Generated space by the harmonic oscillators $e^{iw}$.
$begingroup$
Let $mathbb{T}$ represent the $[0,1)$ interval and for each $winmathbb{Z}$ denote the function $$e_w:mathbb{T}tomathbb{C}$$ such that $$forall xinmathbb{T}, e_w(x)=e^{2pi iwx}.$$ It is known that the set of harmonic oscillators $e_w$ is a complete system in $L_p(mathbb{T})$, i.e. such spaces are "generated" by $big{ e_w : winmathbb{Z} big}$ with coefficients in some specific set of series $ell^qsubsetbig{(c_w)_{winmathbb{N}}: c_winmathbb{C}big}$ since the set of finite sums $big{ sumlimits_{w=0}^{k} c_we_wbig}$ is dense in it, that is
$forall f in L_p(mathbb{T}), exists (c_w)_{winmathbb{N}}in ell^q
Big( limlimits_{kto infty} sumlimits_{w=0}^{k} c_w e_w = f Big)$.
For each $pin (0,infty], (c_w)_{winmathbb{N}}$ lies in a specific set of complex series $ell^q$ where $1=dfrac{1}{p} + dfrac{1}{q}$.
My question is: given the set of all complex sequences ${ (c_w)_{winmathbb{N}} : c_winmathbb{C}}$, What is the space generated by the topological closure of the finite sums $sumlimits_{w=0}^{k} c_w e_w$ ? In other words, What is
$$overline{bigg{ sumlimits_{w=-n}^{n} c_w e_w : ninmathbb{N}, c_winmathbb{C}bigg} } ?$$
Evidently, several of these funcions reach infinite values in various subsets.
convergence exponential-function fourier-series lp-spaces topological-vector-spaces
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
$endgroup$
|
show 3 more comments
$begingroup$
Let $mathbb{T}$ represent the $[0,1)$ interval and for each $winmathbb{Z}$ denote the function $$e_w:mathbb{T}tomathbb{C}$$ such that $$forall xinmathbb{T}, e_w(x)=e^{2pi iwx}.$$ It is known that the set of harmonic oscillators $e_w$ is a complete system in $L_p(mathbb{T})$, i.e. such spaces are "generated" by $big{ e_w : winmathbb{Z} big}$ with coefficients in some specific set of series $ell^qsubsetbig{(c_w)_{winmathbb{N}}: c_winmathbb{C}big}$ since the set of finite sums $big{ sumlimits_{w=0}^{k} c_we_wbig}$ is dense in it, that is
$forall f in L_p(mathbb{T}), exists (c_w)_{winmathbb{N}}in ell^q
Big( limlimits_{kto infty} sumlimits_{w=0}^{k} c_w e_w = f Big)$.
For each $pin (0,infty], (c_w)_{winmathbb{N}}$ lies in a specific set of complex series $ell^q$ where $1=dfrac{1}{p} + dfrac{1}{q}$.
My question is: given the set of all complex sequences ${ (c_w)_{winmathbb{N}} : c_winmathbb{C}}$, What is the space generated by the topological closure of the finite sums $sumlimits_{w=0}^{k} c_w e_w$ ? In other words, What is
$$overline{bigg{ sumlimits_{w=-n}^{n} c_w e_w : ninmathbb{N}, c_winmathbb{C}bigg} } ?$$
Evidently, several of these funcions reach infinite values in various subsets.
convergence exponential-function fourier-series lp-spaces topological-vector-spaces
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
$endgroup$
$begingroup$
What do you mean by 'orthogonal' in $L_p$?.
$endgroup$
– Kavi Rama Murthy
Dec 4 '18 at 23:20
$begingroup$
You are right, I just corrected it! Orthogonality only makes sense when $p=2$.
$endgroup$
– Dr Potato
Dec 4 '18 at 23:23
1
$begingroup$
Closure under what topology?
$endgroup$
– Mike Miller
Dec 6 '18 at 23:13
1
$begingroup$
There is something wrong with your literal formulation. E.g., the way you're writing it, the $e^{iw}$ are just numbers, not functions. Don't you want to talk about functions $psi_n(x)=e^{2pi inx}$ on $[0,1]$, instead? If you rewrite in such terms, your question would be much clearer...
