Let p0, p1, and p2 be the orthogonal polynomials described...

Let $p_0$, $p_1$, and $p_2$ be the orthogonal polynomials described below, where the inner product on $P_4$ is given by evaluation at $-2, -1, 0, 1,$ and $2$. Find the orthogonal projection of $2t^3$ onto $Span\{p_0, p_1, p_2\)$. $p_0(t) = 1$ $p_1(t) = 4t$ $p_2(t) = t^2 - 2$ The orthogonal projection of $2t^3$ onto $Span\{p_0, p_1, p_2\}$ is $oxed{}$. (Simplify your answer.)

Transcript

00:01 Hello students, it is given that the orthogonal polyanodes, here we have to find the orthogonal projection on p equal to 3 t cube.

00:23 Here the given points are minus 2, minus 1, 0, 1 and 2.

00:31 Here we know that the projection is p naught comma p by inner product of p naught comma p naught into p naught plus p1 comma p by p1 comma p1 into p1 plus inner product of p2 comma p by inner product of p2 comma p2 into p2.

01:00 Here the inner product is can be written as f of f comma g equal to f of minus 2 g of minus 2 plus f of minus 1 g of minus 1 plus f of 0 into g of 0 plus f of 1 into g of 1 plus f of 2 into g of 2.

01:27 Here similarly we have to find for p naught, p1 and p2.

01:33 Here the orthogonal projection on p equal to t2 cube.

01:36 Therefore, we can say that as first we have to find p of 3 into q at the given points.

01:46 Therefore, p of minus 2 equal to 3 into minus 2 the whole cube which is minus 24.

01:52 Similarly, by solving this we will get p of minus 1 is minus 3, p of 0 is 0, p of 1 is 3, p of 2 is 24.

02:08 First we have to find p naught comma inner product of p naught comma p.

02:13 That is first inner product of p naught comma p, then inner product of p naught comma p naught.

02:19 Similarly, we have to find it for other values.

02:21 Therefore, we will get here also given that p naught of t is 1.

02:32 Therefore, inner product of p naught comma p equal to p naught of minus 2 into p of minus 2 plus p naught of minus 1 and p of minus 1 plus p naught of 0, p of 0 plus p naught of 1 into p naught p of 1 plus p naught of 2 into p of 2.

03:01 Therefore, by substituting the values we will get 1 into minus 24 plus 1 into minus 3 plus 1 into 3 plus 1 into 0 plus 1 into 24 which is equal to 0.

03:19 Therefore, p naught comma p is 0.

03:23 Next we have to find inner product of p naught comma p naught.

03:26 From the previous method we have to substitute the values.

03:31 Therefore, it will be 1 into 1 plus 1 into 1 plus 1 into 1 plus 1 into 1 plus 1 into 1 therefore it will be 5.

03:45 Next we have to find inner product of p 1 comma p...

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