Adapting Hermite polynomials

In summary, the conversation discusses the use of Hermite polynomials to represent a Normal random variable in generalized polynomial chaos. It is noted that these polynomials represent N(0,1) and if working with a different mean and variance, a coordinate transformation can be used to find new polynomials. There is a question about whether this transformation preserves orthogonality and if the requirement of orthogonality demands a zero mean.
  • #1
Frank Einstein
170
1
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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  • #2
I think you are, but it would be interesting to know how the chaos functions react to coordinate transformations.
 
  • #3
Does your transformation preserve orthogonality? I would think that the requirement of orthogonality would demand zero mean...i.e. for ##i\neq j##$$<\Psi_i \Psi_j>=0$$
 

What are Hermite polynomials?

Hermite polynomials are a set of mathematical functions that are used to solve differential equations and represent certain physical phenomena. They are named after the French mathematician Charles Hermite.

Why are Hermite polynomials important?

Hermite polynomials are important because they have a wide range of applications in physics, engineering, and other fields. They are particularly useful in quantum mechanics, where they are used to describe the wave functions of quantum systems.

How are Hermite polynomials adapted?

Hermite polynomials can be adapted in several ways, such as by changing the coefficients or the degree of the polynomial. They can also be adapted for use in different coordinate systems or for different types of differential equations.

What are some common applications of adapted Hermite polynomials?

Adapted Hermite polynomials are commonly used in signal processing, image processing, and data analysis. They are also used in quantum mechanics, statistical mechanics, and other areas of physics.

What are the benefits of using adapted Hermite polynomials?

Using adapted Hermite polynomials can simplify complex mathematical problems and provide more accurate solutions. They also have a wide range of applications, making them a valuable tool for scientists and engineers.

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