Does Transforming Hermite Polynomials Affect Their Orthogonality?

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SUMMARY

The discussion centers on the application of Hermite polynomials in generalized polynomial chaos for representing Normal random variables. It confirms that Hermite polynomials represent the standard normal distribution N(0,1) and explains how to transform these polynomials for different means and variances using the transformation Z=(X-μ)/σ. The conversation also raises the question of whether such transformations maintain the orthogonality of the polynomials, emphasizing the importance of orthogonality conditions in polynomial chaos expansions.

PREREQUISITES
  • Understanding of Hermite polynomials and their properties
  • Knowledge of normal distributions, specifically N(0,1)
  • Familiarity with polynomial chaos theory
  • Basic concepts of coordinate transformations in probability
NEXT STEPS
  • Research the properties of Hermite polynomials in detail
  • Study the implications of coordinate transformations on polynomial orthogonality
  • Explore generalized polynomial chaos methods in statistical modeling
  • Learn about the relationship between orthogonality and mean/variance in polynomial expansions
USEFUL FOR

Researchers and practitioners in statistics, particularly those working with polynomial chaos methods, as well as mathematicians interested in the properties of orthogonal polynomials and their applications in probability theory.

Frank Einstein
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Hello everyone.

I am working with generalized polynomial chaos. To represent a Normal random variable, the Hermite polynomials are used. However, as far as I understand, these represent N(0,1); if what I have read is correct, if I want to work with any other mean and variance, I shoud simply use the fact that Z=(X-μ)/σ and find the new polynomials (X) substituting the regular ones (Z), which would be: 1, Z, (Z2-1)... which would mean that X1= σZ-μ and X2=σ(Z2-1)-μ...

Can someone please tell me if I am right?

Thanks.
 
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I think you are, but it would be interesting to know how the chaos functions react to coordinate transformations.
 
Does your transformation preserve orthogonality? I would think that the requirement of orthogonality would demand zero mean...i.e. for ##i\neq j##$$<\Psi_i \Psi_j>=0$$
 

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