Quotient of compactly supported functions where denomiator has larger support.
If $ phi, varphi in mathcal{D} $ are given test functions and $ text{Supp} varphi subseteq text{Supp} phi $. I have used many times in my PDE class that the quotient $ varphi / phi $ is well-defined and smooth because $ phi $ vanishes on sets where $ varphi $ vanishes.
I have never quiet thought this through. What is the right point of view in looking at the well-definedness of this function? Is it that we assign the value zero to the expression $0/0$, or perhaps any other reason? Is this statement even correct, in the sense that there has to be some additional assumption on the support? In particular, if $ phi in mathcal{D}$. Then since $ text{Supp} phi' subseteq text{Supp} phi $ , does it make sense to write $ phi' / phi$?
Many thanks in advance!
real-analysis calculus analysis pde
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
|
show 2 more comments
If $ phi, varphi in mathcal{D} $ are given test functions and $ text{Supp} varphi subseteq text{Supp} phi $. I have used many times in my PDE class that the quotient $ varphi / phi $ is well-defined and smooth because $ phi $ vanishes on sets where $ varphi $ vanishes.
I have never quiet thought this through. What is the right point of view in looking at the well-definedness of this function? Is it that we assign the value zero to the expression $0/0$, or perhaps any other reason? Is this statement even correct, in the sense that there has to be some additional assumption on the support? In particular, if $ phi in mathcal{D}$. Then since $ text{Supp} phi' subseteq text{Supp} phi $ , does it make sense to write $ phi' / phi$?
Many thanks in advance!
real-analysis calculus analysis pde
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
You meant $phi$ has no zeros in its support. For $phi'/phi$ try with $phi(x) = e^{-1/x^2}1_{x > 0}$
– reuns
Dec 5 '18 at 11:43
A piece of unrelated advice: $phi$ and $varphi$ are the same symbol. It's confusing do use both, it's like you would use "a" and "$a$" as two symbols. Rather, use $phi$ and $theta$, or $varphi$ and $theta$.
– 5xum
Dec 5 '18 at 11:49
@5xum Mixing $phi,varphi$ is the common notation for functions in $C^infty_c$
– reuns
Dec 5 '18 at 11:52
@reuns Are you sure you aren't talking about mixing $Phi$ and $varphi$?
– 5xum
Dec 5 '18 at 12:01
@Meagain $e^{-1/x^2}1_{x > 0}$ is the first example of a smooth function vanishing for $x < 0$ and $e^{-1/(1-x^2)}1_{|x| < 1}$ is the first example in $C^infty_c$
– reuns
Dec 5 '18 at 12:05
|
show 2 more comments
If $ phi, varphi in mathcal{D} $ are given test functions and $ text{Supp} varphi subseteq text{Supp} phi $. I have used many times in my PDE class that the quotient $ varphi / phi $ is well-defined and smooth because $ phi $ vanishes on sets where $ varphi $ vanishes.
I have never quiet thought this through. What is the right point of view in looking at the well-definedness of this function? Is it that we assign the value zero to the expression $0/0$, or perhaps any other reason? Is this statement even correct, in the sense that there has to be some additional assumption on the support? In particular, if $ phi in mathcal{D}$. Then since $ text{Supp} phi' subseteq text{Supp} phi $ , does it make sense to write $ phi' / phi$?
Many thanks in advance!
real-analysis calculus analysis pde
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
If $ phi, varphi in mathcal{D} $ are given test functions and $ text{Supp} varphi subseteq text{Supp} phi $. I have used many times in my PDE class that the quotient $ varphi / phi $ is well-defined and smooth because $ phi $ vanishes on sets where $ varphi $ vanishes.
I have never quiet thought this through. What is the right point of view in looking at the well-definedness of this function? Is it that we assign the value zero to the expression $0/0$, or perhaps any other reason? Is this statement even correct, in the sense that there has to be some additional assumption on the support? In particular, if $ phi in mathcal{D}$. Then since $ text{Supp} phi' subseteq text{Supp} phi $ , does it make sense to write $ phi' / phi$?
Many thanks in advance!
real-analysis calculus analysis pde
real-analysis calculus analysis pde
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
asked Dec 5 '18 at 11:26
MeagainMeagain
16910
16910
You meant $phi$ has no zeros in its support. For $phi'/phi$ try with $phi(x) = e^{-1/x^2}1_{x > 0}$
– reuns
Dec 5 '18 at 11:43
A piece of unrelated advice: $phi$ and $varphi$ are the same symbol. It's confusing do use both, it's like you would use "a" and "$a$" as two symbols. Rather, use $phi$ and $theta$, or $varphi$ and $theta$.
