4
\documentclass[11pt, a4paper]{report}
    \usepackage{bm}
    \usepackage{amsfonts, graphicx, verbatim, mathtools,amssymb, amsthm, mathrsfs}
    \usepackage{color}
    \usepackage{array}
    \usepackage{setspace}% if you must (for double spacing thesis)
    \usepackage{fancyhdr}
    \usepackage{enumitem}
    \usepackage{tikz}
    \usepackage{parskip}
    \usepackage{lipsum}
    \usepackage{floatrow} 
    \begin{document}  
\newcommand{\iu}{{i\mkern1mu}}
\begin{align*}
\setlength\extrarowheight{3pt}
\noindent\begin{tabular}{c | c c c c c }
    & $0$ & $1$ & $2$ & $3$ & $4$\\
    \cline{1-6}
    $\chi_0$ & $1$ & $1$  &  $1$   & $1$   & $1$\\
    $\chi_1$ & $1$ & $a$  &  $a^2$ & $a^3$ & $a^4$\\
    $\chi_2$ & $1$ & $a^2$ & $a^4$ & $a$   & $a^3$\\
    $\chi_3$ & $1$ & $a^3$ & $a$   & $a^4$ & $a^2$\\
    $\chi_4$ & $1$ & $a^4$ & $a^3$ & $a^2$ & $a$\\
\end{tabular}
\end{align*}

with $a = \exp\{\frac{2\pi\iu}{5}\}$ hence $a^5=1$ with $|G|=5$.

Applying the definition of Fourier transform from definition 3.1.2 we have:
     \doublespacing{
        $\hat{f}(\chi_0)=f(0)+f(1)+f(2)+f(3)+f(4)$\\
        $\hat{f}(\chi_1)=f(0)+af(1)+a^2f(2)+a^3f(3)+a^4f(4)$\\
        $\hat{f}(\chi_2)=f(0)+a^2f(1)+a^4f(2)+af(3)+a^3f(4)$\\
        $\hat{f}(\chi_3)=f(0)+a^3f(1)+af(2)+a^4f(3)+a^2f(4)$\\
        $\hat{f}(\chi_4)=f(0)+a^4f(1)+a^3f(2)+a^2f(3)+af(4)$\\
        }
        Using definition 3.1.3. we can compute the inverse Fourier transform $f(t)$:
        \begin{align*}
        {f}(0)
          &=\frac{1}{5}[ \hat{f}(\chi_0)+\hat{f}(\chi_1)+\hat{f}(\chi_2)+\hat{f}(\chi_3)+\hat{f}(\chi_4)]\\
          &\begin{aligned}[t]
           {}={}&\frac{1}{5}[f(0)+f(1)+f(2)+f(3)+f(4)]\\
           {}+{}&\frac{1}{5}[f(0)+af(1)+a^2f(2)+a^3f(3)+a^4f(4)]\\
           {}+{}&\frac{1}{5}[f(0)+a^2f(1)+a^4f(2)+af(3)+a^3f(4)]\\
           {}+{}&\frac{1}{5}[f(0)+a^3f(1)+af(2)+a^4f(3)+a^2f(4)]\\
           {}+{}&\frac{1}{5}[f(0)+a^4f(1)+a^3f(2)+a^2f(3)+af(4)]
           \end{aligned}
        \\
          &\begin{aligned}[t]
            {}={}&f(0)\\
            {}+{}&\frac{f(1)}{5}[1+a+a^2+a^3+a^4]\\
            {}+{}&\frac{f(2)}{5}[1+a+a^2+a^3+a^4]\\
            {}+{}&\frac{f(3)}{5}[1+a+a^2+a^3+a^4]\\
            {}+{}&\frac{f(4)}{5}[1+a+a^2+a^3+a^4]\\
            {}={}&f(0)
            \end{aligned}
        \end{align*}
        Similarly
        \begin{align*}
        {f}(1)
          &= \frac{1}{5}[\hat{f}(\chi_0)+\frac{1}{a}\hat{f}(\chi_1)+\frac{1}{a^2}\hat{f}(\chi_2)+\frac{1}{a^3}\hat{f}(\chi_3)+\frac{1}{a^4}\hat{f}(\chi_4)]\\
          &\begin{aligned}[t]
           {}={}&f(1)
           \end{aligned}
        \end{align*}
        \begin{align*}
        {f}(2)
          &= \frac{1}{5}[\hat{f}(\chi_0)+a^2\hat{f}(\chi_1)+a^4\hat{f}(\chi_2)+a\hat{f}(\chi_3)+a^3\hat{f}(\chi_4)]\\
          &\begin{aligned}[t]
           {}={}&f(2)
           \end{aligned}
        \end{align*}
        \begin{align*}
        {f}(3)
          &= \frac{1}{5}[\hat{f}(\chi_0)+a^3\hat{f}(\chi_1)+a\hat{f}(\chi_2)+a^4\hat{f}(\chi_3)+a^2\hat{f}(\chi_4)]\\
          &\begin{aligned}[t]
           {}={}&f(3)
           \end{aligned}
        \end{align*}
        \begin{align*}
        {f}(4)
          &= \frac{1}{5}[\hat{f}(\chi_0)+a^4\hat{f}(\chi_1)+a^3\hat{f}(\chi_2)+a^2\hat{f}(\chi_3)+a\hat{f}(\chi_4)]\\
          &\begin{aligned}[t]
           {}={}&f(4)
           \end{aligned}
        \end{align*}
    \end{document} 

