  We discuss the notion of "discrete function bases" with a particular focus on
the discrete basis derived from the Legendre Delay Network (LDN). We
characterize the performance of these bases in a delay computation task, and as
fixed temporal convolutions in neural networks. Networks using fixed temporal
convolutions are conceptually simple and yield state-of-the-art results in
tasks such as psMNIST.
  Main Results
  (1) We present a numerically stable algorithm for constructing a matrix of
DLOPs L in O(qN)
  (2) The Legendre Delay Network (LDN) can be used to form a discrete function
basis with a basis transformation matrix H in O(qN).
  (3) If q < 300, convolving with the LDN basis online has a lower run-time
complexity than convolving with arbitrary FIR filters.
  (4) Sliding window transformations exist for some bases (Haar, cosine,
Fourier) and require O(q) operations per sample and O(N) memory.
  (5) LTI systems similar to the LDN can be constructed for many discrete
function bases; the LDN system is superior in terms of having a finite impulse
response.
  (6) We compare discrete function bases by linearly decoding delays from
signals represented with respect to these bases. Results are depicted in Figure
20. Overall, decoding errors are similar. The LDN basis has the highest and the
Fourier and cosine bases have the smallest errors.
  (7) The Fourier and cosine bases feature a uniform decoding error for all
delays. These bases should be used if the signal can be represented well in the
Fourier domain.
  (8) Neural network experiments suggest that fixed temporal convolutions can
outperform learned convolutions. The basis choice is not critical; we roughly
observe the same performance trends as in the delay task.
  (9) The LDN is the right choice for small q, if the O(q) Euler update is
feasible, and if the low O(q) memory requirement is of importance.
