Fourier series of function with bounded variation is bounded by its norm

Question

We have $\mathbb{T} = \mathbb{R}/\mathbb {Z}.$ We define a space $BV(\mathbb{T}) = \{f| f:\mathbb{T} \to \mathbb{C}, V(f) <\infty \},$ where $V(f)$ denotes the total variation. We also equip this space with norm $\lVert f \rVert_{BV(\mathbb{T})} = \sup_{x \in \mathbb{T}} |f(x)| + V(f).$ I have already shown that $BV(\mathbb{T})$ under this norm is a Banach space. Now the question is, how do we show that for each $N \in \mathbb{N}$ and each $x \in \mathbb{T}$ we have $$ |S_Nf(x)| \leq \lVert f \rVert_{BV(\mathbb{T})},$$ where $S_N$ denotes the $N$-th partial Fourier sum.

I have tried with the following reasoning $$|S_Nf(x)| \leq \lVert S_Nf \rVert_{BV(\mathbb{T})} \leq \lVert S_N \rVert_{op} \lVert f \rVert_{BV(\mathbb{T})} \leq \lVert f \rVert_{BV(\mathbb{T})},$$ which holds if $S_N$ is bounded on $BV(\mathbb{T})$ and has operator norm smaller or equal to 1. However I was unable to prove any of those requirements.

Am I thinking in the right direction or should I rather try to majorize $|S_N f(x)|$ directly by writing it out or by the use Dirichlet's kernel (though, I am not sure how to estimate the total variation of the convolution)? Thank you for your help.

Answer 1

I managed to solve the problem with some help from the Fourier Analysis and Approximation vol. 1 from Butzer and Nessel. First one has to show that $S_nf(x) = \sigma_n f(x) + \frac{1}{n} \sum_{k=-n}^n |k| \hat{f}(k)e^{2\pi i k x},$ where $\sigma_n$ represents the Cesaro sum of $S_n$. This can be checked by simply writing out the definitions. Then it needs to be shown that for $f \in BV(\mathbb{T}),$ it holds $|\hat{f}(k)| \leq \frac{V(f)}{2 \pi |k|}.$ To show this one writes out $|\hat{f}(k)|,$ uses per partes formula for Lebesgue-Stieltjes integral, takes into the account the periodicity of $f$ and $e^{2\pi i k x},$ and then concludes by observing that $|\int_0^1 e^{-2\pi i k x} df(x)| \leq V(f).$ At last by this two results one obtains \begin{align*} |S_nf(x)| &\leq |\sigma_n f(x)| + \frac{1}{n} \sum_{k=-n}^n |k| |\hat{f}(k)| \\ &\leq \lVert f \rVert_\infty + \frac{2n |k|}{n} \frac{V(f)}{2 \pi |k|} \\ &\leq \lVert f \rVert_\infty + V(f) = \lVert f \rVert_{BV(\mathbb{T})}, \end{align*} which completes the proof.