# Orthogonality of Hermite Polynomials

As we know probabilistic Hermite polynomials are orthogonal with respect to the weight function $\frac{1}{\sqrt{2 \pi}} e^{-x^2/2}$ (density of standard normal).

I have a distribution which is a mixture of two Gaussians each having $(Mean, variance) =(1 ,\sigma^2)$ and $(-1,\sigma^2)$ respectively.

what are the changes we have to make ( In weight function and coefficients of Hermite polynomials) so that Hermite polynomials are orthogonal with respect to the mixture of two Gaussian distribution.

• Please clarify: your question basically is asking how to change the "weight function" (among other things) in order to make Hermite polynomials orthogonal with respect to your mixture--which is a fixed weight function. In short, you ask how to change something while leaving it fixed. That makes no sense. What do you really need to do? – whuber Mar 13 '13 at 16:55
• My question is, Is there any way to modify the Hermite polynomials so that they are orthogonal w.r.t the mixture. – Gold Mar 13 '13 at 17:17
• It depends on what you mean by "modify": there will be an orthogonal basis of polynomials relative to the mixture, but it would be a stretch to construe them as "modifications" of the Hermite polynomials. They will be what they will be. So is that what you're looking for: an orthogonal polynomial basis? – whuber Mar 13 '13 at 18:03
• Can you please give the procedure to construct the orthogonal polynomials relative to mixture. – Gold Mar 13 '13 at 18:37
• Use Gram-Schmidt orthogonalization of the monomials (or of any other basis, such as the Hermite polynomials). I posted an example here. That code--which is symbolic--embodies the mathematical principles. It also happens to be executable: if you change limits to range from $-\infty$ to $\infty$ and the kernel k to be the mixture PDF, then apply Orthogonalize to the list of polynomials in vectors, you will have an answer. – whuber Mar 13 '13 at 19:50