Conditions on $f(t)$ so that $\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt$ converges.

Question

Let us consider $$\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt \ \ \ \ (*)$$ for $a,b\in \mathbb R$. If $f\in L^1(-\infty,\infty)$ the integral converges: $$\left|\int_{-\infty}^\infty f(t) \operatorname{sinc}(t-a) \operatorname{sinc}(t-b) dt\right|\leq \int_{-\infty}^\infty \left|f(t)\right| dt$$ But I think this is a too strong condition. What are the conditions to be imposed on $f(t)$ so that (*) converges?

Suggestions are welcome.

Answer 1

Since $|\text{sinc}(t)| \le 1/(\pi |t|)$, it suffices to have $$\int_{-\infty}^\infty \dfrac{|f(x)|\; dx}{1+x^2} < \infty$$