Research proof index

For a fixed spatial grid cube $I$, every frequency parameter lying in one of the prescribed balls centered at the finite net $\mathcal Z(I)$ also lies in at least one of the frequency regions $\Omega_1(I,f)$. Equivalently, the union of those net balls is contained in the union of the first-stage tile frequency regions.

Fix $1<p<2$, an error threshold $\delta>0$, and a measure tolerance $\varepsilon\ge0$. There is an explicit bound $C(\delta,\varepsilon,p)$ such that, whenever a measurable $2\pi$-periodic function $g$ satisfies $\lVert g\rVert_{L^p((0,2\pi])}\le C(\delta,\varepsilon,p)$, the set where the maximal partial Fourier sum $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

For every $\delta,\varepsilon>0$, there is an explicit positive uniform bound $C(\delta,\varepsilon)$ such that, if a measurable $2\pi$-periodic function $g$ satisfies $\lVert g(x)\rVert\le C(\delta,\varepsilon)$ for every $x$, then the set where $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

For a cusp form f of level Γ(n), we have t^s * f(it) → 0 as t → ∞ for any real power s. This follows from the exponential decay of cusp forms at infinity: f = O(exp(-2π τ.im / n)).

Basic properties of derivatives: linearity, Leibniz rule, etc.

D(exp(cz))/exp(cz) = c/(2πi) for any coefficient c.