– 5xum
Dec 5 '18 at 11:49
@5xum Mixing $phi,varphi$ is the common notation for functions in $C^infty_c$
– reuns
Dec 5 '18 at 11:52
@reuns Are you sure you aren't talking about mixing $Phi$ and $varphi$?
– 5xum
Dec 5 '18 at 12:01
@Meagain $e^{-1/x^2}1_{x > 0}$ is the first example of a smooth function vanishing for $x < 0$ and $e^{-1/(1-x^2)}1_{|x| < 1}$ is the first example in $C^infty_c$
– reuns
Dec 5 '18 at 12:05
|
show 2 more comments
You meant $phi$ has no zeros in its support. For $phi'/phi$ try with $phi(x) = e^{-1/x^2}1_{x > 0}$
– reuns
Dec 5 '18 at 11:43
A piece of unrelated advice: $phi$ and $varphi$ are the same symbol. It's confusing do use both, it's like you would use "a" and "$a$" as two symbols. Rather, use $phi$ and $theta$, or $varphi$ and $theta$.
– 5xum
Dec 5 '18 at 11:49
@5xum Mixing $phi,varphi$ is the common notation for functions in $C^infty_c$
– reuns
Dec 5 '18 at 11:52
@reuns Are you sure you aren't talking about mixing $Phi$ and $varphi$?
– 5xum
Dec 5 '18 at 12:01
@Meagain $e^{-1/x^2}1_{x > 0}$ is the first example of a smooth function vanishing for $x < 0$ and $e^{-1/(1-x^2)}1_{|x| < 1}$ is the first example in $C^infty_c$
– reuns
Dec 5 '18 at 12:05
You meant $phi$ has no zeros in its support. For $phi'/phi$ try with $phi(x) = e^{-1/x^2}1_{x > 0}$
– reuns
Dec 5 '18 at 11:43
You meant $phi$ has no zeros in its support. For $phi'/phi$ try with $phi(x) = e^{-1/x^2}1_{x > 0}$
– reuns
Dec 5 '18 at 11:43
A piece of unrelated advice: $phi$ and $varphi$ are the same symbol. It's confusing do use both, it's like you would use "a" and "$a$" as two symbols. Rather, use $phi$ and $theta$, or $varphi$ and $theta$.
– 5xum
Dec 5 '18 at 11:49
A piece of unrelated advice: $phi$ and $varphi$ are the same symbol. It's confusing do use both, it's like you would use "a" and "$a$" as two symbols. Rather, use $phi$ and $theta$, or $varphi$ and $theta$.
– 5xum
Dec 5 '18 at 11:49
@5xum Mixing $phi,varphi$ is the common notation for functions in $C^infty_c$
– reuns
Dec 5 '18 at 11:52
@5xum Mixing $phi,varphi$ is the common notation for functions in $C^infty_c$
– reuns
Dec 5 '18 at 11:52
@reuns Are you sure you aren't talking about mixing $Phi$ and $varphi$?
– 5xum
Dec 5 '18 at 12:01
@reuns Are you sure you aren't talking about mixing $Phi$ and $varphi$?
– 5xum
Dec 5 '18 at 12:01
@Meagain $e^{-1/x^2}1_{x > 0}$ is the first example of a smooth function vanishing for $x < 0$ and $e^{-1/(1-x^2)}1_{|x| < 1}$ is the first example in $C^infty_c$
– reuns
Dec 5 '18 at 12:05
@Meagain $e^{-1/x^2}1_{x > 0}$ is the first example of a smooth function vanishing for $x < 0$ and $e^{-1/(1-x^2)}1_{|x| < 1}$ is the first example in $C^infty_c$
– reuns
Dec 5 '18 at 12:05
|
show 2 more comments
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You meant $phi$ has no zeros in its support. For $phi'/phi$ try with $phi(x) = e^{-1/x^2}1_{x > 0}$
– reuns
Dec 5 '18 at 11:43
A piece of unrelated advice: $phi$ and $varphi$ are the same symbol. It's confusing do use both, it's like you would use "a" and "$a$" as two symbols. Rather, use $phi$ and $theta$, or $varphi$ and $theta$.
– 5xum
Dec 5 '18 at 11:49
@5xum Mixing $phi,varphi$ is the common notation for functions in $C^infty_c$
– reuns
Dec 5 '18 at 11:52
@reuns Are you sure you aren't talking about mixing $Phi$ and $varphi$?
– 5xum
Dec 5 '18 at 12:01
@Meagain $e^{-1/x^2}1_{x > 0}$ is the first example of a smooth function vanishing for $x < 0$ and $e^{-1/(1-x^2)}1_{|x| < 1}$ is the first example in $C^infty_c$
– reuns
Dec 5 '18 at 12:05