FInd a function $f$ which satisfies $af(kx)-bf(x)$ is a function of $k$ and irrevelent to $x$ [on hold]Are there numerical techniques to find undefined points on a 2D function?Find a bounded function with a supporting pointLocally uniformly convex Banach space, which is not uniformly convex.Approximating the constant term of a polynomialIs there an holomorphic function? If that function exists, is it unique?When left inverse of a function is injectiveNonconvex functional show inf is zero but inf is not attainedValue of $f^2(4)+g^2(4)$Function in piecewise linear finite element space which satisfies the divergence-free condition is the zero functionBounded linear function
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FInd a function $f$ which satisfies $af(kx)-bf(x)$ is a function of $k$ and irrevelent to $x$ [on hold]
Are there numerical techniques to find undefined points on a 2D function?Find a bounded function with a supporting pointLocally uniformly convex Banach space, which is not uniformly convex.Approximating the constant term of a polynomialIs there an holomorphic function? If that function exists, is it unique?When left inverse of a function is injectiveNonconvex functional show inf is zero but inf is not attainedValue of $f^2(4)+g^2(4)$Function in piecewise linear finite element space which satisfies the divergence-free condition is the zero functionBounded linear function
$begingroup$
I want to find a function $f$ that satisfies
$af(kx) - bf(x)=c$ where $a,b,c,k$ are constant. In other words, a function that makes the left part irrelevent to $x$.
For example, when $a=b$, we can find a function $f(x)=log(x)$, then $af(kx) - bf(x) = alog(kx)-alog(x)=alog(k)$ which is a constant.
Although it's true that an identity function satisfies the condition, but it's too trivial and not what I want.
functional-analysis
New contributor
$endgroup$
put on hold as too broad by 5xum, Eevee Trainer, Claude Leibovici, uniquesolution, Riccardo.Alestra Mar 11 at 10:21
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 2 more comments
$begingroup$
I want to find a function $f$ that satisfies
$af(kx) - bf(x)=c$ where $a,b,c,k$ are constant. In other words, a function that makes the left part irrelevent to $x$.
For example, when $a=b$, we can find a function $f(x)=log(x)$, then $af(kx) - bf(x) = alog(kx)-alog(x)=alog(k)$ which is a constant.
Although it's true that an identity function satisfies the condition, but it's too trivial and not what I want.
functional-analysis
New contributor
$endgroup$
put on hold as too broad by 5xum, Eevee Trainer, Claude Leibovici, uniquesolution, Riccardo.Alestra Mar 11 at 10:21
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
$begingroup$
If you just want to find examples there are plenty: $f$ can be a constant, it can be the identity function with $b=ak$ etc.
$endgroup$
– Kavi Rama Murthy
Mar 11 at 8:52
$begingroup$
Are $a,b,c,k$ known constants? Are there other limitations to $f$? Right now, this question is waaaay to broad.
$endgroup$
– 5xum
Mar 11 at 8:55
$begingroup$
$f(x)$ should be a monotone function with respect to $x$, and $a,b,c$ are known while $k$ can be solved.
$endgroup$
– troy.tang
Mar 11 at 8:58
$begingroup$
For example, when $a=b$, we can find a $f(x) = log(x)$ which satisfy the equation.
$endgroup$
– troy.tang
Mar 11 at 8:59
$begingroup$
The identity function satisfies the condition, but it's kind of trivial and not what I'm looking for, I'm looking for a function form like $log(x)$
$endgroup$
– troy.tang
Mar 11 at 9:08
|
show 2 more comments
$begingroup$
I want to find a function $f$ that satisfies
$af(kx) - bf(x)=c$ where $a,b,c,k$ are constant. In other words, a function that makes the left part irrelevent to $x$.
For example, when $a=b$, we can find a function $f(x)=log(x)$, then $af(kx) - bf(x) = alog(kx)-alog(x)=alog(k)$ which is a constant.
Although it's true that an identity function satisfies the condition, but it's too trivial and not what I want.
functional-analysis
New contributor
$endgroup$
I want to find a function $f$ that satisfies
$af(kx) - bf(x)=c$ where $a,b,c,k$ are constant. In other words, a function that makes the left part irrelevent to $x$.
