Consider a time series transform $TS$, defined as follows:
$X = (x_{0},x_{1},...,x_{i}, ...,x_{N}) $ is a complex-valued time series,
$$TS x_i = \sum_{j=0}^{j=i} x_j K_j \hspace {0.2 in} K_j = K(t_j,w_j) \hspace {0.2 in} x_j = x(t_j) \hspace {0.2 in} TS x_i = \tilde{X}(w_i) $$
The inverse transform is
$$x_i = \sum_{j=0}^{j=i} \tilde{X_j} K_j^{-1} \hspace {0.5 in} K_j^{-1} = K^{-1}(t_j,w_j) \hspace {0.5 in} \tilde{X_j} = \tilde{X}(w_j) $$
As an example for $K_j$
$$ \displaystyle TSx_i = \sum_{j=0}^{j=i} x_j \exp(\color{red} i \Theta_j) \hspace {0.5 in} K_j = \exp(\color{red} i \Theta_j) \hspace {0.5 in} \Theta_j = \displaystyle \sum_{k=0}^{k=j} w_k \Delta t_k $$
$\tilde{X} = (\tilde X_{0}, \tilde X_{1},...,\tilde X_{i}, ...,\tilde X_{N}) \hspace {0.5 in} \displaystyle \tilde X_i = \sum_{j=0}^{j=i} x_j \exp(\color{red} i \Theta_j) $
$w_k$ is a known time series with $w_k = w(t_{k})$ and $\Delta t_{k} = t_{k+1} - t_{k}$
What is the inverse transform in the above example?