Research proof index

For every continuous, $2\pi$-periodic function $f : \mathbb{R} \to \mathbb{C}$, the symmetric partial Fourier sums $S_N f(x)$ converge to $f(x)$ for almost every $x \in \mathbb{R}$.

Let $T>0$, $1 \le p < \infty$, and let $f$ belong to $L^p((0,T])$. For every $\varepsilon>0$, there is a smooth $T$-periodic function $f_0 : \mathbb{R}\to\mathbb{C}$ such that

$\lVert f-f_0\rVert_{L^p((0,T])} \le \varepsilon.$

Closed ball centered at z with radius z.im/2 is contained in the upper half plane.

For a complex-valued function on the ambient finite group and a nonzero even integer $n$, ordinary self-convolution has no larger normalized $L^n$ norm than self-difference-convolution: $\|f*f\|_n\le\|f\mathbin{\circleddash}f\|_n$.

The $2n$-th Fourier moment of the indicator of a finite set equals its order-$n$ additive energy: $\|\widehat{1_s}\|_{2n}^{2n}=E_n(s)$. This is the standard bridge between Fourier norms and additive tuple counts.

In a conditionally complete linear order, suppose the values $f(i)$ for $i\in s$ are bounded above. If one of those values is exactly $a$, then $a$ is at most the supremum $\sup_{i\in s} f(i)$.