Gram-Schmidt process with Hermite functions on [-1, 1]

Question

Denote by $h_n$ the $n$-th Hermite function. $$ h_n(x) = \frac{(-1)^n }{\sqrt{2^n n! \sqrt{\pi}}} \mathrm{e}^{\frac{x^2}{2}} \frac{\mathrm{d}^n}{\mathrm{d} x^n} \mathrm{e}^{-x^2} $$

I am trying to find the 40th, 41st and 42nd terms in Gram-Schmidt process with Hermite functions $h_n$ on $[-1, 1]$.

I've used the usual procedure for Gram-Schmidt process, but I've been able to calulate only the first 6 terms and then my computer got stuck.

Is there some way to calculate them?

Is this question related to Mathematica (TM)? Can you post your Mathematica code? — Dr. belisarius, Jul 02 '13 at 17:24
Have you seen this? http://functions.wolfram.com/HypergeometricFunctions/HermiteHGeneral/02/ — bill s, Jul 02 '13 at 17:30
Related (duplicate?): http://mathematica.stackexchange.com/q/22040/5 — rm -rf, Jul 02 '13 at 17:32
@ bill s: as I understand, its Hermit polynomial-not function... — Mack, Jul 02 '13 at 17:40
@Mack what is your weight function on the interval to get normalized Hermite polynomials? — Stefan, Jul 02 '13 at 17:58
Maybe you could show us how you used Mathematica to do the first 6 terms? — bill s, Jul 02 '13 at 18:51
To the closers: I don't see any reason to close this as off topic. Mathematica has a function to deal with problems like this(Orthogonalize), so to me it looks like a fine Mathematica topic. — Sjoerd C. de Vries, Jul 02 '13 at 20:22
@SjoerdC.deVries Mma has functions to deal with many problems. But the OP didn't answer the question in the first comment, and there is no code either. — Dr. belisarius, Jul 02 '13 at 20:57
@belisarius That's not nice, but is that sufficient reason to close? Id say the question is answerable quite easily with the information that was provided.. — Sjoerd C. de Vries, Jul 02 '13 at 22:25
Related (possible duplicate?) of Why does Expand not work within a function? — Artes, Jul 03 '13 at 13:25

score 3 · Accepted Answer · edited Jul 02 '13 at 20:18

3

Based on the Wikipedia article you referred, here is a code

u[1] = HermiteH[1, x];
Do[
  u[n] = HermiteH[n, x] - Sum[Integrate[HermiteH[n, x] u[i], 
  {x, -1.0, 1.0}], {i, 1,n-1}], {n, 2, 45}
];

edited Jul 02 '13 at 20:18

Sjoerd C. de Vries

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answered Jul 02 '13 at 19:16

Sumit

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Gram-Schmidt process with Hermite functions on [-1, 1]

1 Answers1