The basis function approach to regression involves using a family of functions or transformations that can be applied to a variable $X$: $b_1(X), b_2(X), \cdots, b_K(X)$. Instead of fitting a standard linear model in $X$, we fit the model: $y_i = \beta_0 + \beta_1b_1(x_i) + \beta_2b_2(x_i) + \cdots + \beta_Kb_K(x_i) + \epsilon_i$. Note that the basis functions $b_1(\cdot), b_2(\cdot), \cdots, b_K(\cdot)$ are fixed and known, meaning they are chosen ahead of time.