Let $ f: \mathbb{T} \rightarrow \mathbb{C} $ .
$f \in C(\mathbb{T})$ is the set of continuous $\mathbb{C}$ valued functions on $\mathbb{T}$ = unit circle in $\mathbb{C}$
Let $ S_N(f) := \sum_{n=-N}^{N} \hat{f}(n)e^{in\theta} $ be the partial sum of the Fourier series of $ f $, where $\hat{f}(n)$ is the "$n$th Fourier coefficient of $f$"
Is the following true or false:
If $S_N(f)$ converges uniformly to a function $g$, then $g = f$
If it is true, is there a well-known theorem that proves this statement?
If it is false, can it be made true by adding extra conditions on $f$ ?
Thank you
Best Answer
This is true.
Indeed, if $(S_N(f))$ converges uniformly to $g$, then its Cesaro means converge also to $g$. But by Fejér's theorem, these means converge to $f$, so by uniqueness of the limit, one has $f=g$.