I want to find the second-order polynomial that best approximates $t^3$, with respect to the norm of the vector space $V$.
I first proved the bracketing map given in the problem was indeed an inner product for $V$.
Secondly, I converted the standard basis for $V$, $\{1, t, t^2\}$, into an orthonormal basis for $V$, using the Gram-Schmidt process and then normalizing the orthogonal set of vectors.
To answer the question of finding the second-order polynomial which best approximates $t^3$, I know that I am supposed to use this orthonormal basis, and use a sum of projection operators to find my polynomial, but I'm not sure how to proceed.
My initial attempt, if it makes any sense at all, is to project $t^3$ onto the subspace spanned by the first orthonormal basis vector, then project $t^3$ onto the subspace spanned by the second orthonormal basis vector, and then project one more time to the third subspace. If this is correct, how I can write out this polynomial explicitly?
Any help would be greatly appreciated. Thanks,
Best Answer
So you have formed an orthonormal basis $B=\{v_1,v_2,v_3\}$ for $V$ via Gram-Schmidt. You have done most of the work. The last thing you need is to find the best approximation of the function $f(t)=t^3$ in terms of $B$, which is done via $$ f(t)\approx\sum_{k=1}^3c_kv_k\quad\text{where}\quad c_k=\langle f,v_k\rangle=\int_a^b f(t)v_k(t)\,dt, $$ where in the last equality I assumed you are using the standard inner product. If not, replace that integral expression with whatever your inner product is.
Yes, your resulting polynomial $\sum_{k=1}^3c_kv_k$ will be explicit.