Representation Proof question

Question

Consider a parametric model governed by the parameter vector $\mathbf{w}$ together with a dataset of input values $\mathbf{x}_1,\ldots,\mathbf{x}_N$ and a nonlinear feature mapping $\phi(\mathbf{x})$. Suppose that the dependence of the error function on $\mathbf{w}$ takes the form $J(\mathbf{w}) = f(\mathbf{w}^\top \phi(\mathbf{x}_1), \ldots, \mathbf{w}^\top \phi(\mathbf{x}_N)) + g(\mathbf{w}^\top \mathbf{w})$ where $g(\cdot)$ is a monotonically increasing function. By writing $\mathbf{w}$ in the form $$ \mathbf{w} = \sum_{n=1}^N \alpha_n \phi(\mathbf{x}_n) + \mathbf{w}_\perp $$ show that the value of $\mathbf{w}$ that minimizes $J(\mathbf{w})$ takes the form of a linear combination of the basis functions $\phi(\mathbf{x}_n)$ for $n = 1, \ldots, N$.