According to the conclusion at the bottom of p. 603 of (say) [Uspensky's paper][1], the Hermite expansion of $f$ converges pointwise to $f$ (under conditions much less restrictive that yours). 

If $f$ is (say) bounded and locally Lipschitz, then the Hermite expansion of $f$ converges to $f$ uniformly on compact sets. This follows e.g. from Theorem 9.1.6 in the book *Orthogonal Polynomials" by Gábor Szegő (1939), by taking the integral in formula (9.1.17) there by parts, noting that the [sine integral function is bounded][2], and then letting $\delta\downarrow0$ in (9.1.17). 

Much more on this is said on pp. 250--251 of Szegő's book, starting from  "From Theorem[...] 9.1.6 there follow the usual theorems on the convergence and the summability of [...] Hermite expansions" on p. 250, where, in particular, the mentioned result by Uspensky is briefly discussed. 


  [1]: https://www.jstor.org/stable/1968401
  [2]: https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral