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I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say:

$$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$

where the convolution between a distribution and a test function is a function that I define as:

$$ T*\varphi \doteqdot x \mapsto \langle T,\tau_x \varphi \rangle $$

With $\tau$ the translation operator, i.e., $\tau_x (t \mapsto \varphi(t))\doteqdot t \mapsto \varphi(t-x) $ .

Does this make any sense? I'm trying to follow what my textbook says but the author is not exactly clear.

  • 0
    What are the brackets? Inner product?2012-12-23
  • 0
    @rlgordonma The brackets is the action of a distribution on a test function.2012-12-23

1 Answers 1