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). 

Much more on this is said on pp. 250--251 of the book *Orthogonal Polynomials" by Gábor Szegő (1939), 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