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Revision as of 03:28, 5 June 2013

2,686 bytes added ,  03:28, 5 June 2013
→‎Fourier Transforms of Boolean Functions
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For ease of reading formulas, let <math>x = (x_1, x_2) = (u, v).\!</math>
 
For ease of reading formulas, let <math>x = (x_1, x_2) = (u, v).\!</math>
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 +
====Table 2.1. Values of &chi;<sub>S</sub>(x)====
    
<pre>
 
<pre>
−
<p align="center">
   
$latex
 
$latex
 
\begin{tabular}{|c||*{4}{c}|}
 
\begin{tabular}{|c||*{4}{c}|}
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\multicolumn{5}{c}{Table 2.1. Values of \( \chi_S(x) \) for \( f : \mathbb{B}^2 \to \mathbb{B} \)} \\[4pt]
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\multicolumn{5}{c}{Table 2.1. Values of \( \boldsymbol{\chi}_\mathcal{S}(x) \) for \( f : \mathbb{B}^2 \to \mathbb{B} \)} \\[4pt]
 
\hline
 
\hline
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\( S \backslash (u, v) \) &amp;
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\( \mathcal{S} \backslash (u, v) \) &amp;
 
\( (1, 1) \) &amp;
 
\( (1, 1) \) &amp;
 
\( (1, 0) \) &amp;
 
\( (1, 0) \) &amp;
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\end{tabular}
 
\end{tabular}
 
&amp;fg=000000$
 
&amp;fg=000000$
−
</p>
   
</pre>
 
</pre>
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====Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables====
    
<pre>
 
<pre>
−
<p align="center">
   
$latex
 
$latex
 
\begin{tabular}{|*{5}{c|}*{4}{r|}}
 
\begin{tabular}{|*{5}{c|}*{4}{r|}}
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\hline
 
\hline
 
~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~\\
 
~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~\\
−
\( L_1 \)&amp;
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\(L_1\)&amp;\(L_2\)&amp;&amp;\(L_3\)&amp;\(L_4\)&amp;
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\( L_2 \)&amp;&amp;
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\(\hat{f}(\varnothing)\)&amp;\(\hat{f}(\{u\})\)&amp;\(\hat{f}(\{v\})\)&amp;\(\hat{f}(\{u,v\})\) \\
−
\( L_3 \)&amp;
  −
\( L_4 \)&amp;
  −
\( \hat{f}(\varnothing) \)&amp;
  −
\( \hat{f}(\{u\})     \)&amp;
  −
\( \hat{f}(\{v\})     \)&amp;
  −
\( \hat{f}(\{u,v\}) \)
  −
\\
   
