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My task is to find three orthogonal polynomials $P_0$, $P_1$ and $P_2$ under the dot product $ = f(-2)g(-2) + f(-1)g(-1) + f(0)g(0) + f(1)g(1) + f(2)g(2)$.

I thought I could start with $P_0(x) = 1$, $P_1(x) = x$, $P_2(x) = x^2$ and then use the Gram-Schimdt orthogonalization process using the dot product defined above. Is this correct, and the easiest way of doing this?

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    That should work. You could also write $P_i$ as a quadratic polynomial with unknown coefficients and write down the system of equations for orthogonality. – 2017-02-04

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