We study the average of sum, in the sense of integral.

Left shift operatorThroughout we consider the Hilbert space $L^2=L^2(\mathbb{R})$, the space of all complex-valued functions with real variable such that $f \in L^2$ if and only if \lVert f \rVert_2^2=\int_{-\infty}^{\infty}|f(t)|^2dm(t)

Suppose $1 < p < \infty$ and $f \in L^p((0,\infty))$ (with respect to Lebesgue measure of course) is a nonnegative func...

Throughout, let $(X,\mathfrak{M},\mu)$ be a measure space where $\mu$ is positive. The questionIf $f$ is of $L^p(\mu)$, which...