Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 1 / c

The optimal approximation is the orthogonal projection onto the linear span of $p_0,\dots ,p_N$. Their orthogonality givesCompleting each square proves both optimality and uniqueness:For real $f$, these are real coefficients. If complex-valued $f$ is allowed while the coefficients remain real, replace the numerator by its real part and use the modulus-squared norm. The question's real formulation corresponds to real $f$.

Here the first three squared norms are one. Using ${\int }_0^{\infty }{x}^j{e}^{-2x}dx=j!/2^{j+1}$,Thus