Chapter 6.8, Problem 14E

In Exercises 5-14, the space is C[0, 2π] with the inner product (6).

14. Suppose the first few Fourier coefficients of some function f in C[0, 2π] are a₀, a₁, a₂, and b₁, b₂, b₃. Which of the following trigonometric polynomials is closer to f? Defend your answer.

$g\left(t\right)=\frac{a_0}{2}+a_1\cos t+a_2\cos 2t+b_1\sin t$

$h\left(t\right)=\frac{a_0}{2}+a_1\cos t+a_2\cos 2t+b_1\sin t+b_2\sin 2t$