I would like assistance with the following problem:
Show that the first two Legendre polynomials, $P_0(x) = 1$ and $P_1(x) = x$ are orthogonal for $x \in [−1,1]$.
Determine constants $\alpha$ and $\beta$ such that the function $h(x) = 1 + \alpha x + \beta x^2$ is orthogonal to both $P_0$ and $P_1$ for $x \in [−1, 1]$.
How does $h(x)$ compare with $P_2(x)$?
My understanding is that the inner product on the space of continuous functions from $[-1, 1]$ to $\mathbb{R}$ is
$$\langle P_0,P_1 \rangle = \int_{[-1,1]}P_0 P_1 r(x) \ dx \tag{1}$$
But then what do I use for the weight function $r(x)$?
I would greatly appreciate it if people could please take the time to clarify this.
Best Answer
Here you should take the weight function $r(x)$ to be just the constant function $1$. There is nothing in the problem statement that explicitly tells you this, but normally if you say that two functions are orthogonal without specifying the weight function that means that the weight function is $1$ (or in other words, there is no weight function and the inner product is just the integral of the product of the two functions). If this convention has not been stated previously and you have only seen a definition of orthogonality with respect to a specified weight function, then the omission of the weight function is just an error in the problem statement.