Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are $\lambda_1\geq\lambda_2\geq\ldots$ and $\{e_i\}$, respectively.
Now we are defining another stochastic process $\{Y_t:t\in [0,1]\}$, where $$Y_t=X_t+X_{t+1}.$$
I want to obtain the eigen values and eigen function of the covariance operator of $\{Y_t:t\in [0,1]\}$ as function of the eigen values and eigen function of the $\{X_t:t\in [0,2]\}$.
I am trying in the following way:
Consider covariance kernel, $K^X(s,t)=Cov(X(s),X(t))$, so by the spectral decomposition we have $$K^X(s,t)=\sum_{j\geq1}\lambda_j\psi_j(s)\psi_j(t).$$
So, the covariance kernel of $\{Y_t\}$ will be $\begin{equation} K^Y(s,t) \\=Cov(Y(s),Y(t))\\ = Cov(X(s),X(t))+Cov(X(s+1),X(t))+Cov(X(s),X(t+1))+Cov(X(s+1),X(t+1))\\ = \sum_{j\geq1}\lambda_j\left(\psi_j(s)\psi_j(t)+\psi_j(s+1)\psi_j(t)+\psi_j(s)\psi_j(t+1)+\psi_j(s+1)\psi_j(t+1)\right)\\ =\sum_{j\geq1}\lambda_j(\psi_j(s)+\psi_j(s+1))(\psi_j(t)+\psi_j(t+1))\\ =\sum_{j\geq1}\lambda_j\phi_j(s)\phi_j(t) \end{equation}$
where, $\phi_j(s)=\psi_j(s)+\psi_j(s+1).$
Now the issue is how to prove or disprove whether $\lambda's$ and $\phi$'s are the eigen values and the eigen function of the process $\{Y_t:t\in [0,1]\}$ or not.
Any help will be gratefully appreciated. Thanks
Added later: More generally my question is: if a covariance kernel $K(s,t)$ is written as $\sum_{j\geq1}\lambda_j\phi_j(s)\phi_j(t)$, then is true that $\phi$'s are orthogonal eigen function corresponding to the eigen values $\lambda$'s of the kernel?