Research proof index

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

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 $T$.