Upper Bounds for Ratios of Lp Norms On Finite Dimensional Spaces Via Spectral Estimates.

摘要

The L sub 2p norm, for positive integers p, of a real (complex) polynomial pi sub n of degree at most n is shown to be the positive 2p-th root of a constrained quadratic (hermitian) form of certain linear operator. The 2p-th root of the spectral radius of this linear operator is shown to give an upper bound for the supremum of the ratio of the L sub 2p norm to the L sub 2 norm of pi sub n, where the supremum is taken over arbitrary real (complex) polynomials pi sub n. The underlying technique is not restricted to polynomials, and a generalization of these results to arbitrary finite dimensional function spaces which satisfy a certain Non-negativity Condition is presented.