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Math Help - Determine Eigenfunctions?

  1. #1
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    Determine Eigenfunctions?

    if g = T(f) and g(x) = integral(f, -infinity, infinity) then prove that every possible lambda is an eigenvalue for T and determine the eigenfunctions corresponding to lambda.

    When I set up the integral = (lambda)(f), i guess I need to take the derivative? How would I take the derivative of an improper integral?
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  2. #2
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    Quote Originally Posted by hashshashin715 View Post
    if g = T(f) and g(x) = integral(f, -infinity, infinity) then prove that every possible lambda is an eigenvalue for T and determine the eigenfunctions corresponding to lambda.

    When I set up the integral = (lambda)(f), i guess I need to take the derivative? How would I take the derivative of an improper integral?
    I am a little confused so let me see if I understand.

    You are given a linear Transformation
    T:X \to \mathbb{R} where X is some space of integrable functions on all of \mathbb{R} and for f \in X
    \displaystyle T(f)=\int_{-\infty}^{\infty}f(x)dx

    Here is my point of confusion for every integrable f(if it is a function of 1 variable) this integral is a constant

    \displaystyle \int_{-\infty}^{\infty}f(x)dx

    But the only constant function that is integrable on all of \mathbb{R} is the zero function f(x) = 0. This is the zero vector of your space so it cannot be an eigenvector!
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  3. #3
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    What you have written is impossible. For any f(x), such that the integral converges, \int_{-\infty}^\infty f(x)dx is a number so there are two ways to interpret this:
    (1) that T(f) is a linear transformation from the set of functions integrable on (-\infty, \infty) to the real numbers. But, in order to have eigenvalues and eigenvectors, a linear transformation must be from some vector space V to itself not from one vector space to another.
    (2) that T(f) is a inear transfromation from the set of functions integrable on (-\infty, \infty) to itself. But then T(f) is a constant function and a non-zero constant function is not in that set.

    Did you mean T(f)= \int_{-\infty}^x f(t)dt? In that case, you have \int_{-\infty}^x f(t)dt= \lambda f, an "integral equation". But, yes, you can differentiate both sides, using the "fundamental theorem of Calculus" on the left to get the differential equation f(x)= \lambda \frac{df}{dx} or \frac{df}{dx}= \frac{1}{\lambda} f. What are the solutions to that equation?
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  4. #4
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    Ok, thanks all I wanted to know was if I can use the fundamental theorem of Calculus on the equation I gave in the OP, because I can get the right answer that way.

    I didn't give the specifics because I really didn't think they mattered. But it seems they do (I never use them).
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