First let me be briefly state the relevant information to my problem:
- $\beta(s) \in C_0^{\infty}([-1,1])$, and $\beta \equiv 1$ around $s=0$. The $\beta$ I'm using is an even function, but it doesn't have to be.
- Let $b(s) = (1-\beta(s))/s$, so $b\in C^{\infty}$. Moreover, $b$ is odd if $\beta$ is even.
- Obviously $b(s) \in L^2(\mathbb{R})$ --- and so $\hat{b}(\sigma)$ exists --- but $b\notin \mathcal{S}(\mathbb{R})$, so $\hat{b}\notin \mathcal{S}(\mathbb{R})$.
- In the following, we take $T\gg1$.
I am studying the behavior of the following integral: $$ F_T(\tau) = \int_{-\infty}^{\infty}\sum \mp b(s)e^{-isT(\tau\pm1/2)}\,ds = \hat{b}(sT(\tau-1/2))-\hat{b}(sT(\tau+1/2)). $$
In particular:
- I am looking to show that $F_T$ is bounded on all of $\mathbb{R}$
- And to analyze any dependence these bounds might have on $T$
Without going into too much of the background for the problem, I am able to show that for any $\tau\neq\pm 1/2$ we have the following bound: $$ |F_T(\tau)| \lesssim \sum T^{-1}|\tau\pm1/2|^{-1}. $$ Hence, for any fixed $\epsilon>0$ we can take $T$ large enough so that $F_T \to 0$ uniformly on $\mathbb{R}\setminus\left\{ \left( \tfrac{-1}{2}-\epsilon,\tfrac{-1}{2}+\epsilon \right) \cup \left( \tfrac{1}{2}-\epsilon,\tfrac{1}{2}+\epsilon\right) \right\}$.
It's the pesky behavior near $\tau=\pm1/2$ that has me stuck. If you look at $F_T(\tau)$ in the right way, you can show that $$ F_T(\tau) \approx \chi(\tau) - (\rho_T * \chi)(\tau), $$ where $\chi$ is the characteristic function of a unit interval, and $\rho_T$ is an approximate identity as $T\to \infty$. In this way, we see that $F_T$ must be bounded, but will not be continuous. Even so, I still have quantitative information about the bounds near $\tau=\pm1/2$, and how they may or may not depend on $T$.
Any help would be much appreciated.
