Approximation by Trigonometric Polynomials

This note includes some basic information about trigonometric approximation.
Trigonometric polynomials are often applied in the approximation of periodic functions. A trigonometric polynomial of order n is a function of the form

\[p(x)=\frac{a_0}{2}+\sum_{j=1}^n[a_j\cos{jx}+b_j\sin{jx}] \]

with \(a_j,b_j\) are real or complex constants. \(p(x)\) is \(2\pi\)-periodic and when approximating a \(\tau-periodic\) function \(f(x)\), we only need to consider the function \(g(x)=f(\frac{\tau x}{2\pi})\).
For simplicity, we will assume the function to be approximated is \(2\pi-periodic\).

An equivalent complex form of \(p(x)\) is wrriten as

\[p(x)=\sum_{j=-n}^n c_j e^{i(jx)} \]

where i is the imaginaty unit.

With Euler's formula, we get

\[\sum_{j=-n}^n c_j e^{i(jx)}=\sum_{j=-n}^n c_j[\cos{jx}+i\sin{jx}]=c_0+\sum_{j=1}^n[(c_j+c_{-j})\cos{jx}+i(c_j-c_{-j})\sin{jx}] \]

and

\[c_j=\frac{1}{2}(a_j-ib_j), \ c_{-j}=\frac{1}{2}(a_j+ib_j). \]

As \(p(x)\) is a real function, it is its own complex conjugate

\[\overline{\sum_{j=-n}^n c_j e^{i(jx)}}=\sum_{j=-n}^n \overline{c_j} e^{-i(jx)}=\sum_{j=-n}^n \overline{c_{-j}} e^{i(jx)}=\sum_{j=-n}^n c_j e^{i(jx)} \]

Therefore, \(a_j\) and \(b_j\) can be simplified to

\[a_j=2 \mathrm{Re} c_j,\ b_j=-2\mathrm{Im} c_j. \]

The following theorem displays the best approximation polynomials.

\(\mathbf{Theorem}\) The partial sum

\[\sum_{j=-n}^n \hat{f}(j)e^{i(jx)} \]

of the Fourier series for \(f(x)\) is the best approximation to \(f(x)\) by trigonometric polynomials of order n with respect to the norm

\[||g||=||g||_2=[\frac{1}{2\pi}\int_{0}^{2\pi}|g(x)|^2 \mathrm{d}x]^{\frac{1}{2}}. \]

In order to calculate \(\hat{f}(j)\), numerical integration will be used. \(\hat{f}_N(j)\) as followings gives an approximation to \(\hat{f}(j)\):

\[\hat{f}_N(j)=\frac{1}{N}\sum_{n=0}^{N-1}f(x_n)e^{-ix_n} \]

where

\[x_n=\frac{2\pi n}{N},\ n=0,1,\cdots,N-1 \]

\(\mathbf{Theorem}\) For any m \(\leq\)n, the mth order trigonometric polynomial

\[p_m(x)=\sum_{j=-m}^{m} \hat{f}_N(j)e^{ijx} \]

is the best approximation to \(f(x)\) by trigonometric polynomials order m with respect to the discrete mean-square norm

\[||g||=\frac{1}{N}(\sum_{j=0}^{N-1}|g(\frac{j2\pi}{N})|^2)^{\frac{1}{2}} \]