Eigenvalues of operator between L^2

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SUMMARY

The discussion focuses on finding the eigenvalues and eigenvectors of the operator M defined on L^2, where (Mf)(t) = ∫(-π, π) sin(y-x)f(x) dx. The participant recognizes that sin(y-x) can be expressed as sin(y)cos(x) - cos(y)sin(x) and identifies that the eigenvectors are contained within the subspace spanned by sin(x) and cos(x). However, they encounter issues with integration leading to a result of zero, prompting questions about their approach and the correct form of f(x).

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  • Understanding of L^2 spaces and their properties
  • Knowledge of eigenvalues and eigenvectors in linear operators
  • Familiarity with Fourier series and trigonometric identities
  • Basic integration techniques in functional analysis
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Mathematicians, students of functional analysis, and anyone studying operator theory and spectral analysis in L^2 spaces.

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Homework Statement



>M: L_2 -> L_2
>
>(Mf)(t) = int(-pi, pi) sin(y-x)f(x) dx
>
>how do i find eigenvalues/vectors of M and what can i use to find
>information about the spectrum?



Homework Equations





The Attempt at a Solution



now i know that sin(y-x) = sinycosx-cosysinx
i also realize that the range is 2dimensional
when i went to construct f i made it = cosy + asinx
so i plugged this in, but when i integrated i got 0. did i integrate wrong? or did i take the wrong approach?
 
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Can you give the steps that led you to the particular form of f(x) that you have stated?
 
because i know that the eigenvector is contained in a subspace spanned by sin(x) and cos(x). i figured that'd be a good f?
 

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