Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{Z}^n, \quad k=1,2,3,\ldots$$Write $H^s(\mathbb{R}^n)$ for the Sobolev space of exponent $s$ and fix $s>n/2$ so that $H^s(\mathbb{R}^n)$ consists of continuous functions by the Sobolev embedding theorem. Define a sequence of decreasing closed subspaces of $H^s(\mathbb{R}^n)$ via$$ S_k=\{ f \in H^s(\mathbb{R}^n): f(x)=0 \,\,\forall x \in L_k\}$$so that $\cap_{k=1}^{\infty}S_k=\{0\}$. Then the sequence of orthogonal projections onto $S_k$ converges strongly to 0 in $H^s(\mathbb{R}^n)$.
My question is: Can one find a sequence of mollifiers$$J_k: L^2(\mathbb{R}^n) \rightarrow H^s(\mathbb{R}^n)$$ such that orthogonal projections onto the preimages $J_k^{-1}(S_k)$ converge strongly to 0 in $L^2(\mathbb{R}^n)$ ?
Note: By a sequence of mollifiers I mean a sequence of operators given by convolution with a sequence of smooth compactly supported functions tending weakly to a delta function.