I have notes which say that the ith fitted value $\hat{Y}_i$ is a linear combination of the response values
$$\hat{Y}_i = \sum_{j=1}h_{ij}Y_j$$
where
$$h_{ij} = \frac 1 n + \frac{(x_i-\overline{x})(x_j - \overline{x})}{S_{xx}} $$
First I'm a little unclear on the exact meanings of the variables. In a linear regression, I thought there was an infinity of fitted values along the line of best fit. Are these just the fitted values at the observed $x_i$ values? And are the observed responses $Y_j$ all observed responses or just those at $x_i$?
Second, I've searched for a proof of this but I can't find one. I have a textbook showing that the coefficients of the regression are linear combinations of response variables (which if these are normal then the coefficient is normal), but that's not this. Can anyone either direct me to one or give the proof?