Fourier Cosine Series — Question 10

Question 10 – Solution

Strategy. Distinguish reconstruction in a finite discrete basis from claims about unmeasured modes or intermediate values.

Step 1: Compute the discrete Gram data. The vectors are $$(1,1,1),\qquad(\sqrt 3/2,0,-\sqrt 3/2),\qquad(1/2,-1,1/2).$$ Their distinct dot products are zero. Dividing squared lengths by three gives squared norms $\boxed{1,1/2,1/2}$.

Step 2: Recover and verify the coefficients. The sample mean gives $A=1$. Dividing each other discrete moment by $1/2$ gives $$\boxed{B=1/\sqrt 3,\qquad C=1,\qquad f=1+\frac 1{\sqrt 3}\cos x+\cos 2x.}$$ The values at the three nodes are $1+1/2+1/2=2$, $1-1=0$ and $1-1/2+1/2=1$. Three nonzero orthogonal vectors are independent, so the coefficient triple is unique within the specified span.

Step 3: Exhibit the invisible smooth freedom. The function $\cos 3x$ vanishes at all nodes, whose tripled arguments are $\pi/2,3\pi/2,5\pi/2$. Hence $\boxed{f_D=f+D\cos 3x,\quad D\in\mathbb R}$ has the same samples. These are distinct smooth functions; reconstruction is not unique among all smooth functions.

Step 4: Check between-node positivity exactly. Since $\cos 2x=2t^2-1$, the reconstructed function is $2t^2+t/\sqrt 3$. Completing the square gives $$\boxed{f=2\left(t+\frac 1{4\sqrt 3}\right)^2-\frac 1{24}.}$$ Its minimum is $-1/24$, attained at the interior point $x=\arccos(-1/(4\sqrt 3))$. Nonnegative samples therefore fail to certify nonnegativity between the nodes, even for this unique finite-span reconstruction.