EDITED: I am going a step up from what I had in the original post.
I am trying to show that the following integral is finite (I boiled down a much longer proof, and am just giving the part I got stuck on). It is known that $\int^\infty_{-\infty} k(x)|x|< \infty$, $\int^\infty_{-\infty} k(x) = c_1 < \infty$, and that $k(x)$ is an even function. Furthermore, it is also known that $k(x)=0$ when $|x|>1$.
$$\int^\infty_{-\infty}\int^\infty_{-\infty} k(x) \left (\frac{cos(\lambda x) - c_1}{\lambda^2} \right ) dx d\lambda $$
I originally tried using the identity:
$$sin^2(x) =\frac{1-cos(2x)}{2}$$
However that gave me,
$$\int_{-\infty}^\infty \int_{-\infty}^\infty \kappa(x) \frac{sin^2(\lambda x/2)}{\lambda^2} dx d\lambda + \int_{-\infty}^\infty \int_{-\infty}^\infty \kappa(x) \frac{c_2}{\lambda^2} dx d\lambda$$
where $c_2+1 = c_1$ ($c_1$ is likely just a bit larger than 1). We can call these the first and second components.
The first component I used the identity $\forall c \in \mathbb{R} \rightarrow \int^\infty_{-\infty} \left ( \frac{\sin(x/c) }{x}\right )^2 dx \approx \frac{\pi}{|c|}$. I gathered that the first component was approximately equal to
$$\frac{\pi}{2}\int_{-\infty}^\infty \int_{-\infty}^\infty \kappa(x) |x| dx d\lambda$$
However the second component escapes me. Is there away to show that my original integral (first one above), is finite?