So I was doing this practice problem where I have to consider the vector space of all polynomials with odd degree. Now I have to find the eigenvalues and eigenvectors corresponding to the linear operator I defined as
$ I(f(x)) = \int_{-x}^x f(y)dy$
When I compute $I(f(x))$ for any polynomial $f(x) := a_n x^n + a_{n-1} x^{n-1} + ..... + a_o x^0$ , and equate it to some scalar $\lambda$ times $ f(x)$ I get to the point where I get
$\lambda(a_n x^n + a_{n-1} x^{n-1} + ..... + a_o x^0) = 0 + 2(a_{n-1}/n) x^n + 0 + 2(a_{n-3}/{(n-1)) x^{n-2} + 0 + ...+ 2 a_0 x + 0}$
Which basically gives me $a_0 = 0, a_2 = 0 ....$ which leads to $a_1 = 0, a_3 = 0, ...$, which I feel is an incorrect solution. Can someone help me find where I am doing it wrong