I know that Hermite polynomials are orthogonal with eachother as follows:
$$\langle H_n,H_m \rangle=\int_{-\infty}^\infty H_n(x) H_m(x) \exp(-x^2) \,\mathrm dx$$
If I define a basis function (the Hermite functions) as follows:
$$ h_n(x) = H_n(x) \exp\left(-\dfrac{x^2}{2} \right)$$
Am I correct to assume that I can represent all continuous functions as follows:
$$f(x) = \Sigma a_n h_n(x)$$
Where:
$$a_n = \int_{-\infty}^\infty f(x) h_n(x) dx$$