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 $\vert u_{k}(x)\vert \le A(x)\epsilon _{k}$, $0<A(x)< \infty$.