Nonnegativity constraints for structured complete systems

Abstract

We investigate pointwise nonnegativity as an obstruction to various types of structured completeness in L p ( R ) L^p(\mathbb {R}) . For example, we prove that if each element of the system { f n } n = 1 ∞ ⊂ L p ( R ) \{f_n\}_{n=1}^\infty \subset L^p(\mathbb {R}) is pointwise nonnegative, then { f n } n = 1 ∞ \{f_n\}_{n=1}^{\infty } cannot be an unconditional basis or unconditional quasibasis (unconditional Schauder frame) for L p ( R ) L^p(\mathbb {R}) . In particular, in L 2 ( R ) L^2(\mathbb {R}) this precludes the existence of nonnegative Riesz bases and frames. On the other hand, there exist pointwise nonnegative conditional quasibases in L p ( R ) L^p(\mathbb {R}) , and there also exist pointwise nonnegative exact systems and Markushevich bases in L p ( R ) L^p(\mathbb {R}) .