Question

# Find the discrete Fourier approximation g_{2}(x) for f(x)f(x) based on the table information.x | 0 | \pif(x) | 0 | 2

Discrete math

Find the discrete Fourier approximation $$g_{2}(x)$$ for $$f(x)f(x)$$ based on the table information.
$$x | 0 | \pi$$
$$f(x) | 0 | 2$$

$$\displaystyle{a}_{{0}}={\frac{{{1}}}{{{2}}}}{\left({f{{\left({0}\right)}}}+{f{{\left(\pi\right)}}}\right)}={\frac{{{1}}}{{{2}}}}{\left({1}+{2}\right)}={\frac{{{3}}}{{{2}}}}$$
$$\displaystyle{a}_{{1}}={\left({f{{\left({0}{\cos{{\left({0}\right)}}}+{f{{\left(\pi\right)}}}{\cos{{\left(\pi\right)}}}\right)}}}={\left({1}{\cos{{\left({0}\right)}}}+{2}{\cos{{\left(\pi\right)}}}\right)}=-{1}\right.}$$
$$\displaystyle{b}_{{1}}={\left({f{{\left({0}\right)}}}{\sin{{\left({0}\right)}}}+{f{{\left(\pi\right)}}}{\sin{{\left(\pi\right)}}}\right)}={\left({1}{\sin{{\left({0}\right)}}}+{2}{\sin{{\left(\pi\right)}}}\right)}={0}$$
$$\displaystyle{g}_{{1}}={\frac{{{3}}}{{{2}}}}-{\cos{{\left({x}\right)}}}$$