6.4.1 1d Fourier analysis and synthesis

To simplify the presentation, we investigate the Fourier transformation with an expansion into sinus and cosinus functions although we have to handle complex coefficients  $\gamma_{k,\ell}$, $\beta_{k,\ell}$ therein.

Denote by $\omega$ the $n$tex2html_wrap_inline^th unit root of 1 (in  $\ensuremath{\mathbb{C}}$) then the matrix of the Fourier transformation  $\ensuremath{{\cal F}}$ is defined as

$$\ensuremath{{\cal F}}_{n} \;=\; \left\{ \ensuremath{{\cal F}}_{j,... ...ft\{ \omega^{j\cdot k}\right\}_{j,k=\overline{0,n-1} \makebox[0pt]{}} \enspace.$$ (6.8)

Together with the normalization  $\tfrac{1}{n}$ we achieve

• for the Fourier analysis : $\qquad\;\; \underline{\gamma} \;:=\; \tfrac{1}{n} \cdot \ensuremath{{\cal F}}\cdot \underline{f}$
• for the Fourier synthesis : $\qquad\underline{u} \;:=\; \ensuremath{{\cal F}}\cdot \underline{b}$ .