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

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 $

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  • $\begingroup$ 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 $(f, g) := \int_0^1 x^2 f(x) g(x) dx$? $\endgroup$ – Travis Willse Mar 11 '19 at 20:53
  • $\begingroup$ Yes trying to find corresponding orthonormal functions. $\endgroup$ – tt z Mar 11 '19 at 21:08
  • $\begingroup$ Perhaps post a statement of the original question, for clarity? $\endgroup$ – Travis Willse Mar 11 '19 at 21:09
  • $\begingroup$ $(f,g)= \int_0^1 x^2 f(x) g(x) dx$ 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 $\phi_0 = 1$ $\endgroup$ – tt z Mar 11 '19 at 21:14
  • $\begingroup$ 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. $\endgroup$ – Travis Willse Mar 11 '19 at 21:17

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