Research proof index

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

Let $f$ be integrable, $R>0$, and $K_T(u)=\sin(Tu)/(\pi u)$ for $u\ne0$, with the project's value $K_T(0)=0$. Removing the part of the convolution outside $[-R,R]$ incurs at most $\left\|\int_{\mathbb R}K_T(u)f(x-u)\,du-\int_{-R}^{R}K_T(u)f(x-u)\,du\right\|\le\frac{1}{\pi R}\int_{\mathbb R}\|f(u)\|\,du.$ The estimate is uniform in $x$ and $T$.

Bound on the q-series ∑ n·qⁿ/(1-qⁿ) that appears in E₂. For ‖q‖ < 1, we have ‖∑ₙ₌₁ n·qⁿ/(1-qⁿ)‖ ≤ ‖q‖/(1-‖q‖)³. The key estimates are: - |1-qⁿ| ≥ 1-|q|ⁿ ≥ 1-|q| for n ≥ 1 - |n·qⁿ/(1-qⁿ)| ≤ n·|q|ⁿ/(1-|q|) - ∑ n·rⁿ = r/(1-r)², so ∑ n·rⁿ/(1-r) = r/(1-r)³

Monotone version of norm_tsum_logDeriv_expo_le: if ‖q‖ ≤ r < 1, then ‖∑ n·qⁿ/(1-qⁿ)‖ ≤ r/(1-r)³. Useful when we have a uniform bound on ‖q‖.

Right multiplication by an element of Bˣ on B ⊗ 𝔸_K does not scale additive Haar measure.

For any positive real r, there's some ρ ∈ ℝˣ such that the haar character of (ρ, 1) ∈ D_f × D_∞ is r.