# Condition for these sequences to be stationary correlated (tipp for integration of exponential functions)

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 $$\gamma$$.