Hello MathstackExchange..
Let $P\in \mathbb{R}_3[X]$ be represented in its canonical form: $P=\sum_{k=0}^3a_kX^k$. Express $P$ by $P(0),P(1),P'(0),P'(1)$.
What's given as a hint is: Give the new basis $(H_k)_{k=0,1,2,3}$ (where $(H_k)$ represents the Hermite Polynomial) Of $\mathbb{R}_3[X]$, that permits that expression and verifies:
for $k=0$: $H_0(0)=1,H_0(1)=0,H'_0(0)=0,H'_0(1)=0$
for $k=1$: $H_1(0)=0,H_1(1)=1,H'_1(0)=0,H'_2(1)=0$
for $k=2$: $H_2(0)=0,H_2(1)=0,H'_2(0)=1,H'_3(1)=0$
for $k=3$: $H_3(0)=0,H_3(1)=0,H'_3(0)=0,H'_4(1)=1$
My problem is I really don't understand what I'm supposed to do.. How am I supposed to get the desired basis with what I'm given? and how to connect it to the expression of $P$? Any hints would be really appreciated.
I get that $P(0)=a_0, P(1)=a_0+a_1+a_2+a_3, P'(0)=a_1, P'(1)=a_1+2a_2+3a_3$
Am I supposed to express the coefficients $a_0,a_1,a_2,a_3$ in terms of $P(0),P(1),P'(0),P'(1)$?
I won't lie this is a homework that i'm not really able to solve.
Sorry if anything was lost with the translation of the problem.
If you solve for the $a_i$ in terms of $P(0), P(1),P'(0), P(1)$, substitute these values into the expression for $P$, and collect like terms, you'll have solved the problem.