# interpolation

• Nov 13th 2008, 04:55 AM
mike_caster
interpolation
f is in C[a,b] , a <= x_0 <= x_1 <= b ; and A is the operator that turns f into
interpolating polynomial of f with degree 1 at the interpolation points x_0 and x_1 ;
i.e.

Af(x) = {(f(x_1)-f(x_0)) / (x_1-x_0)}.x + (x_1.f(x_0) - x_0.f(x_1))

How can we show that A is positive if and only if x_0 = a , x_1 = b ?
• Nov 15th 2008, 10:45 PM
CaptainBlack
Quote:

Originally Posted by mike_caster
f is in C[a,b] , a <= x_0 <= x_1 <= b ; and A is the operator that turns f into
interpolating polynomial of f with degree 1 at the interpolation points x_0 and x_1 ;
i.e.

Af(x) = {(f(x_1)-f(x_0)) / (x_1-x_0)}.x + (x_1.f(x_0) - x_0.f(x_1))

How can we show that A is positive if and only if x_0 = a , x_1 = b ?

Are you sure this is true?

For instance the function $\displaystyle f(x)=1$ is in $\displaystyle C[0,1]$ and $\displaystyle (Af)(x)=1$

$\displaystyle g(x)=\begin{cases} -100x+1 & x \in [0,0.02) \\ -1 & x \in [0.02,0.98) \\ 100(x-0.98)-1 & x \in [0.98,1] \end{cases}$
Now if I have defined this right $\displaystyle g \in C[0,1]$ and $\displaystyle (Ag)(x)=1$, and:
$\displaystyle \langle Ag,g \rangle =\int_a^b (Ag)(x) g(x) dx = \int_a^b g(x) dx<0$