Matrices of minimum norm satisfying certain prescribed band and spectral restrictions -- an extremal characterization of the discrete periodic Laplacian

Abstract

This paper has been motivated by the curiosity that the circulant matrix ${\rm Circ }(1/2, -1/4, 0, \dots, 0,-1/4)$ is the $n\times n$ positive semidefinite, tridiagonal matrix $A$ of smallest Euclidean norm having the property that $Ae = 0$ and $Af = f$, where $e$ and $f$ are, respectively, the vector of all $1$s and the vector of alternating $1$ and $-1$s. It then raises the following question (minimization problem): What should be the matrix $A$ if the tridiagonal restriction is replaced by a general bandwidth $2r + 1$ ($1\leq r \leq \tfrac{n}{2 } -1$)? It is first easily shown that the solution of this problem must still be a circulant matrix. Then the determination of the first row of this circulant matrix consists in solving a least-squares problem having $\tfrac{n}{2} \, - 1$ nonnegative variables (Nonnegative Orthant) subject to $\tfrac{n}{2} - r$ linear equations. Alternatively, this problem can be viewed as the minimization of the norm of an even function vanishing at the points $|i|>r$ of the set $\left\{-\tfrac{n}{2} + 1, \dots, -1, 0, 1, \dots ,\tfrac{n}{2} \right\}$, and whose Fourier-transform is nonnegative, vanishes at zero, and assumes the value one at $\tfrac{n}{2}$. Explicit solutions are given for the special cases of $r=\tfrac{n}{2}$, $r=\tfrac{n}{2} -1$, and $r=2$. The solution for the particular case of $r=2$ can be physically interpreted as the vibrational mode of a ring-like chain of masses and springs in which the springs link both the nearest neighbors (with positive stiffness) and the next-nearest neighbors (with negative stiffness). The paper ends wiih a numerical illustration of the six cases ($1\leq r \leq 6$)corresponding to $n=12$.