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,