Eigenfunctions and eigenvalues of Fourier Transform?

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Discussion Overview

The discussion revolves around the concept of eigenfunctions and eigenvalues in the context of the Fourier Transform, specifically focusing on Hermite Polynomials and their properties as eigenfunctions. Participants explore the mathematical definitions and implications of these concepts within linear transformations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of "Eigenfunctions-values" in relation to the Kernel K(x,t) and its connection to complex exponential functions.
  • Another participant explains that in Linear Algebra, eigenvalues and eigenvectors can be understood in the context of the Fourier transform as a linear transformation on a vector space of integrable functions, noting that the Fourier transform of the nth Hermite Polynomial results in a specific eigenvalue relation.
  • Several participants point out an omission regarding the eigenvalue in a previous definition provided, indicating a need for clarity in mathematical expressions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of eigenfunctions and eigenvalues in the context of the Fourier Transform, and there are multiple viewpoints regarding the definitions and properties discussed.

Contextual Notes

There are unresolved mathematical steps and potential ambiguities in definitions, particularly concerning the relationship between Hermite Polynomials and the Fourier Transform.

eljose
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:rolleyes: :cool: 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?...
 
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In Linear Algebra, an "eigenvalue" and "eigenvector" of a linear transformation, L, are a number, [itex]\lambda[/itex], and vector, v, such that [itex]Av= \lambda v[/itex]. 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 by a moderator:
Note to Halls of Ivy. You left out the eigenvalue in your definition equation.
 
mathman said:
Note to Halls of Ivy. You left out the eigenvalue in your definition equation.
Thanks, I've edited it.
 
mathman said:
Note to Halls of Ivy. You left out the eigenvalue in your definition equation.
Thanks, I've edited it. (I misspelled "lamba" in the TEX)
 

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