For example, when $a=b$, we can find a function $f(x)=log(x)$, then $af(kx) - bf(x) = alog(kx)-alog(x)=alog(k)$ which is a constant.
Although it's true that an identity function satisfies the condition, but it's too trivial and not what I want.
functional-analysis
functional-analysis
New contributor
New contributor
edited 2 days ago
troy.tang
New contributor
asked Mar 11 at 8:50
troy.tangtroy.tang
112
112
New contributor
New contributor
put on hold as too broad by 5xum, Eevee Trainer, Claude Leibovici, uniquesolution, Riccardo.Alestra Mar 11 at 10:21
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as too broad by 5xum, Eevee Trainer, Claude Leibovici, uniquesolution, Riccardo.Alestra Mar 11 at 10:21
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
$begingroup$
If you just want to find examples there are plenty: $f$ can be a constant, it can be the identity function with $b=ak$ etc.
$endgroup$
– Kavi Rama Murthy
Mar 11 at 8:52
$begingroup$
Are $a,b,c,k$ known constants? Are there other limitations to $f$? Right now, this question is waaaay to broad.
$endgroup$
– 5xum
Mar 11 at 8:55
$begingroup$
$f(x)$ should be a monotone function with respect to $x$, and $a,b,c$ are known while $k$ can be solved.
$endgroup$
– troy.tang
Mar 11 at 8:58
$begingroup$
For example, when $a=b$, we can find a $f(x) = log(x)$ which satisfy the equation.
$endgroup$
– troy.tang
Mar 11 at 8:59
$begingroup$
The identity function satisfies the condition, but it's kind of trivial and not what I'm looking for, I'm looking for a function form like $log(x)$
$endgroup$
– troy.tang
Mar 11 at 9:08
|
show 2 more comments
4
$begingroup$
If you just want to find examples there are plenty: $f$ can be a constant, it can be the identity function with $b=ak$ etc.
$endgroup$
– Kavi Rama Murthy
Mar 11 at 8:52
$begingroup$
Are $a,b,c,k$ known constants? Are there other limitations to $f$? Right now, this question is waaaay to broad.
$endgroup$
– 5xum
Mar 11 at 8:55
$begingroup$
$f(x)$ should be a monotone function with respect to $x$, and $a,b,c$ are known while $k$ can be solved.
$endgroup$
– troy.tang
Mar 11 at 8:58
$begingroup$
For example, when $a=b$, we can find a $f(x) = log(x)$ which satisfy the equation.
$endgroup$
– troy.tang
Mar 11 at 8:59
$begingroup$
The identity function satisfies the condition, but it's kind of trivial and not what I'm looking for, I'm looking for a function form like $log(x)$
$endgroup$
– troy.tang
Mar 11 at 9:08
4
4
$begingroup$
If you just want to find examples there are plenty: $f$ can be a constant, it can be the identity function with $b=ak$ etc.
$endgroup$
– Kavi Rama Murthy
Mar 11 at 8:52
$begingroup$
If you just want to find examples there are plenty: $f$ can be a constant, it can be the identity function with $b=ak$ etc.
$endgroup$
– Kavi Rama Murthy
Mar 11 at 8:52
$begingroup$
Are $a,b,c,k$ known constants? Are there other limitations to $f$? Right now, this question is waaaay to broad.
$endgroup$
– 5xum
Mar 11 at 8:55
$begingroup$
Are $a,b,c,k$ known constants? Are there other limitations to $f$? Right now, this question is waaaay to broad.
$endgroup$
– 5xum
Mar 11 at 8:55
$begingroup$
$f(x)$ should be a monotone function with respect to $x$, and $a,b,c$ are known while $k$ can be solved.
$endgroup$
– troy.tang
Mar 11 at 8:58
$begingroup$
$f(x)$ should be a monotone function with respect to $x$, and $a,b,c$ are known while $k$ can be solved.
$endgroup$
– troy.tang
Mar 11 at 8:58
$begingroup$
For example, when $a=b$, we can find a $f(x) = log(x)$ which satisfy the equation.
$endgroup$
– troy.tang
Mar 11 at 8:59
$begingroup$
For example, when $a=b$, we can find a $f(x) = log(x)$ which satisfy the equation.
