Research proof index

Each term n*σ₁(n)*exp(-2πnt) in the q-expansion of negDE₂ has positive real part.

If f is negligible, then fun n => ↑(p.eval n) * f n is negligible for any polynomial p. This is the key lemma for handling polynomial-loss security reductions.

A discrete Minkowski inequality: for p ≥ 1 and a finite sequence a : Fin n → ℝ≥0, the ℓᵖ norm of a is bounded above by the sum of its entries.

Fourier-transform decay from an integrable derivative: for integrable, differentiable g with integrable derivative, ‖𝓕 g w‖ ≤ (∫ ‖deriv g x‖) / (2π·|w|).

For $a>0$ and $1\le R\le B$, the damped sinc integral over the finite tail $[R,B]$ satisfies the uniform estimate $\left|\int_R^B e^{-ax}\operatorname{sinc}(x)\,dx\right|\le\frac{4}{R}.$

The oscillatory-integral form of the decay bound: for 0 < T, ‖∫ g y · exp(T·i·y)‖ ≤ (∫ ‖deriv g x‖) / T.