Nonnegativity constraints for structured complete systems

Abstract:

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