To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$ I wants to understand the integrals of the form
$$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha),  (\mu(0)=0)$$
where $\mu(\alpha)$ is a non decreasing function such that the integral converges for all real $x.$
Trivial Example: If we take, $\mu(\alpha)= \alpha,$ then $\lambda (x)= |x|\frac{\pi}{2}.$
My Questions are:

(1) Can you give few more examples of non decreasing function $\mu(\alpha)$, so that  we know the exact value of $\lambda (x).$ ?
(2) Suppose the above integral ($\lambda (x)$) converges for the given non decreasing function $\mu(\alpha).$  Can we expect(to evolute) to get the exact(precise) values of $\lambda(x), (x\in \mathbb R)$ ?
(3) Why $\lambda (x)$ in the literature it is known as "negative definite  function" ; can you motivates me  bit ? (For instance, Arne Beurling, in his paper, "on the spectral synthesis of bounded functions" called this $\lambda (x)$ as  negative definite functions)
(4) Does there exists any well-known analogue  of ``negative definite functions" defined in $\mathbb R^{2} (\mathbb R^{n})$ ? If not, what one can expect ?

Thanks,
 A: (1) Denote by $\mu(\alpha)$ the Radon-Nikodym derivative $\frac{d\mu}{d\alpha}$.  Notice that if we extend $\mu(\alpha)$ to the negative reals by $\mu(-\alpha)=\mu(\alpha)$, the integral becomes
\begin{equation}-2\int_{-\infty}^{\infty}e^{-i\alpha x}\frac{\mu(\frac{\alpha}{2})}{\alpha^2}d\alpha+\frac{1}{2}\int_{0}^{\infty}\frac{\mu(\alpha)}{\alpha^2}d\alpha.\end{equation}
Therefore, what needs to be computed is the Fourier transform of $\frac{\mu(\frac{\alpha}{2})}{\alpha^2}$. For example, if $\mu(\frac{\alpha}{2})=\alpha^{2-p}$ where $0<p<2$, the Fourier transform is 
\begin{equation}
\frac{2(\sin \frac{\pi p}{2})\Gamma(1-p)}{|\alpha|^{1-p}},
\end{equation}
(after that the original integral is easy to compute), and you can find more examples here. For example, the case $\mu(\alpha)=\alpha^2\log(1+|\alpha|)$ can be computed explicitly (although the result will be a distribution).
(2) Obviously, there is no general method for these integrals, but if you know the function $\mu$ well enough, and if it is analytic, you can compute the integral by the residue theorem. So for exaple if $\mu(\alpha)$ is an even rational function such that the integral converges, it can be computed.
(3) By Bochner's theorem, the Fourier transform of a nonnegative function is positive definite, so its negative is negative definite. 
(4) Negative definite functions can even be defined on any group, see this.
