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Math Help - interpolation

  1. #1
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    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 ?
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  2. #2
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    Quote Originally Posted by mike_caster View Post
    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 f(x)=1 is in C[0,1] and (Af)(x)=1

    Now consider instead

     <br />
g(x)=\begin{cases} -100x+1 & x \in [0,0.02) \\<br />
-1 & x \in [0.02,0.98) \\<br />
100(x-0.98)-1 & x \in [0.98,1] \end{cases}<br />

    Now if I have defined this right g \in C[0,1] and (Ag)(x)=1, and:

     <br />
\langle Ag,g \rangle =\int_a^b (Ag)(x) g(x) dx = \int_a^b g(x) dx<0<br />

    CB
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