- #1
zeebek
- 27
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I am reading the definition in wiki ( nothing better at the moment)
http://en.wikipedia.org/wiki/Lorentz_space
It seems too vague for me, namely what they call "rearrangement function" [itex]f^{*}[/itex]:
[tex]f^{*}: [0, \infty) \rightarrow [0, \infty]; \\
f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}; \\
d_f(\alpha) = \mu(\{x \in X : |f(x)| > \alpha\}).
[/tex]
I am trying to put in words what is written. Is it right:
first for a given [itex]t[/itex] we are looking for all [itex]\alpha[/itex]'s, so that [itex]d_f(\alpha) \leq t[/itex], where [itex]d_f(\alpha) [/itex] is basically a size of the area where [itex]|f(x)| > \alpha [/itex]? Then we take infinum via [itex] \alpha [/itex], so as a result there will be the smallest [itex]d[/itex]?
Still I cannot imagine "geometrically" how is it?
At last, I need just simpler difinition for the case when [itex]f[/itex] is real.
thanks!
http://en.wikipedia.org/wiki/Lorentz_space
It seems too vague for me, namely what they call "rearrangement function" [itex]f^{*}[/itex]:
[tex]f^{*}: [0, \infty) \rightarrow [0, \infty]; \\
f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}; \\
d_f(\alpha) = \mu(\{x \in X : |f(x)| > \alpha\}).
[/tex]
I am trying to put in words what is written. Is it right:
first for a given [itex]t[/itex] we are looking for all [itex]\alpha[/itex]'s, so that [itex]d_f(\alpha) \leq t[/itex], where [itex]d_f(\alpha) [/itex] is basically a size of the area where [itex]|f(x)| > \alpha [/itex]? Then we take infinum via [itex] \alpha [/itex], so as a result there will be the smallest [itex]d[/itex]?
Still I cannot imagine "geometrically" how is it?
At last, I need just simpler difinition for the case when [itex]f[/itex] is real.
thanks!
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