A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2p+1 with a positive integer p, the problem is to construct an n×n real symmetric banded matrix with bandwidth 2p+1 whose m×m leading principal submatrix is a given m×m real symmetric banded matrix with bandwidth 2p+1 and spectrum is a given set of real numbers {λi}i=1n, where the number of distinct real numbers of {λi}i=1n is 2k when m=pk, or 2k+1 when pk