Fourier Cosine Series — Question 8

Question 8 – Solution

Strategy. Linear moments preserve coefficient signs, while a squared-energy measurement can lose them.

Step 1: Solve the coefficient equations. Write $f=A+B\cos x+C\cos 2x$. Orthogonality gives $A=1$, $B=1/2$ and $\pi[A^2+(B^2+C^2)/2]=5\pi/4$. Thus $C^2=1/4$, leaving $$\boxed{f_\pm=1+\tfrac 12\cos x\pm\tfrac 12\cos 2x.}$$ The energy measurement determines the magnitude, but not the sign, of $C$.

Step 2: Test positivity without sampling. With $t=\cos x$, $f_+=t^2+t/2+1/2=(t+1/4)^2+7/16>0$. The other candidate is $f_-=3/2+t/2-t^2$. This concave quadratic reaches its minimum on $[-1,1]$ at an endpoint, with values zero and one. Thus both candidates are nonnegative; positivity does not resolve the ambiguity.

Step 3: Apply the endpoint measurement. At $\pi$, $f_+(\pi)=1$ while $f_-(\pi)=0$. Hence the unique candidate within the specified span is $\boxed{f=f_-}$. Its mean and first moment are the prescribed ones, and its squared norm is $\pi(1+1/8+1/8)=5\pi/4$, verifying all four data.

Step 4: Remove the model restriction. The distinct function $$\boxed{g=1+\tfrac 12\cos x-\tfrac 12\cos 4x}$$ has the same mean, first cosine moment and squared norm by orthogonality. Also $g(\pi)=0$ and $g\geq 1-1/2-1/2=0$ everywhere. Thus these measurements and nonnegativity do not give uniqueness among all continuous functions. The finite-span hypothesis was essential.