Hermite Functions (show hermite function belongs in schwartz class )

In summary, the conversation is about determining whether the hermite function belongs to the Schwartz class. The person has a definition for the Schwartz class and has attempted to use the n+1 formula for the hermite function, but ended up with a value of 0 for the n=2 term. They also mention using a hint to find the Fourier transform of the hermite function when n=0. They are seeking guidance on how to proceed with their task.
  • #1
jac7
21
0
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 forumla 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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  • #2
sorry here is the attachment
 

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1. What are Hermite Functions?

Hermite Functions are a type of mathematical function that are named after the French mathematician Charles Hermite. These functions are used in the field of Gaussian mechanics and are a special case of the more general class of hypergeometric functions.

2. What is the Schwartz Class?

The Schwartz Class, also known as the space of rapidly decreasing functions, is a space of functions that decay faster than any polynomial as the variable approaches infinity. This class is important in mathematical analysis and is often used in the study of Hermite Functions.

3. How are Hermite Functions related to the Schwartz Class?

Hermite Functions belong in the Schwartz Class because they decay rapidly as the variable approaches infinity, making them ideal candidates for functions in this space. In fact, Hermite Functions are a complete orthogonal system in the Schwartz Class.

4. How are Hermite Functions used in science?

Hermite Functions are used in a variety of scientific fields, including quantum mechanics, statistical mechanics, and signal processing. They are particularly useful in solving differential equations and in modeling wave-like phenomena.

5. Are Hermite Functions applicable to real-world problems?

Yes, Hermite Functions have many practical applications in real-world problems. They are commonly used in the study of quantum mechanics and are also used in engineering and physics for modeling and analyzing data. They have been proven to be effective and accurate in a wide range of real-world scenarios.

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