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Generalized Tenney dual norms and Tp tuning space

Revision as of 12:10, 6 August 2012 by Wikispaces>mbattaglia1 (**Imported revision 356537294 - Original comment: **)

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=Dual Norms= 
Given any [[Generalized Tenney Norms and Tp Interval Space|Tp norm]] on an interval space **Tp<span style="font-size: 10px; vertical-align: super;">G</span>** associated with a group **G**, we can define a corresponding **dual Tq* norm** on the dual space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** which satisfies the following identity:

[[math]]
||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}
[[math]]

for all f in **Tq<span style="font-size: 10px; vertical-align: super;">G</span>***. This normed space, for which the group of vals on **G** comprise the lattice of covectors with integer coefficients, is called **Tq* Tuning Space.**

==Prime Power Interval Groups== 
In the simplest case where **G** has as its chosen basis only primes and prime powers, || · ||**<span style="font-size: 10px; vertical-align: sub;">Tp</span>** is given by

[[math]]
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}
[[math]]

for diagonal weighting matrix **W<span style="font-size: 80%; vertical-align: sub;">G</span>**. Then the dual norm || · ||**<span style="font-size: 10px; vertical-align: sub;">Tq*</span>** on **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** is given for f in **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** by

[[math]]
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}
[[math]]

where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.

The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log<span style="font-size: 80%; vertical-align: sub;">2</span>(**G**<span style="font-size: 80%; vertical-align: sub;">n</span>), where **G**<span style="font-size: 10px; vertical-align: sub;">n</span> is the nth basis element in **G**. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.

For **G** with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.

==Arbitrary Interval Groups== 
For an arbitrary group **G** with its chosen basis containing intervals other than primes and prime powers, || · ||**<span style="font-size: 10px; vertical-align: sub;">Tp</span>** is given by

[[math]]
\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}
[[math]]

for a [[Subgroup Mapping Matrices (V-maps)|V-map]] **V<span style="font-size: 80%; vertical-align: sub;">G</span>** representing **G** in some full-limit **L** and a diagonal weighting matrix **W<span style="font-size: 10px; vertical-align: sub;">L</span>** for **L**. Then if **Tp<span style="font-size: 10px; vertical-align: super;">L</span>** represents the full-limit interval space that **G** is embedded in, and **Tq<span style="font-size: 10px; vertical-align: super;">L</span>*** is the dual space, the dual norm || · ||**<span style="font-size: 10px; vertical-align: sub;">Tq </span>**on **Tq<span style="font-size: 80%; vertical-align: super;">G</span>*** is given by

[[math]]
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}
[[math]]

Note that this is the quotient norm induced on the space **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**), where ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the [[@http://www.math.unl.edu/~s-bbockel1/928/node25.html|Hahn-Banach theorem]], which demonstrates that the space **Tq<span style="font-size: 10px; vertical-align: super;">G</span>*** must be isometrically isomorphic to **Tq<span style="font-size: 10px; vertical-align: super;">L</span>***/ker(**V<span style="font-size: 10px; vertical-align: sub;">G</span>**).

Original HTML content:

<html><head><title>Generalized Tenney Dual Norms and Tp Tuning Space</title></head><body><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:5 -->Dual Norms</h1>
 Given any <a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20Interval%20Space">Tp norm</a> on an interval space <strong>Tp<span style="font-size: 10px; vertical-align: super;">G</span></strong> associated with a group <strong>G</strong>, we can define a corresponding <strong>dual Tq* norm</strong> on the dual space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* which satisfies the following identity:<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}&lt;br/&gt;[[math]]
 --><script type="math/tex">||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
for all f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>*. This normed space, for which the group of vals on <strong>G</strong> comprise the lattice of covectors with integer coefficients, is called <strong>Tq* Tuning Space.</strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc1"><a name="Dual Norms-Prime Power Interval Groups"></a><!-- ws:end:WikiTextHeadingRule:7 -->Prime Power Interval Groups</h2>
 In the simplest case where <strong>G</strong> has as its chosen basis only primes and prime powers, || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tp</span></strong> is given by<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
for diagonal weighting matrix <strong>W<span style="font-size: 80%; vertical-align: sub;">G</span></strong>. Then the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq*</span></strong> on <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* is given for f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* by<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
where the coefficients p in Tp and q in Tq* satisfy the relationship 1/p + 1/q = 1.<br />
<br />
The dual of any Tp norm is very similar to the dual of the ordinary Lp norm. The crucial difference to be noted is that the weighting for covectors in tuning space is the inverse of the weighting for vectors in interval space; simple primes are weighted less in interval space but more in tuning space. Unlike the weighting matrix for interval space, the weighting matrix on tuning space is a diagonal matrix in which the nth entry in the diagonal is 1/log<span style="font-size: 80%; vertical-align: sub;">2</span>(<strong>G</strong><span style="font-size: 80%; vertical-align: sub;">n</span>), where <strong>G</strong><span style="font-size: 10px; vertical-align: sub;">n</span> is the nth basis element in <strong>G</strong>. We denote such inverse weighted norms with an asterisk, so that the inverse-Tenney weighted Linf norm in tuning space is Tinf*.<br />
<br />
For <strong>G</strong> with basis of only primes and prime powers, the dual of the T1 norm is the Tinf* norm, the dual of the Tinf norm is the T1* norm, and the dual of the T2 norm is the T2* norm.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc2"><a name="Dual Norms-Arbitrary Interval Groups"></a><!-- ws:end:WikiTextHeadingRule:9 -->Arbitrary Interval Groups</h2>
 For an arbitrary group <strong>G</strong> with its chosen basis containing intervals other than primes and prime powers, || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tp</span></strong> is given by<br />
<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]&lt;br/&gt;
\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \| \vec{v} \right \|_{\textbf{T1}}^\textbf{G} = \left \| \mathbf{W_L} \cdot \mathbf{V_\textbf{G}} \cdot \vec{v} \right \|_\textbf{1}</script><!-- ws:end:WikiTextMathRule:3 --><br />
<br />
for a <a class="wiki_link" href="/Subgroup%20Mapping%20Matrices%20%28V-maps%29">V-map</a> <strong>V<span style="font-size: 80%; vertical-align: sub;">G</span></strong> representing <strong>G</strong> in some full-limit <strong>L</strong> and a diagonal weighting matrix <strong>W<span style="font-size: 10px; vertical-align: sub;">L</span></strong> for <strong>L</strong>. Then if <strong>Tp<span style="font-size: 10px; vertical-align: super;">L</span></strong> represents the full-limit interval space that <strong>G</strong> is embedded in, and <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>* is the dual space, the dual norm || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tq </span></strong>on <strong>Tq<span style="font-size: 80%; vertical-align: super;">G</span></strong>* is given by<br />
<br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]&lt;br/&gt;
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}&lt;br/&gt;[[math]]
 --><script type="math/tex">\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \inf_{n \in \text{ker}(\mathbf{V_G})} \left \{ \left \| (f-n) \cdot \mathbf{W}_\mathbf{L}^{-1} \right \|_\textbf{Tq*}^\mathbf{L} \right \}</script><!-- ws:end:WikiTextMathRule:4 --><br />
<br />
Note that this is the quotient norm induced on the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>), where ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>) is the set of vals in the V-map that are restricted away. This result is due to a corollary of the <a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank">Hahn-Banach theorem</a>, which demonstrates that the space <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>* must be isometrically isomorphic to <strong>Tq<span style="font-size: 10px; vertical-align: super;">L</span></strong>*/ker(<strong>V<span style="font-size: 10px; vertical-align: sub;">G</span></strong>).</body></html>
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