$endgroup$
– paul garrett
Dec 7 '18 at 0:31
1
$begingroup$
That ${ u_n}$ is dense in $X,|.|$ doesn't mean that for any element $f in X$ there is a sequence of complex numbers $(c_n)$ such that $lim_{N to infty} |f-sum_{n=1}^N c_n u_n| = 0$ ! It means that for any element $f in X$ there is $(c_{n,N})$ such that $lim_{N to infty} |f-sum_{n=1}^N c_{n,N} u_n| = 0$ (try with $u_n(x) = x^n$ then $sum_{n=1}^infty c_n u_n$ converges in $L^q([0,1])$ iff it converges uniformly to an analytic function for $|x| le r < 1$)
$endgroup$
– reuns
Dec 7 '18 at 12:55
|
show 3 more comments
$begingroup$
Let $mathbb{T}$ represent the $[0,1)$ interval and for each $winmathbb{Z}$ denote the function $$e_w:mathbb{T}tomathbb{C}$$ such that $$forall xinmathbb{T}, e_w(x)=e^{2pi iwx}.$$ It is known that the set of harmonic oscillators $e_w$ is a complete system in $L_p(mathbb{T})$, i.e. such spaces are "generated" by $big{ e_w : winmathbb{Z} big}$ with coefficients in some specific set of series $ell^qsubsetbig{(c_w)_{winmathbb{N}}: c_winmathbb{C}big}$ since the set of finite sums $big{ sumlimits_{w=0}^{k} c_we_wbig}$ is dense in it, that is
$forall f in L_p(mathbb{T}), exists (c_w)_{winmathbb{N}}in ell^q
Big( limlimits_{kto infty} sumlimits_{w=0}^{k} c_w e_w = f Big)$.
For each $pin (0,infty], (c_w)_{winmathbb{N}}$ lies in a specific set of complex series $ell^q$ where $1=dfrac{1}{p} + dfrac{1}{q}$.
My question is: given the set of all complex sequences ${ (c_w)_{winmathbb{N}} : c_winmathbb{C}}$, What is the space generated by the topological closure of the finite sums $sumlimits_{w=0}^{k} c_w e_w$ ? In other words, What is
$$overline{bigg{ sumlimits_{w=-n}^{n} c_w e_w : ninmathbb{N}, c_winmathbb{C}bigg} } ?$$
Evidently, several of these funcions reach infinite values in various subsets.
convergence exponential-function fourier-series lp-spaces topological-vector-spaces
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
$endgroup$
Let $mathbb{T}$ represent the $[0,1)$ interval and for each $winmathbb{Z}$ denote the function $$e_w:mathbb{T}tomathbb{C}$$ such that $$forall xinmathbb{T}, e_w(x)=e^{2pi iwx}.$$ It is known that the set of harmonic oscillators $e_w$ is a complete system in $L_p(mathbb{T})$, i.e. such spaces are "generated" by $big{ e_w : winmathbb{Z} big}$ with coefficients in some specific set of series $ell^qsubsetbig{(c_w)_{winmathbb{N}}: c_winmathbb{C}big}$ since the set of finite sums $big{ sumlimits_{w=0}^{k} c_we_wbig}$ is dense in it, that is
$forall f in L_p(mathbb{T}), exists (c_w)_{winmathbb{N}}in ell^q
Big( limlimits_{kto infty} sumlimits_{w=0}^{k} c_w e_w = f Big)$.
For each $pin (0,infty], (c_w)_{winmathbb{N}}$ lies in a specific set of complex series $ell^q$ where $1=dfrac{1}{p} + dfrac{1}{q}$.
