FACTORIZATION OF TOTALLY POSITIVE, SYMMETRICAL, PERIODIC, BANDED MATRICES WITH APPLICATIONS

We show that a totally positive, symmetric, periodic, banded matrix A can be factored in a symmetric manner into positive one-banded periodic factors and deduce from this that A has positive eigenvalues with certain restrictions on their coincidence and on the sign changes of the corresponding eigenvectors. From this are derived spectral properties of certain positive, periodic spline operators.