Undergrad Generalizing the translation operator

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The discussion focuses on the effects of generalized translation operators, specifically ##e^{g(p)\partial_p}## and ##e^{g(p)\partial_p^2}##, on a function ##f(p)##. It establishes that the operator ##e^{a\partial_p}## translates the function by a constant shift, while ##e^{a\partial_p^2}## corresponds to the Weierstrass transform. The challenge arises in determining the action of the more complex operators involving a function ##g(p)##. Participants suggest expressing these operators in a Taylor series to analyze their effects on ##f(p)##. Understanding these generalizations is crucial for advancing the application of translation operators in various mathematical contexts.
BlackHole213
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If I have the operator, ##e^{a\partial_p}## acting on ##f(p)##, I know that $$e^{a\partial_p}f(p)=f(p+a)\,.$$
If I have ##e^{a\partial_p^2}f(p)##, this is just the Weierstrass transform of ##f(p)##. However, what happens if I have a general operator, ##e^{g(p)\partial_p}## or ##e^{g(p)\partial_p^2}##. How would I know what ##e^{g(p)\partial_p}## or ##e^{g(p)\partial_p^2}## does to ##f(p)##?
 
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Write it out in a Taylor series.
 

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