My question is: given the set of all complex sequences ${ (c_w)_{winmathbb{N}} : c_winmathbb{C}}$, What is the space generated by the topological closure of the finite sums $sumlimits_{w=0}^{k} c_w e_w$ ? In other words, What is
$$overline{bigg{ sumlimits_{w=-n}^{n} c_w e_w : ninmathbb{N}, c_winmathbb{C}bigg} } ?$$
Evidently, several of these funcions reach infinite values in various subsets.
convergence exponential-function fourier-series lp-spaces topological-vector-spaces
convergence exponential-function fourier-series lp-spaces topological-vector-spaces
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
edited Jan 21 at 17:18
Dr Potato
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
asked Dec 4 '18 at 23:11
Dr PotatoDr Potato
394
394
$begingroup$
What do you mean by 'orthogonal' in $L_p$?.
$endgroup$
– Kavi Rama Murthy
Dec 4 '18 at 23:20
$begingroup$
You are right, I just corrected it! Orthogonality only makes sense when $p=2$.
$endgroup$
– Dr Potato
Dec 4 '18 at 23:23
1
$begingroup$
Closure under what topology?
$endgroup$
– Mike Miller
Dec 6 '18 at 23:13
1
$begingroup$
There is something wrong with your literal formulation. E.g., the way you're writing it, the $e^{iw}$ are just numbers, not functions. Don't you want to talk about functions $psi_n(x)=e^{2pi inx}$ on $[0,1]$, instead? If you rewrite in such terms, your question would be much clearer...
$endgroup$
– paul garrett
Dec 7 '18 at 0:31
1
$begingroup$
That ${ u_n}$ is dense in $X,|.|$ doesn't mean that for any element $f in X$ there is a sequence of complex numbers $(c_n)$ such that $lim_{N to infty} |f-sum_{n=1}^N c_n u_n| = 0$ ! It means that for any element $f in X$ there is $(c_{n,N})$ such that $lim_{N to infty} |f-sum_{n=1}^N c_{n,N} u_n| = 0$ (try with $u_n(x) = x^n$ then $sum_{n=1}^infty c_n u_n$ converges in $L^q([0,1])$ iff it converges uniformly to an analytic function for $|x| le r < 1$)
$endgroup$
– reuns
Dec 7 '18 at 12:55
|
show 3 more comments
$begingroup$
What do you mean by 'orthogonal' in $L_p$?.
$endgroup$
– Kavi Rama Murthy
Dec 4 '18 at 23:20
$begingroup$
You are right, I just corrected it! Orthogonality only makes sense when $p=2$.
$endgroup$
– Dr Potato
Dec 4 '18 at 23:23
1
$begingroup$
Closure under what topology?
$endgroup$
– Mike Miller
Dec 6 '18 at 23:13
1
$begingroup$
There is something wrong with your literal formulation. E.g., the way you're writing it, the $e^{iw}$ are just numbers, not functions. Don't you want to talk about functions $psi_n(x)=e^{2pi inx}$ on $[0,1]$, instead? If you rewrite in such terms, your question would be much clearer...
$endgroup$
– paul garrett
Dec 7 '18 at 0:31
1
$begingroup$
That ${ u_n}$ is dense in $X,|.|$ doesn't mean that for any element $f in X$ there is a sequence of complex numbers $(c_n)$ such that $lim_{N to infty} |f-sum_{n=1}^N c_n u_n| = 0$ ! It means that for any element $f in X$ there is $(c_{n,N})$ such that $lim_{N to infty} |f-sum_{n=1}^N c_{n,N} u_n| = 0$ (try with $u_n(x) = x^n$ then $sum_{n=1}^infty c_n u_n$ converges in $L^q([0,1])$ iff it converges uniformly to an analytic function for $|x| le r < 1$)
$endgroup$
– reuns
Dec 7 '18 at 12:55
$begingroup$
What do you mean by 'orthogonal' in $L_p$?.
$endgroup$
– Kavi Rama Murthy
Dec 4 '18 at 23:20
$begingroup$
What do you mean by 'orthogonal' in $L_p$?.
$endgroup$
– Kavi Rama Murthy
Dec 4 '18 at 23:20
$begingroup$
You are right, I just corrected it! Orthogonality only makes sense when $p=2$.
$endgroup$
– Dr Potato
Dec 4 '18 at 23:23
$begingroup$
You are right, I just corrected it! Orthogonality only makes sense when $p=2$.
