What conditions required for functions $f$ and $g$ for $int^{cx} f(t) , dt = int^{x} h(t) ,dt $ for all $xin...
$begingroup$
suppose I have two functions $f$ and $g$ both mapping from $mathbb{R}$ to $mathbb{R}_{++}$.
I wonder what conditions do I need to impose on $f$ and $g$ so that
$$int^{cx}_{-infty} f(t) , dt = int^{x}_{-infty} h(t) ,dt, $$
for some constant $c$ and for all $xin mathbb{R}$.
One example this equation holds is that
$$f(t) = left{begin{array}{cc} 2t &text{if } t>0\
0 &text{otherwise} end{array} right. ~~g(t) = left{begin{array}{cc} 1/2t &text{if } t>0\
0 &text{otherwise} end{array} right.$$
Then, choose $c=1/4$.
Likewise, my guess is that $f(t)$ and $g(t)$ has to be linear(?) functions to hold my assertion. I wonder if my guess is correct or not.
calculus
asked Jan 21 at 22:36
user1292919user1292919
770512
$endgroup$
add a comment |
$begingroup$
suppose I have two functions $f$ and $g$ both mapping from $mathbb{R}$ to $mathbb{R}_{++}$.
I wonder what conditions do I need to impose on $f$ and $g$ so that
$$int^{cx}_{-infty} f(t) , dt = int^{x}_{-infty} h(t) ,dt, $$
for some constant $c$ and for all $xin mathbb{R}$.
One example this equation holds is that
$$f(t) = left{begin{array}{cc} 2t &text{if } t>0\
0 &text{otherwise} end{array} right. ~~g(t) = left{begin{array}{cc} 1/2t &text{if } t>0\
0 &text{otherwise} end{array} right.$$
Then, choose $c=1/4$.
Likewise, my guess is that $f(t)$ and $g(t)$ has to be linear(?) functions to hold my assertion. I wonder if my guess is correct or not.
calculus
asked Jan 21 at 22:36
user1292919user1292919
770512
$endgroup$
add a comment |
$begingroup$
suppose I have two functions $f$ and $g$ both mapping from $mathbb{R}$ to $mathbb{R}_{++}$.
I wonder what conditions do I need to impose on $f$ and $g$ so that
$$int^{cx}_{-infty} f(t) , dt = int^{x}_{-infty} h(t) ,dt, $$
for some constant $c$ and for all $xin mathbb{R}$.
One example this equation holds is that
$$f(t) = left{begin{array}{cc} 2t &text{if } t>0\
0 &text{otherwise} end{array} right. ~~g(t) = left{begin{array}{cc} 1/2t &text{if } t>0\
0 &text{otherwise} end{array} right.$$
Then, choose $c=1/4$.
Likewise, my guess is that $f(t)$ and $g(t)$ has to be linear(?) functions to hold my assertion. I wonder if my guess is correct or not.
calculus
asked Jan 21 at 22:36
user1292919user1292919
770512
$endgroup$
suppose I have two functions $f$ and $g$ both mapping from $mathbb{R}$ to $mathbb{R}_{++}$.
I wonder what conditions do I need to impose on $f$ and $g$ so that
$$int^{cx}_{-infty} f(t) , dt = int^{x}_{-infty} h(t) ,dt, $$
for some constant $c$ and for all $xin mathbb{R}$.
One example this equation holds is that
$$f(t) = left{begin{array}{cc} 2t &text{if } t>0\
0 &text{otherwise} end{array} right. ~~g(t) = left{begin{array}{cc} 1/2t &text{if } t>0\
0 &text{otherwise} end{array} right.$$
Then, choose $c=1/4$.
Likewise, my guess is that $f(t)$ and $g(t)$ has to be linear(?) functions to hold my assertion. I wonder if my guess is correct or not.
calculus
calculus
asked Jan 21 at 22:36
user1292919user1292919
770512
asked Jan 21 at 22:36
user1292919user1292919
770512
asked Jan 21 at 22:36
user1292919user1292919
770512
asked Jan 21 at 22:36
user1292919user1292919
770512
asked Jan 21 at 22:36
user1292919user1292919
770512
770512
add a comment |
add a comment |
1 Answer
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$begingroup$
Use substitution $t mapsto ct$, so first integral becomes $int_{-infty}^x c f(ct), dt$.
Then, since the equality is true for every $x$, it must be $c f(cx) = h(x)$ a.e.
answered Jan 21 at 22:44
HarnakHarnak
1,299512
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add a comment |
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$begingroup$
Use substitution $t mapsto ct$, so first integral becomes $int_{-infty}^x c f(ct), dt$.
Then, since the equality is true for every $x$, it must be $c f(cx) = h(x)$ a.e.
answered Jan 21 at 22:44
HarnakHarnak
1,299512
$endgroup$
add a comment |
$begingroup$
Use substitution $t mapsto ct$, so first integral becomes $int_{-infty}^x c f(ct), dt$.
Then, since the equality is true for every $x$, it must be $c f(cx) = h(x)$ a.e.
answered Jan 21 at 22:44
HarnakHarnak
1,299512
$endgroup$
add a comment |
$begingroup$
Use substitution $t mapsto ct$, so first integral becomes $int_{-infty}^x c f(ct), dt$.
Then, since the equality is true for every $x$, it must be $c f(cx) = h(x)$ a.e.
answered Jan 21 at 22:44
HarnakHarnak
1,299512
$endgroup$
Use substitution $t mapsto ct$, so first integral becomes $int_{-infty}^x c f(ct), dt$.
Then, since the equality is true for every $x$, it must be $c f(cx) = h(x)$ a.e.
answered Jan 21 at 22:44
HarnakHarnak
1,299512
edited Jan 21 at 23:06
answered Jan 21 at 22:44
HarnakHarnak
1,299512
answered Jan 21 at 22:44
HarnakHarnak
1,299512
answered Jan 21 at 22:44
HarnakHarnak
1,299512
1,299512
add a comment |
add a comment |
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