Research proof index

Let $f$ be $2\pi$-periodic and belong to $L^q((0,2\pi])$ for some $q>1$. Given thresholds $\delta,\varepsilon>0$, there is an index $N_0$ such that the set where the tail error $\sup_{N>N_0}\lVert f(x)-S_Nf(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

For a continuous, $2\pi$-periodic function $f$ and any $\delta,\varepsilon>0$, there is an index $N_0$ such that the set of $x\in(0,2\pi]$ for which $\sup_{N>N_0}\lVert f(x)-S_Nf(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

If a tile $p$ lies in the $n$th minimal layer of a set of tiles $A$, then there is a tile $p'$ in the zeroth minimal layer with $p'\le p$, and the scale of $p$ is at least the scale of $p'$ plus $n$.

Let $V$ be the intersection of the kernels of a set of additive characters $\Delta$. Averaging $f$ over $V$ equals the average of $\widehat f(\psi)$ over all characters, with the Fourier coefficient retained precisely when $\psi$ lies in the additive subgroup generated by $\Delta$.

Reduce a power of I * w via I ^ 4 = 1. -/ lemma I_mul_npow (w : ℂ) (n : ℕ) : (I * w) ^ n = I ^ (n % 4) * w ^ n := by rw [mul_pow, I_pow_eq_pow_mod] /- Functional equation of $F$

Functional equation of $F$ restricted to the imaginary axis.