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.Originally Posted by hashshashin715
You are given a linear Transformation
where
is some space of integrable functions on all of
and for
Here is my point of confusion for every integrable(if it is a function of 1 variable) this integral is a constant
But the only constant function that is integrable on all ofis the zero function
. This is the zero vector of your space so it cannot be an eigenvector!
What you have written is impossible. For any f(x), such that the integral converges,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 onto 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
Did you mean? In that case, you have
, an "integral equation". But, yes, you can differentiate both sides, using the "fundamental theorem of Calculus" on the left to get the differential equation
or
. What are the solutions to that equation?
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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