Does Transforming Hermite Polynomials Affect Their Orthogonality?

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.
 
Mathematics news on Phys.org
  • #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##$$<\Psi_i \Psi_j>=0$$
 

FAQ: Does Transforming Hermite Polynomials Affect Their Orthogonality?

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.

Similar threads

I Defining Legendre polynomials in (1,2)
Replies
2
Views
1K
I Associated Legendre polynomials: complex vs real argument
Replies
1
Views
3K
I Polynomial Division: Solve with Mike's Help!
Replies
4
Views
1K
I Proving Hermite polynomials satisfy Hermite's equation
Replies
1
Views
3K
I Matrix of columns of polynomials coefficients invertibility
Replies
2
Views
1K
Orthogonality Relationship for Legendre Polynomials
Replies
5
Views
2K
C/C++ How to Implement Polynomial Operations in C++?
Replies
5
Views
2K
Can you help me prove the integral for Hermite polynomials?
Replies
2
Views
2K
Expansion of Cos(x) in Hermite polynomials
Replies
4
Views
3K
Why does polynomial long division work?
Replies
4
Views
4K

Hot Threads

Recent Insights

Back
Top