A Fourier multiplier mapping $L^{\infty}(\mathbb{T})$ into $C(\mathbb{T})$ corresponds to a function from $L^1(\mathbb{T})$

Question

How can I prove that a Fourier multiplier sequence $\lbrace{m_n\rbrace}_{n=-\infty}^{\infty}$ mapping $L^{\infty}(\mathbb{T})$ into $C(\mathbb{T})$ corresponds to a function from $L^1(\mathbb{T})$?

This question is part of a proof I'm reading, and the book refers to Zygmund's Trigonometric series when doing this statement, but it does not specify any chapter or page where I can find the result.

Answer 1

say your filter is $h(x) = \sum_n m(n) e^{2 i \pi n x}$, consider the function $f(x) = \text{sign}(h(-x))$ which is $L^\infty([0,1[)$.

$$f \ast h(0) = \int_0^1 f(0-y) h(y) dy = \int_0^1 |h(y)| dy$$