How do I find the eigenvalues of the operator $\int_{-1}^1(1-3tz)\,x(z)\,dz$?

Question

I'm having trouble finding all the eigenvalues of the following operator:

$$Tx(t)=\int_{-1}^1(1-3tz)\,x(z)\,dz$$

In the examples I've seen, there's no systematic way that I learnt to do this. Can someone show me the correct way through this example?

Answer 1

Hint: We want functions $x(t)$ such that $Tx(t) = ax(t)$ for some scalar $a$.

But, $Tx(t) = \displaystyle\int_{-1}^{1}(1-3tz)x(z)\,dz = \left[\int_{-1}^{1}x(z)\,dz\right]-\left[\int_{-1}^{1}3zx(z)\,dz\right]t$.

Thus, $Tx(t)$ is always a polynomial in $t$ with degree at most $1$.

So if $x(t)$ is an eigenfunction, then $x(t)$ must also be a polynomial in $t$ with degree at most $1$.