On the sign distributions of Hilbert space frames

Abstract

We show that the positive and negative parts $$ u_{k}^{\pm }$$ of any frame in a real $$ L^{2}$$ space with respect to a continuous measure have both “infinite $$ l^{2}$$ masses”: (1) always, $$ \sum _{k}u_{k}^{\pm }(x)^{2}=\infty $$ almost everywhere (in particular, there exist no positive frames, nor Riesz bases), but (2) $$ \sum _{k=1}^{n}(u_{k}^{+}(x)-u_{k}^{-}(x))^{2}$$ can grow “locally” as slow as we wish (for $$ n\longrightarrow \infty $$ ), and (3) it can happen that $$ \sum _{k=1}^{n}u_{k}^{-}(x)^{2}= o(\sum _{k=1}^{n}u_{k}^{+}(x)^{2})$$ , and vice versa, as $$ n\longrightarrow \infty $$ on a set of positive measure. Property (1) for the case of an orthonormal basis in $$ L^{2}(0,1)$$ was settled earlier (V. Ya. Kozlov, 1948) using completely different (and more involved) arguments. Our elementary treatment includes also the case of unconditional bases in a variety of Banach spaces, as well as the case of complex valued spaces and frames. For property (2), we show that, moreover, whatever is a monotone sequence $$ \epsilon _{k}>0$$ satisfying $$ \sum _{k}\epsilon ^{2}_{k}= \infty $$ there exists an orthonormal basis $$ (u_{k})_{k}$$ in $$ L^{2}$$ such that $$ \vert u_{k}(x)\vert \le A(x)\epsilon _{k}$$ , $$ 0<A(x)< \infty $$ .