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We define,

$$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for every real $x$. Some authors call $\lambda (x)$ a negative definite function. (This is proved by von Neumann, J.; Schoenberg, I. J. in 1941; see, Fourier integrals and metric geometry)

But here I am interested to learn some method(techniques) (if exists) to know about behaviour of the function $\lambda(x)$.

My Question: (1) How to choose a non-decreasing function $\mu (\alpha)$ such that $\int_{\mathbb R} \frac{1}{\lambda(x)} dx < \infty.$ ? (2) Suppose given $f\in L^{1}(\mathbb R)$; which is also continuous on $\mathbb R$ and vanishing at infinity. Can we expect to choose $\mu(\alpha)$ (may be in terms of $f$) so that $\int_{\mathbb R} |f(x)| \lambda (x) + \frac{1}{\lambda (x)} dx < \infty $ ? If not, can we produce some counter example ?

(I am also looking for any references or suggestions concerning this)


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