# (Legendre)orthogonal functions with weighted function

$$(f,g)= \int_0^1 x^2 f(x) g(x) dx$$

I need to find corresponding orthagonal function $$\phi_0,\phi_1$$ where $$\phi_0 = 1$$

and $$w(x) = x^2$$ (weight function)

$$\phi_1(x) = (x - \alpha_0)\phi_0$$

Where $$\alpha_0 = \frac{(\phi_0,x\phi)}{||\phi||^2}$$

so I did $$\alpha_0 = \frac{(\phi,x\phi)}{||\phi||^2} = \frac{\int_0^1 x^2.xdx}{\int_0^1 1.1dx} = \frac{1/4}{1} = 1/4$$ so

$$\phi_1 =x - 1/4$$

is this the right place and time to put weight fucntion $$x^2$$ in side the integral multiplying $$xdx$$

problem is when I check this function $$\int_{0}^1 w(x)\phi_1 \phi_2$$ its not equal to zero. ie $$\int_{0}^1 x^2.1.(x - \frac{1}{4})dx \ne 0$$

• It's not entirely clear what's going on here. Are you trying to produce a basis of some vector space of polynomials orthonormal with respect to the given inner product $(f, g) := \int_0^1 x^2 f(x) g(x) dx$? – Travis Willse Mar 11 '19 at 20:53
• Yes trying to find corresponding orthonormal functions. – tt z Mar 11 '19 at 21:08
• Perhaps post a statement of the original question, for clarity? – Travis Willse Mar 11 '19 at 21:09
• $(f,g)= \int_0^1 x^2 f(x) g(x) dx$ is inner production function. Determined the corresponding orthogonal function $\phi_0 \phi_1. \phi_2$<question ends> I know from other exercise that we start with $\phi_0 = 1$ – tt z Mar 11 '19 at 21:14
• I meant, edit your original post to include it. But either way---what are the constraints on $\phi_1, \phi_2$? There's no way to answer this question without more context. – Travis Willse Mar 11 '19 at 21:17