Nonnegativity of a discrete quadratic functional in terms of the (strengthened) legendre and jacobi conditions

This paper contains a complete characterization of the nonnegativity of a discrete quadratic functional with one endpoint allowed to vary. In particular, we derive the exact form and explain the role of the (strengthened) Legendre condition in the discrete calculus of variations. Under this condition, the nonnegativity of the quadratic functional is equivalent to each of the following conditions: the nonexistence of intervals conjugate to 0, the existence of a certain conjoined basis of the associated Jacobi difference equation, the nonnegativity of certain recurrence matrices, and, under a natural additional assumption, the existence of a symmetric solution to the Riccati matrix difference equation. Moreover, an extension of the discrete Legendre condition is derived for the given discrete variational problem.