Math 6610 • Analysis of Numerical Methods I

\textbf{a)} What is the best least squares approximation to \(f(x)=\cos 3x\) by
functions of the form
\[
  a_0+a_1\cos x+a_2\cos^2 x+b_1\sin x+b_2\sin^2 x
\]
with the inner product given by
\[
  (v,w)_C=\frac{1}{2\pi}\int_0^{2\pi}v(x)\overline{w(x)}\,dx
\]

\textbf{b)} What if the inner product is
\[
  (v,w)_D=\frac{1}{J}\sum_{j=0}^{J-1}v(x_j)\overline{w(x_j)}
\]
with \(x_j=jh\), \(h=2\pi/J\).

\textbf{c)} Briefly discuss the normal equations (do they form a well- or
ill-conditioned system?), the Gram-Schmidt process, QR factorization in the
context of parts \textbf{a)} and \textbf{b)}. Observe the connection with problem
\#2.