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Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying $$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$ $$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \...

Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$ Can we demonstrate that following integral is ...