A Characterization of Compact Multipliers
Gregory F. Bachelis, Louis Pigno · Transactions of the American Mathematical Society · 1972
Let G be a compact abelian group and $\varphi$ a complex-valued function defined on the dual $\Gamma$. The main result of this paper is that $\varphi$ is a compact multiplier of type $(p,q),1 \leqq p 0$ there corresponds a finite set $K \subset \Gamma$ such that $|\sum {a_\gamma }{b_\gamma }\varphi (\gamma )| 0$ there corresponds a finite set $K \subset \Gamma$ such that $|\sum {b_\gamma }\varphi (\gamma )| < \varepsilon$ whenever $Q = \sum {b_\gamma }\gamma$ is a trigonometric polynomial satisfying ${\left \| Q \right \|_1} \leqq 1$ and ${b_\gamma } = 0$ for $\gamma \in K$.