Bounds for the remainder of the Fourier series of the fractional-part function

Question

Let $\psi(x) = \{x\} - 1/2$, where $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. It is known that $\psi$ has the Fourier series representation $$ \psi(x) = -\dfrac{1}{\pi}\sum_{n=1}^\infty \dfrac{\sin(2\pi n x)}{n} $$ valid for $x \in \mathbb{R}\setminus \mathbb{Z}$. I was wondering how well the partial sums approximate $\psi(x)$? For instance, can one show that for $N \geq 1$, $$ \sum_{n > N}\dfrac{\sin(2\pi n x)}{n} = O\left( \dfrac{1}{N}\right) $$ with the implied constant independent of $x$? If not, can one find exponential series of the form $$ S_N(x) = \sum_{1 \leq |n| \leq N} a_n e^{inx} $$ with $a_n = O(1/n)$ and such that $\psi(x) -S_N(x)= O(1/N)$ uniformly in $x$? Thanks.

Answer 1

The space of bounded continuous functions is closed under the sup-norm. Wikipedia calls this the Uniform limit theorem. So you cannot approximate a discontinuous function $\psi$ with continuous functions $S_N$ in the sense of $\|\psi-S_N\|_{\infty}\to 0$.

Intuitively, if $S_N$ takes some value $x$ at the discontinuity $0$, there is going to be an error of nearly $|1/2-x|$ at small negative values, and nearly $|-1/2-x|$ at small positive values, and one of these is nearly at least $1/2$.

For Fourier series the minimum of the partial sums doesn't even converge to $-\tfrac 1 2$, the minimum of $\psi$ - this is the Gibbs phenomenon.