= Homogeneous Banach space on the circle
{title2=$\|\tau_tf\|_B=\|f\|_B,\quad\|\tau_tf-f\|_B\longrightarrow0$}
A <Banach space> $B\subseteq L^1(\mathbb T)$ is homogeneous when its inclusion into $L^1$ is continuous, circle translations act isometrically, and each orbit is continuous in the $B$ norm. One commonly normalizes $\|f\|_1\leq\|f\|_B$. The <continuous functions> with the <supremum norm>, and $L^p(\mathbb T)$ for $1\leq p<\infty$, are examples. Full $L^\infty$ fails translation continuity. If $g$ is continuous, the <Bochner integral> $f*g=\int g(t)\tau_tf\,dt/(2\pi)$ belongs to $B$ and satisfies $\|f*g\|_B\leq\|f\|_B\|g\|_1$. Approximation by <continuous functions> extends this to every $g\in L^1$.
Back to article page
