- #1
mmzaj
- 107
- 0
Hi
i'm looking for some class of functions [tex]\phi(t) [/tex] that satisfy :
[tex]\int_T \ t^n \phi(t) \, dt = \left( \int_T \ t \phi(t) \, dx \right)^n ; n=0,1,2,3 ...[/tex]
from what i understand - if I'm not mistaken - the problem transforms to finding a set of measure spaces whose measure [tex]\ ds =\phi(t)dt[/tex] , and nth norm of [tex] t [/tex]
[tex]\left\|t\right\|_n = \left( \int \ t^n\ ds \right)^\frac{1}{n} [/tex]
satisfy :
1 - [tex]\int ds =1[/tex]
2 - [tex]\left\|t\right\|_n=\left\|t\right\|_1 ; n=2,3,4 ... [/tex]
obviously this problem is appropriately studied in [tex] L^p [/tex] spaces .
if I'm not mistaken , can you help me please , and if not , would you please advise .
i'm looking for some class of functions [tex]\phi(t) [/tex] that satisfy :
[tex]\int_T \ t^n \phi(t) \, dt = \left( \int_T \ t \phi(t) \, dx \right)^n ; n=0,1,2,3 ...[/tex]
from what i understand - if I'm not mistaken - the problem transforms to finding a set of measure spaces whose measure [tex]\ ds =\phi(t)dt[/tex] , and nth norm of [tex] t [/tex]
[tex]\left\|t\right\|_n = \left( \int \ t^n\ ds \right)^\frac{1}{n} [/tex]
satisfy :
1 - [tex]\int ds =1[/tex]
2 - [tex]\left\|t\right\|_n=\left\|t\right\|_1 ; n=2,3,4 ... [/tex]
obviously this problem is appropriately studied in [tex] L^p [/tex] spaces .
if I'm not mistaken , can you help me please , and if not , would you please advise .
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