Fourier Sine Series — Question 10

Question 10 – Solution

Strategy. Orthogonality makes the cost of omitting a mode explicit, so optimal selection depends on coefficient size rather than frequency alone.

Step 1: Identify the coefficient energy. The sine functions have squared norm $\pi/2$ and are mutually orthogonal. Thus $b_1=1,b_2=1/2,b_3=1/3,b_{10}=2$, with all other coefficients zero. Consequently $$\boxed{\|f\|_2^2=\frac\pi 2\left(1+\frac 14+\frac 19+4\right)=\frac{193\pi}{72}.}$$

Step 2: Evaluate the conventional truncation. The first-three-frequency sum is $S_3=\sin x+\tfrac 12\sin 2x+\tfrac 13\sin 3x$. Its residual is $2\sin 10x$, so $\boxed{\|f-S_3\|_2^2=2\pi}$. Frequency order alone discards the largest-amplitude component here.

Step 3: Optimize the three selected modes. For any selected frequencies, orthogonality expresses the error as $\pi/2$ times the sum of squared coefficient differences on selected modes plus the squared coefficients of omitted modes. Each retained coefficient must therefore equal the true coefficient. To minimize the omitted energy, retain the three largest magnitudes, uniquely $2,1,1/2$, at frequencies $10,1,2$. Hence $$\boxed{g_*=2\sin 10x+\sin x+\tfrac 12\sin 2x,\qquad \|f-g_*\|_2^2=\frac\pi{18}.}$$ The squared error of $S_3$ is 36 times larger. Adding a zero-coefficient mode cannot improve the selection and using fewer than three omits additional energy.

Step 4: Interpret incomplete spectral information. The measured first-three energy is $(\pi/2)(1+1/4+1/9)=49\pi/72$. Subtracting it from the known total leaves $\boxed{2\pi}$ of unseen energy. Its location is not determined: $S_3+2\sin 10x$ and $S_3+2\sin 100x$ have the same given coefficients and norm. Total energy constrains the squared size of the tail, not its frequency distribution.