Generalized Tenney dual norms and Tp tuning space
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- This revision was by author mbattaglia1 and made on 2012-08-06 11:15:49 UTC.
- The original revision id was 356530718.
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Original Wikitext content:
=Dual Norms=
Given any [[Generalized Tenney Norms and Tp spaces|Tp norm]] on an interval space **Tp<span style="font-size: 10px; vertical-align: sub;">G</span>** associated with a group **G**, we can define a corresponding **dual norm** on the dual space **Tp<span style="font-size: 10px; vertical-align: sub;">G</span>*** which satisfies the following identity:
[[math]]
||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}
[[math]]
for all f in **Tp<span style="font-size: 10px; vertical-align: sub;">G</span>***.
In the simplest case where **G** has as its chosen basis only primes and prime powers, and hence || · ||**<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 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 **Tp<span style="font-size: 10px; vertical-align: sub;">G</span>*** is given 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.Original HTML content:
<html><head><title>Generalized Tenney Dual Norms and Tp Tuning Space</title></head><body><!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc0"><a name="Dual Norms"></a><!-- ws:end:WikiTextHeadingRule:3 -->Dual Norms</h1>
Given any <a class="wiki_link" href="/Generalized%20Tenney%20Norms%20and%20Tp%20spaces">Tp norm</a> on an interval space <strong>Tp<span style="font-size: 10px; vertical-align: sub;">G</span></strong> associated with a group <strong>G</strong>, we can define a corresponding <strong>dual norm</strong> on the dual space <strong>Tp<span style="font-size: 10px; vertical-align: sub;">G</span></strong>* which satisfies the following identity:<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}<br/>[[math]]
--><script type="math/tex">||f|| = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
for all f in <strong>Tp<span style="font-size: 10px; vertical-align: sub;">G</span></strong>*.<br />
<br />
In the simplest case where <strong>G</strong> has as its chosen basis only primes and prime powers, and hence || · ||<strong><span style="font-size: 10px; vertical-align: sub;">Tp</span></strong> is given by<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
\left \| \vec{v} \right \|_{\textbf{Tp}}^\textbf{G} = \left \| \mathbf{W_G} \cdot \vec{v} \right \|_\textbf{p}<br/>[[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 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>Tp<span style="font-size: 10px; vertical-align: sub;">G</span></strong>* is given by<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]<br/>
\left \|f \right \|_{\textbf{Tq*}}^\textbf{G} = \left \| f \cdot \mathbf{W}_\mathbf{G}^{-1} \right \|_\textbf{q}<br/>[[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.</body></html>