# Eigenfunctions and eigenvalues of Fourier Transform?

1. Jul 11, 2006

### eljose

:grumpy: I have a question..yesterday at Wikipedia i heard about the "Hermite Polynomials2 as Eigenfunctions of fourier (complex?) transform with Eigenvalues i^{n} and i^{-n}...could someone explain what it refers with that?...when it says "Eigenfunctions-values" it refers to the Kernel K(x,t) that is a complex exponential function?...

2. Jul 11, 2006

### HallsofIvy

Staff Emeritus
In Linear Algebra, an "eigenvalue" and "eigenvector" of a linear transformation, L, are a number, $\lambda$, and vector, v, such that $Av= \lambda v$. We can think of the set of (integrable) functions as a vector space and the Fourier transform is a linear transformation on that set. The Hermite Polynomials have the property that the Fourier transform of the nth Hermite Polynomial, Hn, is
F(Hn)= inHn.

Last edited: Jul 12, 2006
3. Jul 11, 2006

### mathman

Note to Halls of Ivy. You left out the eigenvalue in your definition equation.

4. Jul 12, 2006

### HallsofIvy

Staff Emeritus
Thanks, I've edited it.

5. Jul 12, 2006

### HallsofIvy

Staff Emeritus
Thanks, I've edited it. (I misspelled "lamba" in the TEX)