A Fourier transform related problem.

Question

$\def\sgn{\operatorname{sgn}}$Given that $f(x)=\sgn(x)+\cos\pi x+\sin5\pi x (-1 \le x<1), g(x)=\alpha \cos\pi x+\beta \sin 5\pi x$ and $E(\alpha,\beta)=\int_{-1}^1|f(x)-g(x)|^2dx$, where $\alpha$ and $\beta$ are both real numbers, how can I express $E(\alpha,\beta)$ by using a quadratic function of $\alpha$ an $\beta$ ? Can somebody help me? Thanks a lot.

Answer 1

\begin{align*} h(x) &:= f(x)-g(x)\\ &= \text{sgn}(x) + (1-\alpha)\cos(\pi x) + (1-\beta)\sin(5\pi x) \end{align*}

Option 1

If you compute the Fourier coefficients of $h$, then you can use Parseval's theorem to evaluate the integral.

Option 2

Directly compute \begin{align*} h(x)^2 =& \text{sgn}(x)^2 + (1-\alpha)^2\cos^2(\pi x) + (1-\beta)^2\sin^2(5\pi x)\\ & + 2(1-\alpha)\text{sgn}(x)\cos(\pi x) + 2(1-\beta)\text{sgn}(x)\sin(5\pi x) + 2(1-\alpha)(1-\beta)\cos(\pi x)\sin(5\pi x) \end{align*}

Since some of these terms ($\text{sgn}(x)\cos(\pi x)$ and $\cos(\pi x)\sin(5\pi x)$) are odd and we are integrating over a symmetric interval about 0,

\begin{align*} \int_{-1}^1 h(x)^2 dx &= \int_{-1}^1 1 + (1-\alpha)^2\cos^2(\pi x) + (1-\beta)^2\sin^2(5\pi x) + 2(1-\beta)\text{sgn}(x)\sin(5\pi x) dx \end{align*}

Then split the integral into two pieces, corresponding to the intervals $(-1,0)$ and $(0,1)$. Use trig identities to evaluate.