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Band matrices are often called Toeplitz matrices.
In the subject of Time Series Analysis are found
spectral factorization methods that require computations
proportional to the dimension of the matrix squared.
These calculations can often be terminated early with a reasonable partial result.
Two Toeplitz methods, the Levinson method
and the Burg method,
are described in my first textbook, FGDP.
Our interest is multidimensional data sets,
the matrices of interest are truely huge and the cost
of Toeplitz methods is proportional to the square of the matrix size.
Thus, before we find Toeplitz methods
especially useful, we may need to find
ways to take advantage of the sparsity of our filters.