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Let [tex]\{ \phi_0,\phi_1,...,\phi_n\}[/tex] othogonal set of polynomials on [a,b] n>0, with a weight function w(x)

[tex]\int_{a}^b w(x)\phi_n Q_k (x) \; dx = 0 [/tex]

for any polynomail [tex]Q_k(x) [/tex] of degree k<n ?

My work :

I think there is a problem in the question since if we take [tex]x^2,x^3 [/tex] on the interval [-1,1] they are orthogonal

but if we take x

[tex]\int_{-1}^{1} x(x^3 ) \; dx \neq 0 [/tex]

**prove that**[tex]\int_{a}^b w(x)\phi_n Q_k (x) \; dx = 0 [/tex]

for any polynomail [tex]Q_k(x) [/tex] of degree k<n ?

My work :

I think there is a problem in the question since if we take [tex]x^2,x^3 [/tex] on the interval [-1,1] they are orthogonal

but if we take x

[tex]\int_{-1}^{1} x(x^3 ) \; dx \neq 0 [/tex]

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