- #1
psie
- 315
- 40
- Homework Statement
- Determine the polynomial ##p## of degree at most ##1## that minimizes ##\int_0^2|e^x-p(x)|^2 dx##. (Hint: first find an orthogonal basis for a suitably chosen space of polynomials of degree ##1##.)
- Relevant Equations
- ##L^2## norm, inner product, least squares approximation.
I'm posting to inquire about a possible typo in the given answer in the back of the book, or if maybe I did something wrong, because my answer does not agree with the one stated.
So the exercise is about finding the least squares approximation. The norm is the ##L^2## norm and the corresponding inner product is $$\langle f, g\rangle=\int_0^2 f(x)g(x)dx.$$ I choose the polynomials ##v_1=1## and ##v_2=x## as a basis and make them orthogonal according to Gram-Schmidt, i.e. I find that ##u_1=1## and ##u_2=x-1## are orthogonal to each other under the inner product above. Put ##u=e^x##, then the orthogonal projection of ##u## on the subspace spanned by ##u_1,u_2## is given by $$P(u)=\frac{\langle u, u_1\rangle}{\langle u_1,u_1\rangle}u_1+\frac{\langle u, u_2\rangle}{\langle u_2,u_2\rangle}u_2.$$ Now, \begin{align}\langle u_1,u_1\rangle&=\int_0^2 dx=2 \nonumber \\ \langle u_2,u_2\rangle&=\int_0^2(x-1)^2dx=\int_0^2(x^2-2x+1)dx=\frac23, \nonumber\end{align}and\begin{align}
\langle u,u_1\rangle&=\int_0^2 e^xdx=e^2-1 \nonumber \\
\langle u,u_2\rangle&=\int_0^2(x-1)e^x dx=2. \nonumber
\end{align}
Plugging these into the equation for ##P(u)##, gives $$P(u)=3x+\frac12(e^2-1),$$but the answer given is ##3x+\frac12(e^2-7)##. Is there a way to check one's answer to know if you got the right one?
So the exercise is about finding the least squares approximation. The norm is the ##L^2## norm and the corresponding inner product is $$\langle f, g\rangle=\int_0^2 f(x)g(x)dx.$$ I choose the polynomials ##v_1=1## and ##v_2=x## as a basis and make them orthogonal according to Gram-Schmidt, i.e. I find that ##u_1=1## and ##u_2=x-1## are orthogonal to each other under the inner product above. Put ##u=e^x##, then the orthogonal projection of ##u## on the subspace spanned by ##u_1,u_2## is given by $$P(u)=\frac{\langle u, u_1\rangle}{\langle u_1,u_1\rangle}u_1+\frac{\langle u, u_2\rangle}{\langle u_2,u_2\rangle}u_2.$$ Now, \begin{align}\langle u_1,u_1\rangle&=\int_0^2 dx=2 \nonumber \\ \langle u_2,u_2\rangle&=\int_0^2(x-1)^2dx=\int_0^2(x^2-2x+1)dx=\frac23, \nonumber\end{align}and\begin{align}
\langle u,u_1\rangle&=\int_0^2 e^xdx=e^2-1 \nonumber \\
\langle u,u_2\rangle&=\int_0^2(x-1)e^x dx=2. \nonumber
\end{align}
Plugging these into the equation for ##P(u)##, gives $$P(u)=3x+\frac12(e^2-1),$$but the answer given is ##3x+\frac12(e^2-7)##. Is there a way to check one's answer to know if you got the right one?
