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In chapter 9.4 of Strauss PDE, we try to find the solution of 3D Diffusion Equation. And there is a exercise which is closely relating to the proof, as following:

$$\lim_{t\to0} \iiint_{R^3} S_3(X-X',t)\phi(X')~dX'=\phi(X)\tag{6}$$

  1. Prove that (6) is valid for products of the form $\phi(X)=φ(x)ψ(y)ζ(z)$ and hence for any finite sum of such products
  2. Deduce (6) for any bounded continuous function $\phi(X)$. You may use the fact that there is a sequence of finite sums of products as in previous part which converges uniformly to $\phi(X)$.

I had finished part 1. But in part 2, I am very confused about it. Why the Author said that "there is a sequence of finite sums of products as in previous part which converges uniformly to $\phi(X)$" ???

Thanks a lot!

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  • $\begingroup$ He wants you to approximate $\phi$ by functions $\phi_j(X) = \varphi(x)\psi(y)\zeta(z)$, apply part 1, then take the limit as $j \to \infty$. $\endgroup$
    – Kakashi
    Commented Jul 10 at 2:20
  • $\begingroup$ I means how to prove that the sequence of finite sums converge uniformly to the initial condition? $\endgroup$
    – 郭冠廷
    Commented Jul 10 at 9:23
  • $\begingroup$ The existence of finite sums of products that converge uniformly to $\phi(X)$ is an analysis result. One way of proof is the Stone-Weierstrass theorem. $\endgroup$
    – Kakashi
    Commented Jul 10 at 14:47
  • $\begingroup$ But Stone-Weierstrass theorem only work for compact set, not the whole space. How does the theorem lead to the result? $\endgroup$
    – 郭冠廷
    Commented Jul 10 at 16:08
  • $\begingroup$ Good point. There may be a way to salvage Strauss's proof. But you could just copy the 1-d proof, as it is essentially the same proof for higher dimensions. $\endgroup$
    – Kakashi
    Commented Jul 10 at 19:23

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