$endgroup$
– troy.tang
Mar 11 at 8:59
$begingroup$
The identity function satisfies the condition, but it's kind of trivial and not what I'm looking for, I'm looking for a function form like $log(x)$
$endgroup$
– troy.tang
Mar 11 at 9:08
$begingroup$
The identity function satisfies the condition, but it's kind of trivial and not what I'm looking for, I'm looking for a function form like $log(x)$
$endgroup$
– troy.tang
Mar 11 at 9:08
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
In general, $f(x)=fracca-b$, the constant function is the only solution defined for all $x$.
Plug in $x=1$, then you can rearrange the equation to $f(k)=frac1a(c+bcdot f(1))$, so $f$ has to be constant. You can compute the constant by plugging in $k=1$.
There can be more solutions for specific values of $a,b,c,k$ and maybe also other solutions that are only defined for some values of $x$.
$endgroup$
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In general, $f(x)=fracca-b$, the constant function is the only solution defined for all $x$.
Plug in $x=1$, then you can rearrange the equation to $f(k)=frac1a(c+bcdot f(1))$, so $f$ has to be constant. You can compute the constant by plugging in $k=1$.
There can be more solutions for specific values of $a,b,c,k$ and maybe also other solutions that are only defined for some values of $x$.
$endgroup$
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
add a comment |
$begingroup$
In general, $f(x)=fracca-b$, the constant function is the only solution defined for all $x$.
Plug in $x=1$, then you can rearrange the equation to $f(k)=frac1a(c+bcdot f(1))$, so $f$ has to be constant. You can compute the constant by plugging in $k=1$.
There can be more solutions for specific values of $a,b,c,k$ and maybe also other solutions that are only defined for some values of $x$.
$endgroup$
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
add a comment |
$begingroup$
In general, $f(x)=fracca-b$, the constant function is the only solution defined for all $x$.
Plug in $x=1$, then you can rearrange the equation to $f(k)=frac1a(c+bcdot f(1))$, so $f$ has to be constant. You can compute the constant by plugging in $k=1$.
There can be more solutions for specific values of $a,b,c,k$ and maybe also other solutions that are only defined for some values of $x$.
$endgroup$
In general, $f(x)=fracca-b$, the constant function is the only solution defined for all $x$.
Plug in $x=1$, then you can rearrange the equation to $f(k)=frac1a(c+bcdot f(1))$, so $f$ has to be constant. You can compute the constant by plugging in $k=1$.
There can be more solutions for specific values of $a,b,c,k$ and maybe also other solutions that are only defined for some values of $x$.
answered Mar 11 at 9:20
quaraguequarague
436210
436210
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
add a comment |
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Given that the OP accepts $f(x) = log(x)$ as a solution, they may permit $c$ to depend on $k$
$endgroup$
– preferred_anon
Mar 11 at 9:56
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
$begingroup$
Yes, in fact. I want to find a non-trivial function that satisfies $af(kx)-bf(x)$ is irrelevant to $x$. When $a=b$, given a $f(x)=log(x)$, then it becomes $a log(kx)-a log(x)=a log(k)$ which is a constant. The problem is what's the function will be when $a neq b$?
$endgroup$
– troy.tang
Mar 11 at 11:32
add a comment |
4
$begingroup$
If you just want to find examples there are plenty: $f$ can be a constant, it can be the identity function with $b=ak$ etc.
$endgroup$
– Kavi Rama Murthy
Mar 11 at 8:52
$begingroup$
Are $a,b,c,k$ known constants? Are there other limitations to $f$? Right now, this question is waaaay to broad.
$endgroup$
– 5xum
Mar 11 at 8:55
$begingroup$
$f(x)$ should be a monotone function with respect to $x$, and $a,b,c$ are known while $k$ can be solved.
$endgroup$
– troy.tang
Mar 11 at 8:58
$begingroup$
For example, when $a=b$, we can find a $f(x) = log(x)$ which satisfy the equation.
$endgroup$
– troy.tang
Mar 11 at 8:59
$begingroup$
The identity function satisfies the condition, but it's kind of trivial and not what I'm looking for, I'm looking for a function form like $log(x)$
$endgroup$
– troy.tang
Mar 11 at 9:08