Exercise 3.5
This is a simple exercise of plugging the expression for \hat{\beta} in the formula of the ith fitted value:
\hat{y}_i = x_i \hat{\beta} = x_i \left( \sum_{i'=1}^n x_{i'} y_{i'} \right) / \left( \sum_{j=1}^n x_j^2 \right) = \sum_{i'=1}^n \left( \frac{x_i x_{i'} }{\sum_{j=1}^n x_j^2} y_{i'} \right)
and comparing to the expression
\hat{y}_i = \sum_{i'=1}^n a_{i'} y_{i'}
to obtain:
a_{i'} = \frac{x_i x_{i'} }{\sum_{j=1}^n x_j^2}