Exponential integral of translation invariant function
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Consider function $f: (mathbb{R}^{d})^{n} rightarrow mathbb{R}$ with spatial invariance property of the form :
$f(x_1,x_2,...,x_n) = f(x_1 + zeta, x_2 + zeta,..., x_n + zeta)$ for $zeta in mathbb{R}^{d} $.
I want to show that $int limits_{(mathbb{R}^{d})^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n ,=, infty $.
I would like some help.
My only idea is to fix ball $B(0,r) subseteq mathbb{R}^{d}$ and $zeta in mathbb{R}^{d}$ and then compute $int limits_{{B(0,r)}^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n , = int limits_{{B(zeta,r)}^{n}} exp(,f(y_1 - zeta,...,y_n - zeta) ,) dy_1..dy_n = $
$ int limits_{{B(zeta,r)}^{n}} exp(,f(y_1,...,y_n) ,) dy_1..dy_n , $, where the first equality comes from setting $x_i = y_i - zeta $ and the second from the spatial invariance property.
So we get that the integral of this strictly positive f over any ball is the same. But does it imply that the integral over $(mathbb{R}^{d})^{n}$ is $infty$ ?
multivariable-calculus vector-analysis
asked Nov 27 at 17:52
vl.ath
317
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Consider function $f: (mathbb{R}^{d})^{n} rightarrow mathbb{R}$ with spatial invariance property of the form :
$f(x_1,x_2,...,x_n) = f(x_1 + zeta, x_2 + zeta,..., x_n + zeta)$ for $zeta in mathbb{R}^{d} $.
I want to show that $int limits_{(mathbb{R}^{d})^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n ,=, infty $.
I would like some help.
My only idea is to fix ball $B(0,r) subseteq mathbb{R}^{d}$ and $zeta in mathbb{R}^{d}$ and then compute $int limits_{{B(0,r)}^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n , = int limits_{{B(zeta,r)}^{n}} exp(,f(y_1 - zeta,...,y_n - zeta) ,) dy_1..dy_n = $
$ int limits_{{B(zeta,r)}^{n}} exp(,f(y_1,...,y_n) ,) dy_1..dy_n , $, where the first equality comes from setting $x_i = y_i - zeta $ and the second from the spatial invariance property.
So we get that the integral of this strictly positive f over any ball is the same. But does it imply that the integral over $(mathbb{R}^{d})^{n}$ is $infty$ ?
multivariable-calculus vector-analysis
asked Nov 27 at 17:52
vl.ath
317
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider function $f: (mathbb{R}^{d})^{n} rightarrow mathbb{R}$ with spatial invariance property of the form :
$f(x_1,x_2,...,x_n) = f(x_1 + zeta, x_2 + zeta,..., x_n + zeta)$ for $zeta in mathbb{R}^{d} $.
I want to show that $int limits_{(mathbb{R}^{d})^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n ,=, infty $.
I would like some help.
My only idea is to fix ball $B(0,r) subseteq mathbb{R}^{d}$ and $zeta in mathbb{R}^{d}$ and then compute $int limits_{{B(0,r)}^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n , = int limits_{{B(zeta,r)}^{n}} exp(,f(y_1 - zeta,...,y_n - zeta) ,) dy_1..dy_n = $
$ int limits_{{B(zeta,r)}^{n}} exp(,f(y_1,...,y_n) ,) dy_1..dy_n , $, where the first equality comes from setting $x_i = y_i - zeta $ and the second from the spatial invariance property.
So we get that the integral of this strictly positive f over any ball is the same. But does it imply that the integral over $(mathbb{R}^{d})^{n}$ is $infty$ ?
multivariable-calculus vector-analysis
asked Nov 27 at 17:52
vl.ath
317
Consider function $f: (mathbb{R}^{d})^{n} rightarrow mathbb{R}$ with spatial invariance property of the form :
$f(x_1,x_2,...,x_n) = f(x_1 + zeta, x_2 + zeta,..., x_n + zeta)$ for $zeta in mathbb{R}^{d} $.
I want to show that $int limits_{(mathbb{R}^{d})^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n ,=, infty $.
I would like some help.
My only idea is to fix ball $B(0,r) subseteq mathbb{R}^{d}$ and $zeta in mathbb{R}^{d}$ and then compute $int limits_{{B(0,r)}^{n}} exp(,f(x_1,...,x_n) ,) dx_1..dx_n , = int limits_{{B(zeta,r)}^{n}} exp(,f(y_1 - zeta,...,y_n - zeta) ,) dy_1..dy_n = $
$ int limits_{{B(zeta,r)}^{n}} exp(,f(y_1,...,y_n) ,) dy_1..dy_n , $, where the first equality comes from setting $x_i = y_i - zeta $ and the second from the spatial invariance property.
So we get that the integral of this strictly positive f over any ball is the same. But does it imply that the integral over $(mathbb{R}^{d})^{n}$ is $infty$ ?
multivariable-calculus vector-analysis
multivariable-calculus vector-analysis
asked Nov 27 at 17:52
vl.ath
317
asked Nov 27 at 17:52
vl.ath
317
asked Nov 27 at 17:52
vl.ath
317
asked Nov 27 at 17:52
vl.ath
317
asked Nov 27 at 17:52
vl.ath
317
317
add a comment |
add a comment |
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