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===Fourier Transforms of Boolean Functions===
 
===Fourier Transforms of Boolean Functions===
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<pre>
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Re: <a href="http://rjlipton.wordpress.com/2013/05/21/twin-primes-are-useful/" target="_blank">Another Problem</a>
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<div style="margin-left:30px;">
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<p>The problem is concretely about Boolean functions $latex {f}$ of $latex {k}$ variables, and seems not to involve prime numbers at all.  For any subset $latex {S}$ of the coordinates, the corresponding Fourier coefficient is given by:</p>
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<p align="center">
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$latex \displaystyle \hat{f}(S) = \frac{1}{2^k} \sum_{x \in \mathbb{Z}_2^k} f(x)\chi_S(x)$</p>
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<p>where $latex {\chi_S(x)}$ is $latex {-1}$ if $latex {\sum_{i \in S} x_i}$ is odd, and $latex {+1}$ otherwise.</p>
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</div>
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<b>Note to Self.</b>  I need to play around with this concept a while.
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Begin with a survey of concrete examples, perhaps in tabular form.
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$latex {k = 1}$
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&hellip;
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$latex {k = 2}$
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For ease of reading formulas, let $latex {x = (x_1, x_2) = (u, v)}.$
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<p align="center">
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$latex
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\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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\hline
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\( S \backslash (u, v)  \) &amp;
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\( (1, 1) \) &amp;
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\( (1, 0) \) &amp;
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\( (0, 1) \) &amp;
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\( (0, 0) \)
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\\
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\hline\hline
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\( \varnothing \) &amp; \( +1 \) &amp; \( +1 \) &amp; \( +1 \) &amp; \( +1 \) \\
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\( \{ u \} \)    &amp; \( -1 \) &amp; \( -1 \) &amp; \( +1 \) &amp; \( +1 \) \\
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\( \{ v \} \)    &amp; \( -1 \) &amp; \( +1 \) &amp; \( -1 \) &amp; \( +1 \) \\
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\( \{ u, v \} \)  &amp; \( +1 \) &amp; \( -1 \) &amp; \( -1 \) &amp; \( +1 \) \\
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\hline
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\end{tabular}
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&amp;fg=000000$
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</p>
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<p align="center">
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$latex
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\begin{tabular}{|*{5}{c|}*{4}{r|}}
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\multicolumn{9}{c}{Table 2.2. 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;
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\( L_2 \)&amp;&amp;
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\( L_3 \)&amp;
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\( L_4 \)&amp;
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\( \hat{f}(\varnothing) \)&amp;
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\( \hat{f}(\{u\})    \)&amp;
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\( \hat{f}(\{v\})    \)&amp;
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\( \hat{f}(\{u,v\}) \)
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\\
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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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\(f_{1}\)&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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\(f_{3}\)&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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\(f_{4}\)&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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\(f_{5}\)&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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\(f_{6}\)&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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\(f_{7}\)&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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\(f_{8}\)&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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\(f_{9}\)&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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\(f_{10}\)&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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\(f_{11}\)&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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\(f_{12}\)&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;
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\( 0 \)&amp;
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\( 0 \)
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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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\(f_{15}\)&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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\\
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\hline
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\end{tabular}&amp;fg=000000$
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</p>
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<i>To be continued &hellip;</i>
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<h3>Notes</h3>
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<ul><li><a href="http://inquiryintoinquiry.com/2013/05/31/special-classes-of-propositions/" target="_blank">Special Classes of Propositions</a></li></ul>
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<h3>References</h3>
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<table border="0" style="border-width:0;">
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<tr>
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<td style="border-top:1px solid white;">21 May 2013</td>
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<td style="border-top:1px solid white;"><a href="http://rjlipton.wordpress.com/2013/05/21/twin-primes-are-useful/" target="_blank">Twin Primes Are Useful</a></td></tr>
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<tr>
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<td style="border-top:1px solid white;">08 Nov 2012</td>
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<td style="border-top:1px solid white;"><a href="http://rjlipton.wordpress.com/2012/11/08/the-power-of-guessing/" target="_blank">The Power Of Guessing</a></td></tr>
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<tr>
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<td style="border-top:1px solid white;">05 Jan 2011</td>
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<td style="border-top:1px solid white;"><a href="http://rjlipton.wordpress.com/2011/01/05/fourier-complexity-of-cirtemmys-boolean-functions/" target="_blank">Fourier Complexity Of Symmetric Boolean Functions</a></td></tr>
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<tr>
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<td style="border-top:1px solid white;">19 Nov 2010</td>
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<td style="border-top:1px solid white;"><a href="http://rjlipton.wordpress.com/2010/11/19/is-complexity-theory-on-the-brink/" target="_blank">Is Complexity Theory On The Brink?</a></td></tr>
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<tr>
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<td style="border-top:1px solid white;">18 Sep 2009</td>
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<td style="border-top:1px solid white;"><a href="http://rjlipton.wordpress.com/2009/09/18/why-believe-that-pnp-is-impossible/" target="_blank">Why Believe That P=NP Is Impossible?</a></td></tr>
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<tr>
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<td style="border-top:1px solid white;">04 Jun 2009</td>
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<td style="border-top:1px solid white;"><a href="http://rjlipton.wordpress.com/2009/06/04/the-junta-problem/" target="_blank">The Junta Problem</a></td></tr>
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</table>
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</pre>
    
==Work 2==
 
==Work 2==
Jon Awbrey
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