(8) |

The two-dimensional matrix of coefficients for the Laplacian operator
is shown in (8),
where,
on a Cartesian space, *h*=0,
and in the helix geometry, *h*=-1.
(A similar partitioned matrix arises from packing
a cylindrical surface into a array.)
Notice that the partitioning becomes transparent for the helix, *h*=-1.
With the partitioning thus invisible, the matrix
simply represents one-dimensional convolution
and we have an alternative analytical approach,
one-dimensional Fourier Transform.
We often need to solve sets of simultaneous equations
with a matrix similar to (8).
The method we use is triangular factorization.

Although the autocorrelation has mostly zero values, the factored autocorrelation from (5) has a great number of nonzero terms, but fortunately they seem to be converging rapidly (in the middle) so truncation (of the middle coefficients) seems reasonable. I wish I could show you a larger matrix, but all I can do is to pack the signal into shifted columns of a lower triangular matrix like this:

(9) |

6/2/1998