Hermite Functions (show hermite function belongs in schwartz class )

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SUMMARY

The discussion centers on demonstrating that Hermite functions belong to the Schwartz class. The user has a definition of the Schwartz class but struggles with the application of the Hermite function's recursive formula, particularly for the n=2 term, which resulted in zero. The user also references the Fourier transform of the Hermite function at n=0 as a hint for their proof. Guidance is sought to clarify the steps needed to establish this inclusion definitively.

PREREQUISITES
  • Understanding of Hermite functions and their properties
  • Familiarity with the Schwartz class definition
  • Knowledge of Fourier transforms, specifically for Hermite functions
  • Experience with recursive formulas in mathematical functions
NEXT STEPS
  • Study the properties of Hermite functions in detail
  • Review the definition and characteristics of the Schwartz class
  • Learn about the Fourier transform of Hermite functions
  • Explore recursive relationships in mathematical functions
USEFUL FOR

Mathematicians, physicists, and students studying functional analysis or quantum mechanics who need to understand the properties of Hermite functions and their relation to the Schwartz class.

jac7
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I have been given this question (in the attachment).

I have a deifnition for what it means for a function to be in the schwartz class, but I don't know how to start showing that the hermite function belongs to it?
I have attempted to write out the first couple of terms using the n+1 formula for the hermite functions but i ended up getting 0 for n=2 term!

I also know, using the hint, when n=0, what the Fourier transform of the hermite function at n=0 is.

If someone could please give me some guidence, it would be a great help!
 
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sorry here is the attachment
 

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