hofa.norm#
This module contains the Numpy implementation for computing Gowers norms on NumPy ndarrays
Conventions#
The following conventions are used for functions and data structures in this file:
Any finite abelian group \(Z\) is isomorphic to \(Z/n_1Z \times Z/n_2Z \times ... \times Z/n_kZ\). Therefore, a function \(f: Z \to \mathbb{C}\) is represented as a
numpy.ndarraytensor with shape(n_1, n_2, ..., n_k).A \(Z\)-matrix, where \(Z = Z/n_1Z \times ... \times Z/n_kZ\), is represented as a
numpy.ndarraytensor with shape(n_1, ..., n_k, n_1, ..., n_k).
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.
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.
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