~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~\\
 
~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~\\
 
\hline
 
\hline
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\\
 
\\
 
\hline
 
\hline
−
\end{tabular}&amp;fg=000000$
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\end{tabular}
−
</p>
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&amp;fg=000000$
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</pre>
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 +
====Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables====
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 +
<pre>
 +
$latex
 +
\begin{tabular}{|*{5}{c|}*{4}{r|}}
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\multicolumn{9}{c}{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt]
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\hline
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~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~\\
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\(L_1\)&amp;\(L_2\)&amp;&amp;\(L_3\)&amp;\(L_4\)&amp;
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\(\hat{f}(\varnothing)\)&amp;\(\hat{f}(\{u\})\)&amp;\(\hat{f}(\{v\})\)&amp;\(\hat{f}(\{u,v\})\) \\
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~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~&amp;~\\
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\hline
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&amp;&amp; \(u =\)&amp; 1 1 0 0&amp;&amp;&amp;&amp;&amp; \\
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&amp;&amp; \(v =\)&amp; 1 0 1 0&amp;&amp;&amp;&amp;&amp; \\
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\hline
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\(f_{0}\)&amp;
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\(f_{0000}\)&amp;&amp;
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0 0 0 0&amp;
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\((~)\)&amp;
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\(0\)&amp;
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\(0\)&amp;
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\(0\)&amp;
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\(0\)
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\\
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\hline
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\(f_{1}\)&amp;
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\(f_{0001}\)&amp;&amp;
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0 0 0 1&amp;
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\((u)(v)\)&amp;
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\(1/4\)&amp;
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\(1/4\)&amp;
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\(1/4\)&amp;
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\(1/4\)
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\\
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\(f_{2}\)&amp;
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\(f_{0010}\)&amp;&amp;
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0 0 1 0&amp;
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\((u)~v~\)&amp;
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\( 1/4\)&amp;
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\( 1/4\)&amp;
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\(-1/4\)&amp;
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\(-1/4\)
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\\
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\(f_{4}\)&amp;
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\(f_{0100}\)&amp;&amp;
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0 1 0 0&amp;
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\(~u~(v)\)&amp;
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\( 1/4\)&amp;
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\(-1/4\)&amp;
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\( 1/4\)&amp;
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\(-1/4\)
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\\
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\(f_{8}\)&amp;
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\(f_{1000}\)&amp;&amp;
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1 0 0 0&amp;
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\(~u~~v~\)&amp;
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\( 1/4\)&amp;
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\(-1/4\)&amp;
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\(-1/4\)&amp;
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\( 1/4\)
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\\
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\hline
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\(f_{3}\)&amp;
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\(f_{0011}\)&amp;&amp;
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0 0 1 1&amp;
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\((u)\)&amp;
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\(1/2\)&amp;
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\(1/2\)&amp;
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\( 0 \)&amp;
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\( 0 \)
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\\
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\(f_{12}\)&amp;
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\(f_{1100}\)&amp;&amp;
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1 1 0 0&amp;
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\(u\)&amp;
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\( 1/2\)&amp;
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\(-1/2\)&amp;
 +
\( 0 \)&amp;
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\( 0 \)
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\\
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\hline
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\(f_{6}\)&amp;
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\(f_{0110}\)&amp;&amp;
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0 1 1 0&amp;
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\((u,~v)\)&amp;
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\( 1/2\)&amp;
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\( 0 \)&amp;
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\( 0 \)&amp;
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\(-1/2\)
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\\
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\(f_{9}\)&amp;
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\(f_{1001}\)&amp;&amp;
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1 0 0 1&amp;
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\(((u,~v))\)&amp;
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\(1/2\)&amp;
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\( 0 \)&amp;
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\( 0 \)&amp;
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\(1/2\)
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\\
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\hline
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\(f_{5}\)&amp;
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\(f_{0101}\)&amp;&amp;
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0 1 0 1&amp;
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\((v)\)&amp;
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\(1/2\)&amp;
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\( 0 \)&amp;
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\(1/2\)&amp;
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\( 0 \)
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\\
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\(f_{10}\)&amp;
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\(f_{1010}\)&amp;&amp;
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1 0 1 0&amp;
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\(v\)&amp;
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\( 1/2\)&amp;
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\( 0 \)&amp;
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\(-1/2\)&amp;
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\( 0 \)
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\\
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\hline
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\(f_{7}\)&amp;
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\(f_{0111}\)&amp;&amp;
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0 1 1 1&amp;
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\((u~~v)\)&amp;
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\( 3/4\)&amp;
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\( 1/4\)&amp;
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\( 1/4\)&amp;
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\(-1/4\)
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\\
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\hline
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\(f_{11}\)&amp;
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\(f_{1011}\)&amp;&amp;
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1 0 1 1&amp;
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\((~u~(v))\)&amp;
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\( 3/4\)&amp;
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\( 1/4\)&amp;
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\(-1/4\)&amp;
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\( 1/4\)
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\\
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\(f_{13}\)&amp;
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\(f_{1101}\)&amp;&amp;
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1 1 0 1&amp;
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\(((u)~v~)\)&amp;
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\( 3/4\)&amp;
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\(-1/4\)&amp;
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\( 1/4\)&amp;
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\( 1/4\)
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\\
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\(f_{14}\)&amp;
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\(f_{1110}\)&amp;&amp;
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1 1 1 0&amp;
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\(((u)(v))\)&amp;
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\( 3/4\)&amp;
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\(-1/4\)&amp;
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\(-1/4\)&amp;
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\(-1/4\)
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\\
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\hline
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\(f_{15}\)&amp;
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\(f_{1111}\)&amp;&amp;
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1 1 1 1&amp;
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\(((~))\)&amp;
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\(1\)&amp;
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\(0\)&amp;
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\(0\)&amp;
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\(0\)
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\\
 +
\hline
 +
\end{tabular}
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&amp;fg=000000$
 
</pre>
 
</pre>
  
Jon Awbrey
12,080

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