How can I reduce the spacing in this? Where it says "using definition....." I would like to move this up to $\hat{f}(\chi_4)$. Also I want to reduce the spacing of $f(1) = ... = f(1)$ and $f(2) = ... = f(2)$ etc.

Edit: I've attached all 3 pages so you guys get the full picture of what the issue is.

  • Unrelated to the issue, but switching from tabular to array you can remove all the repeated $ signs. – leandriis May 1 at 16:37
4

With this simpler code, it can all fit on a single page. I loaded nccmath for its medium-sized fractions, which look better for coefficients, in my opinion:

\documentclass[11pt, a4paper]{report}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{bm}
\usepackage{nccmath}
\usepackage{amsfonts, graphicx, verbatim, mathtools,amssymb, amsthm, mathrsfs}
\usepackage{color}
\usepackage{array}
\usepackage{setspace}% if you must (for double spacing thesis)
\usepackage{fancyhdr}
\usepackage{enumitem}
\usepackage{tikz}
\usepackage{parskip}
\usepackage{lipsum}
\usepackage{floatrow}

\begin{document}

\newcommand{\iu}{{i\mkern1mu}}
\[
\setlength\extrarowheight{3pt}
    \begin{array}{c | c c c c c }
    & 0 & 1 & 2 & 3 & 4\\
    \cline{1-6}
    \chi_0 & 1 & 1 & 1 & 1 & 1\\
    \chi_1 & 1 & a & a^2 & a^3 & a^4\\
    \chi_2 & 1 & a^2 & a^4 & a & a^3\\
    \chi_3 & 1 & a^3 & a & a^4 & a^2\\
    \chi_4 & 1 & a^4 & a^3 & a^2 & a\\
    \end{array}
\]

with $a = \exp\bigl\{\frac{2\pi \iu}{5}\bigr\}$, hence $a^5=1$ with $|G|=5$.

