Tp tuning: Difference between revisions

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||Ƹ||, the norm of the full ''p''-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full ''p''-limit [[JIP]], must equal the T''p'' tuning for S*. Thus to find the T''p'' tuning of S for the group G, we may first find the T''p'' tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.

Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.

Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique. It is also easy to see that this can only be non-unique if p=0 or p=Infinity, so that we may get a unique Lp tuning (called the "TIPTOP" tuning for p=Infinity) by simply taking the limit as p approaches our value.

== T2 tuning ==

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 ||Ƹ||, the norm of the full ''p''-limit error map, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full ''p''-limit [[JIP]], must equal the T''p'' tuning for S*. Thus to find the T''p'' tuning of S for the group G, we may first find the T''p'' tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.
-Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.
+Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique. It is also easy to see that this can only be non-unique if p=0 or p=Infinity, so that we may get a unique Lp tuning (called the "TIPTOP" tuning for p=Infinity) by simply taking the limit as p approaches our value.
 == T2 tuning ==