$endgroup$
– Dr Potato
Dec 4 '18 at 23:23
1
1
$begingroup$
Closure under what topology?
$endgroup$
– Mike Miller
Dec 6 '18 at 23:13
$begingroup$
Closure under what topology?
$endgroup$
– Mike Miller
Dec 6 '18 at 23:13
1
1
$begingroup$
There is something wrong with your literal formulation. E.g., the way you're writing it, the $e^{iw}$ are just numbers, not functions. Don't you want to talk about functions $psi_n(x)=e^{2pi inx}$ on $[0,1]$, instead? If you rewrite in such terms, your question would be much clearer...
$endgroup$
– paul garrett
Dec 7 '18 at 0:31
$begingroup$
There is something wrong with your literal formulation. E.g., the way you're writing it, the $e^{iw}$ are just numbers, not functions. Don't you want to talk about functions $psi_n(x)=e^{2pi inx}$ on $[0,1]$, instead? If you rewrite in such terms, your question would be much clearer...
$endgroup$
– paul garrett
Dec 7 '18 at 0:31
1
1
$begingroup$
That ${ u_n}$ is dense in $X,|.|$ doesn't mean that for any element $f in X$ there is a sequence of complex numbers $(c_n)$ such that $lim_{N to infty} |f-sum_{n=1}^N c_n u_n| = 0$ ! It means that for any element $f in X$ there is $(c_{n,N})$ such that $lim_{N to infty} |f-sum_{n=1}^N c_{n,N} u_n| = 0$ (try with $u_n(x) = x^n$ then $sum_{n=1}^infty c_n u_n$ converges in $L^q([0,1])$ iff it converges uniformly to an analytic function for $|x| le r < 1$)
$endgroup$
– reuns
Dec 7 '18 at 12:55
$begingroup$
That ${ u_n}$ is dense in $X,|.|$ doesn't mean that for any element $f in X$ there is a sequence of complex numbers $(c_n)$ such that $lim_{N to infty} |f-sum_{n=1}^N c_n u_n| = 0$ ! It means that for any element $f in X$ there is $(c_{n,N})$ such that $lim_{N to infty} |f-sum_{n=1}^N c_{n,N} u_n| = 0$ (try with $u_n(x) = x^n$ then $sum_{n=1}^infty c_n u_n$ converges in $L^q([0,1])$ iff it converges uniformly to an analytic function for $|x| le r < 1$)
$endgroup$
– reuns
Dec 7 '18 at 12:55
|
show 3 more comments
0
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$begingroup$
What do you mean by 'orthogonal' in $L_p$?.
$endgroup$
– Kavi Rama Murthy
Dec 4 '18 at 23:20
$begingroup$
You are right, I just corrected it! Orthogonality only makes sense when $p=2$.
$endgroup$
– Dr Potato
Dec 4 '18 at 23:23
1
$begingroup$
Closure under what topology?
$endgroup$
– Mike Miller
Dec 6 '18 at 23:13
1
$begingroup$
There is something wrong with your literal formulation. E.g., the way you're writing it, the $e^{iw}$ are just numbers, not functions. Don't you want to talk about functions $psi_n(x)=e^{2pi inx}$ on $[0,1]$, instead? If you rewrite in such terms, your question would be much clearer...
$endgroup$
– paul garrett
Dec 7 '18 at 0:31
1
$begingroup$
That ${ u_n}$ is dense in $X,|.|$ doesn't mean that for any element $f in X$ there is a sequence of complex numbers $(c_n)$ such that $lim_{N to infty} |f-sum_{n=1}^N c_n u_n| = 0$ ! It means that for any element $f in X$ there is $(c_{n,N})$ such that $lim_{N to infty} |f-sum_{n=1}^N c_{n,N} u_n| = 0$ (try with $u_n(x) = x^n$ then $sum_{n=1}^infty c_n u_n$ converges in $L^q([0,1])$ iff it converges uniformly to an analytic function for $|x| le r < 1$)
$endgroup$
– reuns
Dec 7 '18 at 12:55