integer representing the order of the Gowers norm.
- Returns:
The \(U^k\) norm of
f.- Return type:
float