Nonnegativity constraints for structured complete systems (Q2790715)

The main results of the paper show that pointwise nonnegativity is an obstruction to unconditional basis and frames in \(L^p[0,1].\) More precisely, fix \(1 < p < \infty\) and let \(\frac{1}{p}+\frac{1}{q} = 1\). Suppose that \(\left\{f_n\right\}_{n=1}^\infty\subset L^p[0,1]\) and \(\left\{g_n\right\}_{n=1}^\infty\subset L^q[0,1]\) satisfy NEWLINE\[NEWLINE A\|g\|_q^q \leq \int_0^1\biggl(\sum_{n=1}^\infty|\left<g,f_n\right>|^2 |g_n(t)|^2\biggr)^\frac{q}{2}\;dt \leq B\|g\|_q^q\;\;\forall g\in L^q[0,1] NEWLINE\]NEWLINE for some constants \(0 < A\leq B < \infty.\) Then \(f_n\) cannot be a.e. nonnegative for each \(n\geq 1\). This is the content of Theorem 4.1. As a consequence, for \(1 < p < \infty\) there does not exist any unconditional basis for \(L^p[0,1]\) consisting of nonnegative a.e. functions. Nor is there any frame for \(L^2[0,1]\) formed by nonnegative functions. These results are generalized in Section 5, were the nonexistence of a nonnegative unconditional quasibases for \(L^p[0,1]\) is proved. The paper also presents examples of Markushevich or conditional quasibasis for \(L^p[0,1]\) consisting of pointwise nonnegative functions. The obtained results are easily extended to \(L^p({\mathbb R})\).