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?...Thanks, I've edited it.mathman said:
Thanks, I've edited it. (I misspelled "lamba" in the TEX)mathman said:
A Fourier Transform is a mathematical operation that decomposes a function into its constituent frequencies. It converts a function from its original domain (e.g. time) to a representation in the frequency domain.
Eigenfunctions are special functions that, when transformed by a linear operator, remain unchanged except for a scalar multiple (eigenvalue). In the context of Fourier Transform, eigenfunctions are the basis functions that are used to represent a given function in the frequency domain.
Eigenfunctions are important because they allow us to represent a function in the frequency domain as a linear combination of these basis functions. This simplifies the analysis of complex functions and makes it easier to manipulate them. Eigenvalues, on the other hand, give us information about the amplitude and phase of each frequency component in the transformed function.
Eigenfunctions in Fourier Transform are the complex exponential functions, while the eigenvalues are the corresponding frequencies. These can be calculated using the Fourier Transform integral formula or by using the discrete Fourier Transform algorithm.
In addition to simplifying the analysis of complex functions, the eigenfunctions and eigenvalues in Fourier Transform have physical significance as they represent the different frequency components of a signal or function. This is useful in fields such as signal processing, image processing, and quantum mechanics.