Sequence of integrals
Letting $finC^0[0,1]$, compute the limit of the sequence:
$a_n=int_0^1int_0^1...int_0^1 prod_(i=1)^n (2x_i)*f((sum_(i=1)^n x_i)/n) dx_1dx_2...dx_n $
$a_n=int_0^1int_0^1...int_0^1 prod_(i=1)^n (2x_i)*f((sum_(i=1)^n x_i)/n) dx_1dx_2...dx_n $
Risposte
ok ok.... 
....
well my approach at the moment is simply expanding every sum and evaluate the single terms
...
I think I can generalize for every f polynomial... but it needs some work... hope to have time this week-end! pherpaps it won't bring to the solution, but it can be a nice exercise!!
.... well my approach at the moment is simply expanding every sum and evaluate the single terms
... I think I can generalize for every f polynomial... but it needs some work... hope to have time this week-end! pherpaps it won't bring to the solution, but it can be a nice exercise!!
"Thomas":
luca, I hope u are still alive...
to answer your question:
probably it would work better if you try to generalize with $f$ polynomial; your approach is interesting
luca, I hope u are still alive... too much chocolate???
mmm... u mean trying to get a direct calculation? I can try... I can make it right now for a second degree polynomial, just to see what I get:
$f(x)=ax^2+bx+c$
first there is the "linear part", which brings $2int_0^1(bx+c)dx=2/3b+c$... then the second degree monomial brings
$a/(2n)+2/9(n*(n-1))/n^2$
putting all together the limit seems to be
$2/3b+c+2/9a$
but this is not the result I wanted, there is an exceeding term that I don't like (this term corrisponds to $a(2int_0^1x^2dx)^2/(2!)$(I observe this with the aim of a future generalization )... I can check my calculation tomorrow anyway
pheraps I can try to generalize for every polinomyal f (it doesn't seem necessary to calculate "everything" every time), but first I'd like to know if it's worth...
what do u think, luca?
$f(x)=ax^2+bx+c$
first there is the "linear part", which brings $2int_0^1(bx+c)dx=2/3b+c$... then the second degree monomial brings
$a/(2n)+2/9(n*(n-1))/n^2$
putting all together the limit seems to be
$2/3b+c+2/9a$
but this is not the result I wanted, there is an exceeding term that I don't like (this term corrisponds to $a(2int_0^1x^2dx)^2/(2!)$(I observe this with the aim of a future generalization
)... I can check my calculation tomorrow anyway pheraps I can try to generalize for every polinomyal f (it doesn't seem necessary to calculate "everything" every time), but first I'd like to know if it's worth...
what do u think, luca?
It could be the right way... are you able to proceed with polynomial $f(*)$?
I was trying to solve this but I got stuck... (this expression is surely wrong but i like it )...
I could only find that when f is linear (unluckily I think it means only of the form $ax+b$) the sequence is identically equal to $2int_0^1f(y)ydy$... this made me believe that the solution is the same when f is non-linear... but at the moment I am not able to prove it... I need to prove something like that: taken a sequence ${x_i}$ where $0
I also tried with changes of variables, but the extrema of the integrals are quite hard to deal with....
)...I could only find that when f is linear (unluckily I think it means only of the form $ax+b$) the sequence is identically equal to $2int_0^1f(y)ydy$... this made me believe that the solution is the same when f is non-linear... but at the moment I am not able to prove it... I need to prove something like that: taken a sequence ${x_i}$ where $0
I also tried with changes of variables, but the extrema of the integrals are quite hard to deal with....
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