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. For property 2), we show that, moreover, whatever is a monotone sequence $ ε_{k}>0$ satisfying $ \sum _{k}ε^{2}_{k}=\, \infty $ there exists an orthonormal basis $ (u_{k})_{k\, }$in $ L^{2}$ such that $ \vert u_{k}(x)\vert \leq \, A(x)ε_{k}$, $ 0<A(x)<\, \infty $.