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From Chapter 1, Exercise 14, Strichartz's book: Distribution theory & fourier transforms

  1. Suppose $f$ & $g$ are distributions such that $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi \rangle = 0$. Show that $\langle f,\phi \rangle =c\langle g,\phi\rangle$ for some constant c.

Does the question mean that if the condition is true for some test functions $\phi$ the result $\langle f,\phi \rangle=c \langle g,\phi\rangle$ can be deduced for all such test functions $\phi$? I think that one has to exploit the continuity of distributions like one has for functions wherein continuous functions are completely defined if they are defined on a dense subset. But apart from that I don't know how to proceed.

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The assumption: $\forall \phi $ (in the test class) $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi \rangle = 0$.

The desired conclusion: $\exists c$ such that $\forall \phi$ the equality $\langle f,\phi \rangle =c\langle g,\phi\rangle$ holds.

Plan of attack:

  1. Dispose of the trivial case in which $\forall \phi $ $\langle g,\phi\rangle=0$.
  2. Pick $\phi_0$ such that $\langle g,\phi_0\rangle\ne 0$. Let $c=\langle f,\phi_0 \rangle /\langle g,\phi_0\rangle$; this is the only choice that could work, right?
  3. If $\phi$ is any test function, let $b=\langle g,\phi \rangle /\langle g,\phi_0\rangle$ and observe that $\langle g,\phi-b\phi_0\rangle=0$. The rest should be easy.
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  • $\begingroup$ I couldn't understand why we are interested in case 3? Isn't it enough to look for case 1&2? $\endgroup$ – Serkan Yaray Nov 9 '15 at 19:14
  • $\begingroup$ These are not cases, they are steps of a proof. I like to split proofs into small enumerated chunks. Step 1: dispose of trivial case. Step 2: pick a candidate for constant $c$. Step 3: prove that the candidate works. $\endgroup$ – user147263 Nov 9 '15 at 19:16
  • $\begingroup$ Thanks, but I couldn't understand how we observed $\langle g,\phi-b\phi_0\rangle=0$ $\endgroup$ – Serkan Yaray Nov 11 '15 at 14:16

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