Riesz Representation Thereom for Polynomials with real coefficients problem

Question

Find a polynomial q(x) $\in$ P$_2$($\Bbb R$) Such that $ p ( 1/4 ) = $$\int_0^1 p(x)q(x) \,dx$$ $. I'm sorry to ask this question, but I've been working on it for some time. The inner product on P$_2$($\Bbb R$) is given by$ <f,g>= $$\int_0^1 fg \,dx$$. $ I have aimed to utilize the Reisz Representation Thereom to solve this problem. To save time, an orthonomal basis for this space is given by $\beta$ = { $1$, $\sqrt 12$ ($x-1/2$), $\sqrt180$( $ $$x^2$- x + 1/6 $ $ }. Call these vectors $s_1$, $s_2$, $s_3$. Now, should I use the formula q(x) = $\sum S_i S_i$(1/4), where the sum runs from i=0 to 3 ? Is this complelety rong? Any Hints? Thanks for your time as always, and don't waste your time with it too long, thanks.

Answer 1

Doesn't your question boil down to solving $d,f,g$ in terms of $a,b,c$ in the following equation?

$$\int_0^1p(x)q(x) \, dx =p(\frac{1}{4}) \\\int_0^1(ax^2+bx+c)(dx^2+ex+f) \, dx =a(\frac{1}{4})^2+b(\frac{1}{4})+c \\\ cf+ \frac{(ce+bf)}{2}+\frac{cd+be+af}{3}+\frac{bd+ae}{4}+\frac{ad}{5}\, =\frac{a}{16}+\frac{b}{4}+c$$

Note that $p(x)=ax^2+bx+c$ and $q(x)=dx^2+ex+f$

This is just a linear equation in 3 unknowns.