An inner product <f,g> is defined as the integral from -1 to 1 of f(x)g(x).
Given an orthonormal basis for the set of polynomials of degree less than or equal to 2, i.e.,
w1 = (rt 2)/2,
w2 = (rt 6)/2,
w3 = [(x^2 - 1/3) 3(rt10)] / 4,
Find the polynomial of degree at most two which is closest to sin (pi x).
I tried to minimize the norm of v - a linear combination of w1, w2 and w2, and I arrived at the answer 3/pi, which seems wrong. I checked my answer using an approximation through taylor series and as expected I obtained pi.
I can't figure out where I went wrong - all the integration seems to be correct; any help would be much appreciated. Thanks!
What you can say without any computation is that as and are even functions (and we are working with the interval and the usual inner product) that their coefficients in the expansion of will be zero. So your approximation will be:
The attachment shows a plot of the function and the fitted polynomial (line).