I'm using Muscalu and Schlag's textbook to study harmonic analysis by myself, where they give the following exercise about multipliers' estimate:
Consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies: $$\sum_{n \in \mathbb{Z}}|m_{n}-m_{n-1}| \leq B, \ \lim_{n \rightarrow -\infty}m_{n} = 0$$ where $B > 0$. Now let's define a multiplier operator $T$: $$Tf(x)= \sum_{n \in \mathbb{Z}}m_{n}\hat{f}(n)e^{2\pi inx}$$ Then for any trigonometric polynomial $f$ and any $p \in (1,\infty)$, there exists some constant $C_p > 0$, such that: $$||Tf||_{L^p} \leq C_{p}B||f||_{L^p}$$ Any ideas on this? What I have tried is to try generalizing the claim to functions in $L^p$. However, it seems that I can't find a proof or a counter-example for the general case....