Applying the definition of Fourier transform from definition 3.1.2 we have:
    \begin{fleqn}
    \begin{align*}
        \hat{f}(\chi_0) & =f(0)+f(1)+f(2)+f(3)+f(4) \\
        \hat{f}(\chi_1) & =f(0)+af(1)+a^2f(2)+a^3f(3)+a^4f(4) \\
        \hat{f}(\chi_2) & =f(0)+a^2f(1)+a^4f(2)+af(3)+a^3f(4) \\
        \hat{f}(\chi_3) & =f(0)+a^3f(1)+af(2)+a^4f(3)+a^2f(4) \\
        \hat{f}(\chi_4) & =f(0)+a^4f(1)+a^3f(2)+a^2f(3)+af(4)
    \end{align*}
    \end{fleqn}
        Using definition 3.1.3. we can compute the inverse Fourier transform $f(t)$:
\allowdisplaybreaks
        \begin{align*}
        {f}(0)
          &=\mfrac{1}{5}\bigl[ \hat{f}(\chi_0)+\hat{f}(\chi_1)+\hat{f}(\chi_2)+\hat{f}(\chi_3)+\hat{f}(\chi_4)\bigr]\\
          & = \begin{aligned}[t]
           &\mfrac{1}{5}\bigl[f(0)+f(1)+f(2)+f(3)+f(4)]\\
            & + \mfrac{1}{5}\bigl[f(0)+af(1)+a^2f(2)+a^3f(3)+a^4f(4)\bigr]\\
            & + \mfrac{1}{5}\bigl[f(0)+a^2f(1)+a^4f(2)+af(3)+a^3f(4)\bigr]\\
            & + \mfrac{1}{5}\bigl[f(0)+a^3f(1)+af(2)+a^4f(3)+a^2f(4)\bigr]\\
            & + \mfrac{1}{5}\bigl[f(0)+a^4f(1)+a^3f(2)+a^2f(3)+af(4)\bigr]
           \end{aligned}\\
      & =f(0) \begin{aligned}[t] 
            & + \mfrac{f(1)}{5}[1+a+a^2+a^3+a^4]\\
            & + \mfrac{f(2)}{5}[1+a+a^2+a^3+a^4]\\
            & + \mfrac{f(3)}{5}[1+a+a^2+a^3+a^4]\\
            & + \mfrac{f(4)}{5}[1+a+a^2+a^3+a^4]
            \end{aligned}\\
        & = f(0)
        \shortintertext{Similarly:}
        {f}(1)
          &= \mfrac{1}{5}\Bigl[\hat{f}(\chi_0)+\mfrac{1}{a}\hat{f}(\chi_1)+\mfrac{1}{a^2}\hat{f}(\chi_2)+\mfrac{1}{a^3}\hat{f}(\chi_3)+\mfrac{1}{a^4}\hat{f}(\chi_4)\Bigr]\\
             & = f(1) \\[1.5ex]
         f(2)
         &= \mfrac{1}{5}\bigl[\hat{f}(\chi_0)+a^2\hat{f}(\chi_1)+a^4\hat{f}(\chi_2)+a\hat{f}(\chi_3)+a^3\hat{f}(\chi_4)\bigr] \\
          & = f(2) \\[1.5ex]
        f(3)
         &= \mfrac{1}{5}\bigl[\hat{f}(\chi_0)+a^3\hat{f}(\chi_1)+a\hat{f}(\chi_2)+a^4\hat{f}(\chi_3)+a^2\hat{f}(\chi_4)\bigr] \\
         & = f(3) \\[1.5ex]
        f(4)
          &= \mfrac{1}{5}\bigl[\hat{f}(\chi_0)+a^4\hat{f}(\chi_1)+a^3\hat{f}(\chi_2)+a^2\hat{f}(\chi_3)+a\hat{f}(\chi_4)\bigr] \\
          & = f(4)
        \end{align*}

    \end{document} 

enter image description here

  • this is nice however you missed out the remaining section for $f(0)$ aha! – Maths May 2 at 11:03
  • Oh! yes. I'll fix it in a moment – Bernard May 2 at 11:08
  • I had to slightly modify the code to make it fit on a single page (replaced \intertext with shortintertext, and loading nccmath before mathtools to make it work). – Bernard May 2 at 11:24
  • there's still some code of f(0) missing aha. its where I group f(1) ... f(4) as f(1)[1+a+...+a^4] etc. Also, I don't mind if it runs over two pages. I don't want the text to be squashed, all I wanted is to make use of the empty white space :) – Maths May 2 at 11:26
  • Refer to my code in the question, you'll see the part you missed :) – Maths May 2 at 11:35
3

You should avoid \\ on the last line of alignments. Perhaps the following is closer to what you want:

Sample output

\documentclass[11pt, a4paper]{report}

\usepackage{amsmath,array}

\begin{document}

\newcommand{\iu}{{i\mkern1mu}}
\begin{equation*}
  \setlength\extrarowheight{3pt}
  \begin{tabular}{c | c c c c c }
    & $0$ & $1$ & $2$ & $3$ & $4$\\
    \cline{1-6}
    $\chi_0$ & $1$ & $1$  &  $1$   & $1$   & $1$\\
    $\chi_1$ & $1$ & $a$  &  $a^2$ & $a^3$ & $a^4$\\
    $\chi_2$ & $1$ & $a^2$ & $a^4$ & $a$   & $a^3$\\
    $\chi_3$ & $1$ & $a^3$ & $a$   & $a^4$ & $a^2$\\
    $\chi_4$ & $1$ & $a^4$ & $a^3$ & $a^2$ & $a$\\
  \end{tabular}
\end{equation*}
with $a = \exp\{\frac{2\pi\iu}{5}\}$ hence $a^5=1$ with $|G|=5$.

