Why is convolution with Fejer kernel enough to prove this result?

Question

Exercise 2.9 in Katznelson's Introduction to Harmonic Analysis reads as follows.

Show that for $f \in L^{1}(\mathbb{T})$ the norm of the operator $ F: g \mapsto f * g$ on $L^{1}(\mathbb{T})$ is $||f||_{L^{1}}$.

I can see easily enough how $||f * g)||_{1} \leq ||f||_{1}||g||_{1} \ \forall \ g \in L^{1}(\mathbb{T})$ and thus $||F||_{\text{operator}}$ is bounded by using Fubini's theorem, but was struggling to prove the actual equality.

Then my professor told me that the solution was just to recall that for the Fejer summability kernel $K_{n}$,

$$ f * K_{n} \to f \text{ as } n \to \infty$$

and so $||f * K_{n}||_{L^1} \to ||f||_{L^1}$, therefore $||F||_{\text{operator}} = ||f||_{L^1}$.

My question is, why is it enough to just show this? I have looked at my definitions for Fejer's kernel and summability kernels, and can't make the logical step from trying to show "$\sup_{||g||_{L^{1}} \leq 1} ||f * g||_{L^{1}} = ||f||_{1}$" being equivalent to showing "$||f * K_{n}||_{L^1} \to ||f||_{L^1}$", and so think I must be missing something very obvious.

Answer 1

Note that $\|K_n\|_{L^1}=1$ (using Katznelson's normalisation). Let the operator norm of $g\mapsto f*g$ be $A$. We know $A\le \|f\|_{L^1}$. Then $\|f\ast K_n\|_{L_1}\le A\|K_n\|_{L^1}=A$. But $\|f\ast K_n\|_{L^1}\to \|f\|_{L^1}$, so in the limit, $\|f\|_{L^1}\le A$.

Answer 2

I was staring at this question thinking "Why is this true?", then I realized the maxim: we would let $\delta_{0}$ into $L^{1}$ if only we could.

Pretend momentarily that $\delta_{0} \in L^{1}(\mathbb{T})$ ($\delta_{0}$ being the Dirac mass at $0 \in \mathbb{T}$). We know $K_{n} \rightharpoonup \delta_{0}$ weakly$*$ in $C(\mathbb{T})^{*}$. Let's pretend we could upgrade that to $L^{1}(\mathbb{T})$. Then $$\|f\|_{L^{1}(\mathbb{T})} = \|f * \delta_{0}\|_{L^{1}(\mathbb{T})}$$ and $\|\delta_{0}\|_{L^{1}(\mathbb{T})} = 1$ since $K_{n} \to \delta_{0}$ so $\|f\|_{L^{1}(\mathbb{T})} = \|F\|$ immediately.

Instead, we have to replace $\delta_{0}$ with its approximation $(K_{n})_{n \in \mathbb{N}}$ and take limits $$\|f\|_{L^{1}(\mathbb{T})} = \lim_{n \to \infty} \|f * K_{n}\|_{L^{1}(\mathbb{T})}.$$ The accepted answer is rigorous, but I hope this helps.