This is true.  

By a partition of unity, the proof can be reduced to the case when the test functions have their supports in a unit cube and the result follows from a more or or less straightforward manipultation with the corresponding Fourier series. 

See, for instance, Theorem 4.3.1 in ["Introduction to the Theory of Distributions"][1] by Friedlander and Joshi  (или задачи 423 и 430 в книге Кириллов А.А., Гвишиани А.Д. *"Теоремы и задачи функционального анализа"*).


  [1]: http://books.google.co.uk/books?id=9O-pXIVUEFMC&printsec=frontcover&dq=Introduction+to+the+Theory+of+Distributions&hl=en&ei=vmhhTIimAZS7jAe9-OyvCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false