Optimal $$C^\infty $$-approximation of functions with exponentially or sub-exponentially integrable derivative

Abstract

AbstractWe discuss Meyers-Serrin’s type results for smooth approximations of functions $$b=b(t,x): \mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}^m$$ b = b ( t , x ) : R × R n → R m , with convergence of an energy of the form $$\begin{aligned} \int _{\mathbb {R}}\int _{\mathbb {R}^n} w(t,x) \varphi \left( |Db(t,x)|\right) \textrm{d} x \textrm{d} t\,, \end{aligned}$$ ∫ R ∫ R n w ( t , x ) φ | D b ( t , x ) | d x d t , where $$w>0$$ w > 0 is a suitable weight function, and $$\varphi :[0,\infty )\rightarrow [0, \infty )$$ φ : [ 0 , ∞ ) → [ 0 , ∞ ) is a convex function with $$ \varphi (0)=0$$ φ ( 0 ) = 0 having exponential or subexponential growth.