Thread: 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 ?

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 is in and

Now consider instead



Now if I have defined this right and , and:



CB