xi = [ -2 -1 0 1 2 ];
f  = [  1  1 1 1 1 ];
g  = xi.^2 - 2;
uTrue  = exp( -(xi.^2)./8 );
uTaylor = 1 - (xi.^2)/8;

dot( f, g )

a = dot( uTrue, f )/dot( f, f );
b = dot( uTrue, g )/dot( g, g );

uApprox = a * f + b * g;

figure; hold on;
plot( xi, uApprox, '-or' );
plot( xi, uTrue, '-og' );
plot( xi, uTaylor, '-ob' );
title( 'Approximation Comparison' );