When we think of functions, we refer to their oscillations or periodic components in terms of frequencies. The Fourier transform allows us to decompose a function into its constituent frequencies, providing a way to analyze the function in the frequency domain.

For operators that act on infinite-dimensional function spaces, is there an analog of the frequency concept? Is there a transform or methodology that lets us analyze operators in a similar "frequency-centric" manner?

I'm curious about both theoretical constructs and any potential applications such a perspective might offer.

  • $\begingroup$ If you think of waves of different frequencies as providing a "basis" of the vector space of functions, then using this basis to express operators lets you express operators as a "matrix" whose rows and columns are indexed by frequencies. This is roughly the idea behind Heisenberg's "matrix mechanics" approach to quantum mechanics: en.wikipedia.org/wiki/Matrix_mechanics $\endgroup$ Oct 28, 2023 at 19:43
  • $\begingroup$ The Fourier transform of the Wigner transform can be seen as a Fourier transform of the operator. $\endgroup$
    – LL 3.14
    Oct 29, 2023 at 8:08
  • $\begingroup$ One observation: the algebra of operators is generally non commutative, thus this is possibly an obstacle in the construction of a complete analogue of Fourier analysis for functions of an operator variable (which, if I have not erroneously understand, it is the sense of the question in the OP). $\endgroup$ Nov 4, 2023 at 9:01


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