Eigenfunctions of an Integral Operator

In summary, the author is trying to find an equation for the function Kf(t) = \lambda f(t), but is having difficulty.
  • #1
Kreizhn
743
1

Homework Statement



If there are any eigenvalues for the following integral operator, calculate them

[tex] Kf(t) = \int_0^1 (1+st) f(s) \ ds [/tex]

The Attempt at a Solution



I've tried making this into a differential equation, to no avail. I've also just tried solving the equation [itex] Kf(t) = \lambda f(t) [/itex] though that also didn't lead anywhere. I'm not sure if there's some special way of attacking problem like this. Does anybody have any ideas?
 
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  • #2
It's a lot easier than you are think it is. Your integral expression is a linear function of t, no matter what f is.
 
  • #3
not too sure.. but how about noticing the t depndence of the function... does the fact
[tex]\int_0^1 f(s) ds [/tex]
and
[tex]\int_0^1 sf(s) ds [/tex]
are constants help..
 
  • #4
I had tried looking for eigenfunctions of the form [itex] a + bt [/itex] but for some reason got an inconsistent system of equations. I'll go back and look at it again.
 
  • #5
i haven't tried, but for non-zero b
(a+bt) = b(a/b+t)

so if (a+bt) is an eigenfunction, then so is (a/b+t) = c+t which ma be easier to deal with
 
  • #6
So if try a function of the form [itex] f(t) = t + a [/itex] and try [itex] K f(t) = \lambda f(t) [/itex] I get the following system of equations
[tex] \begin{align*}
\frac13 + \frac a2 &= \lambda \\
\frac12 + a &= \lambda a
\end{align*}
[/tex]

I can solve this to get [itex] a = 1.868517092, \lambda = 1.267591879 [/itex]. But it seems odd that an integral operator would only have one eigenvalue in it's spectrum. Furthermore, how can I be sure that this is exhaustive? How would I find other eigenvalues and eigenfunctions?
 
  • #7
the only other option is a constant, which i don't think works as you end up with a t dependence
 
  • #8
Indeed, so you're saying that the only possible eigenfunctions are of the form [itex] f(t) = t+a [/itex] since for all other functions, [itex] Kf(t) [/itex] will be first order polynomial in t.
 
  • #9
Kreizhn said:
So if try a function of the form [itex] f(t) = t + a [/itex] and try [itex] K f(t) = \lambda f(t) [/itex] I get the following system of equations
[tex] \begin{align*}
\frac13 + \frac a2 &= \lambda \\
\frac12 + a &= \lambda a
\end{align*}
[/tex]

I can solve this to get [itex] a = 1.868517092, \lambda = 1.267591879 [/itex]. But it seems odd that an integral operator would only have one eigenvalue in it's spectrum. Furthermore, how can I be sure that this is exhaustive? How would I find other eigenvalues and eigenfunctions?

Don't you get a quadratic equation in lambda? Aren't there two eigenvalues?
 
  • #10
Yes. I was being lazy an had plugged this into a Maple to solve numerically. But it isn't hard to do algebraically. I realized this, but figured the thread was dead so it wasn't important to post it :P
 
  • #11
Kreizhn said:
Yes. I was being lazy an had plugged this into a Maple to solve numerically. But it isn't hard to do algebraically. I realized this, but figured the thread was dead so it wasn't important to post it :P

Sure. Just checking...
 

Related to Eigenfunctions of an Integral Operator

What is an integral operator?

An integral operator is a type of mathematical operator that maps a function to another function by using an integral as its operation. It can be represented by an integral sign and a kernel function.

What are eigenfunctions of an integral operator?

Eigenfunctions of an integral operator are functions that, when operated on by the integral operator, result in a scalar multiple of the original function. In other words, they are the special functions that remain unchanged or are only multiplied by a constant when operated on by the integral operator.

What is the significance of eigenfunctions of an integral operator?

The eigenfunctions of an integral operator are important because they form a complete set of functions that can be used as a basis for representing other functions. This allows for the simplification and solution of complex mathematical problems.

How are eigenfunctions of an integral operator calculated?

Eigenfunctions of an integral operator can be calculated by solving a special type of integral equation, known as the eigenvalue problem, which involves finding the values of the kernel function that result in scalar multiples of the original function.

What are some applications of eigenfunctions of an integral operator?

Eigenfunctions of an integral operator have various applications in mathematics, physics, and engineering. They are commonly used in solving differential equations, analyzing systems in quantum mechanics, and in signal processing and image analysis.

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