Applying the definition of Fourier transform from Definition~3.1.2 we
have:
\begin{align*}
  \hat{f}(\chi_0) &=f(0)+f(1)+f(2)+f(3)+f(4),\\
  \hat{f}(\chi_1) &=f(0)+af(1)+a^2f(2)+a^3f(3)+a^4f(4),\\
  \hat{f}(\chi_2) &=f(0)+a^2f(1)+a^4f(2)+af(3)+a^3f(4),\\
  \hat{f}(\chi_3) &=f(0)+a^3f(1)+af(2)+a^4f(3)+a^2f(4),\\
  \hat{f}(\chi_4) &=f(0)+a^4f(1)+a^3f(2)+a^2f(3)+af(4).
\end{align*}
Using Definition~3.1.3 we can compute the inverse Fourier transform
$f(t)$:
\begin{align*}
  f(0)
  &=\frac{1}{5}[ \hat{f}(\chi_0) + \hat{f}(\chi_1) + \hat{f}(\chi_2) +
    \hat{f}(\chi_3) + \hat{f}(\chi_4)]\\
  &=\frac{1}{5}[f(0)+f(1)+f(2)+f(3)+f(4)]\\
  &\qquad + \frac{1}{5}[f(0)+af(1)+a^2f(2)+a^3f(3)+a^4f(4)]\\
  &\qquad + \frac{1}{5}[f(0)+a^2f(1)+a^4f(2)+af(3)+a^3f(4)]\\
  &\qquad + \frac{1}{5}[f(0)+a^3f(1)+af(2)+a^4f(3)+a^2f(4)]\\
  &\qquad + \frac{1}{5}[f(0)+a^4f(1)+a^3f(2)+a^2f(3)+af(4)]
  \\
  &= f(0)\\
  &\qquad + \frac{f(1)}{5}[1+a+a^2+a^3+a^4]\\
  &\qquad +\frac{f(2)}{5}[1+a+a^2+a^3+a^4]\\
  &\qquad +\frac{f(3)}{5}[1+a+a^2+a^3+a^4]\\
  &\qquad +\frac{f(4)}{5}[1+a+a^2+a^3+a^4]\\
  &=f(0).
\end{align*}
Similarly
\begin{align*}
  f(1)
  &= \frac{1}{5}\Bigl[\hat{f}(\chi_0) + \frac{1}{a}\hat{f}(\chi_1) +
    \frac{1}{a^2}\hat{f}(\chi_2) + \frac{1}{a^3}\hat{f}(\chi_3) +
    \frac{1}{a^4}\hat{f}(\chi_4)\Bigr]\\
  &=f(1),\\
  f(2)
  &= \frac{1}{5}[\hat{f}(\chi_0) + a^2\hat{f}(\chi_1) +
    a^4\hat{f}(\chi_2) + a\hat{f}(\chi_3) + a^3\hat{f}(\chi_4)]\\
  &=f(2), \\
  f(3)
  &= \frac{1}{5}[\hat{f}(\chi_0) + a^3\hat{f}(\chi_1) +
    a\hat{f}(\chi_2) + a^4\hat{f}(\chi_3) + a^2\hat{f}(\chi_4)]\\
  &=f(3),\\
  f(4)
  &= \frac{1}{5}[\hat{f}(\chi_0) + a^4\hat{f}(\chi_1) +
    a^3\hat{f}(\chi_2) + a^2\hat{f}(\chi_3) + a\hat{f}(\chi_4)]\\
  & =f(4).
\end{align*}
\end{document}
  • why did you push f(0) outwards? it wasn't necessary. but thanks for your solution – Maths May 1 at 15:42
  • 1
    The +'s should not be under the =, but to the right of it as they belong to that side of the equation. Whether you want to indent by \qquad as I did, or the smaller \quad is a matter of taste. – Andrew Swann May 1 at 18:28

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