I have two sequences $s$ and $r$ defined as :

  • $s = \{s_n\}_{n \in \mathbb{Z}}$ where $s_n(t) = (M_{\beta}^n s)(t) = s(t) e^{int}$ with arbitrary $s \in L^2(\mathbb{R})$ and $\beta > 0$
  • $r = \{r_n\}_{n \in \mathbb{Z}}$ where $r_n(t) = (M_{\alpha}^n r)(t) = r(t) e^{int}$ with arbitrary $r \in L^2(\mathbb{R})$ and $\alpha > 0$

I am trying to understand necessary and sufficient conditions on $\alpha, \beta$ such that both sequences are stationary correlated.

My thoughts

  1. Obviously, stationary correlated means dependence on indices differenc only such that that $\langle s_n, r_n \rangle = \langle s, r \rangle$
  2. I can thus write : $\langle s_n, r_n \rangle = \int_{\mathbb{R}} s \cdot e^{i n \beta t} \cdot \overline{r} \cdot e^{- i n \alpha t} dt = \int_{\mathbb{R}} s \cdot \overline{r} \cdot e^{i n (\beta - \alpha) t} dt $
  3. So we have $\langle s_n, r_n \rangle = \langle s, r \rangle = \int_{\mathbb{R}} s \cdot \overline{r} dt$ for example if $\alpha = \beta$

So that would be one very obvious condition, in fact $s$ and $r$ would be identical then. However, are there other conditions? I am thinking of something like

$\int_{\mathbb{R}} x(t) \cdot e^{i \gamma t} dt = \int_{\mathbb{R}} x(t) dt$ for some special conditions on $\gamma$.

Please help, thank you! :-)



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