Mixing
If $(a_n)_nsubseteq Bbb{R}$ and $leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n)$, then $Tin...
$begingroup$
Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$
My trial
It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.
$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?
functional-analysis distribution-theory
asked Feb 1 at 13:48
MichealMicheal
26511
$endgroup$
add a comment |
$begingroup$
Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$
My trial
It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.
$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?
functional-analysis distribution-theory
asked Feb 1 at 13:48
MichealMicheal
26511
$endgroup$
$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06
$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08
1
$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09
$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13
add a comment |
$begingroup$
Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$
My trial
It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.
$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?
functional-analysis distribution-theory
asked Feb 1 at 13:48
MichealMicheal
26511
$endgroup$
Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$
My trial
It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.
$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?
functional-analysis distribution-theory
functional-analysis distribution-theory
asked Feb 1 at 13:48
MichealMicheal
26511
asked Feb 1 at 13:48
MichealMicheal
26511
edited Feb 1 at 14:00
Micheal
asked Feb 1 at 13:48
MichealMicheal
26511
asked Feb 1 at 13:48
MichealMicheal
26511
asked Feb 1 at 13:48
MichealMicheal
26511
26511
$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06
$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08
1
$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09
$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13
add a comment |
$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06
$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08
1
$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09
$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13
$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06
$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06
$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08
$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08
1
1
$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09
$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09
$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13
$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13
add a comment |
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$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06
$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08
1
$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09
$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13