Mixing
Norm convergence of Fourier series in the dual of a Sobolev space
$begingroup$
If $mathbb{T}$ is the 1-torus and $1<p<infty$, then for every $f$ in the Sobolev space $W^{1,p}(mathbb{T})$ we have that the Fourier series of $f$ converges in the $W^{1,p}(mathbb{T})$ to $f$.
From this, we can deduce that the Fourier series of every $fin W^{-1,p}(mathbb{T}):=(W^{1,p}(mathbb{T}))'$ converges in the weak* topology of $W^{-1,p}(mathbb{T})$ to $f$. In fact, denoting $e_n(t):=e^{int}$ so that $hat f(n):=langle f,e_{-n}rangle$, we have that:
$$forall fin W^{-1,p}(mathbb{T}),forall varphi in W^{1,p}(mathbb{T}), langlesum_{n=-N}^Nhat{f}(n)e_n,varphirangle = sum_{n=-N}^Nhat{f}(n)hatvarphi(-n) \= langle f,sum_{n=-N}^Nhat{varphi}(-n)e_{-n}rangle = langle f,sum_{n=-N}^Nhat{varphi}(n)e_{n}rangle to langle f,varphirangle, Nto+infty.$$
What about the convergence in the norm topology of $W^{-1,p}(mathbb{T})$? I.e.
Is it true that $forall fin W^{-1,p}(mathbb{T}), |sum_{n=-N}^N hat f(n)e_n-f|_{W^{-1,p}(mathbb{T})}to 0, Nto+infty?$
I know from the fact that the Fourier transform is an isometry between $W^{-1,2}(mathbb{T})$ and ${ainmathbb{C}^{mathbb{Z}} | sum_{ninmathbb{Z}backslash{0}}frac{|a_n|^2}{|n|^2}<+infty}$ that the result is true for $p=2$, but what if $pin(1,2)cup(2,+infty)$?
fourier-series sobolev-spaces
asked Feb 2 at 23:18
BobBob
1,7051725
$endgroup$
add a comment |
$begingroup$
If $mathbb{T}$ is the 1-torus and $1<p<infty$, then for every $f$ in the Sobolev space $W^{1,p}(mathbb{T})$ we have that the Fourier series of $f$ converges in the $W^{1,p}(mathbb{T})$ to $f$.
From this, we can deduce that the Fourier series of every $fin W^{-1,p}(mathbb{T}):=(W^{1,p}(mathbb{T}))'$ converges in the weak* topology of $W^{-1,p}(mathbb{T})$ to $f$. In fact, denoting $e_n(t):=e^{int}$ so that $hat f(n):=langle f,e_{-n}rangle$, we have that:
$$forall fin W^{-1,p}(mathbb{T}),forall varphi in W^{1,p}(mathbb{T}), langlesum_{n=-N}^Nhat{f}(n)e_n,varphirangle = sum_{n=-N}^Nhat{f}(n)hatvarphi(-n) \= langle f,sum_{n=-N}^Nhat{varphi}(-n)e_{-n}rangle = langle f,sum_{n=-N}^Nhat{varphi}(n)e_{n}rangle to langle f,varphirangle, Nto+infty.$$
What about the convergence in the norm topology of $W^{-1,p}(mathbb{T})$? I.e.
Is it true that $forall fin W^{-1,p}(mathbb{T}), |sum_{n=-N}^N hat f(n)e_n-f|_{W^{-1,p}(mathbb{T})}to 0, Nto+infty?$
I know from the fact that the Fourier transform is an isometry between $W^{-1,2}(mathbb{T})$ and ${ainmathbb{C}^{mathbb{Z}} | sum_{ninmathbb{Z}backslash{0}}frac{|a_n|^2}{|n|^2}<+infty}$ that the result is true for $p=2$, but what if $pin(1,2)cup(2,+infty)$?
fourier-series sobolev-spaces
asked Feb 2 at 23:18
BobBob
1,7051725
$endgroup$
add a comment |
$begingroup$
If $mathbb{T}$ is the 1-torus and $1<p<infty$, then for every $f$ in the Sobolev space $W^{1,p}(mathbb{T})$ we have that the Fourier series of $f$ converges in the $W^{1,p}(mathbb{T})$ to $f$.
From this, we can deduce that the Fourier series of every $fin W^{-1,p}(mathbb{T}):=(W^{1,p}(mathbb{T}))'$ converges in the weak* topology of $W^{-1,p}(mathbb{T})$ to $f$. In fact, denoting $e_n(t):=e^{int}$ so that $hat f(n):=langle f,e_{-n}rangle$, we have that:
$$forall fin W^{-1,p}(mathbb{T}),forall varphi in W^{1,p}(mathbb{T}), langlesum_{n=-N}^Nhat{f}(n)e_n,varphirangle = sum_{n=-N}^Nhat{f}(n)hatvarphi(-n) \= langle f,sum_{n=-N}^Nhat{varphi}(-n)e_{-n}rangle = langle f,sum_{n=-N}^Nhat{varphi}(n)e_{n}rangle to langle f,varphirangle, Nto+infty.$$
What about the convergence in the norm topology of $W^{-1,p}(mathbb{T})$? I.e.
Is it true that $forall fin W^{-1,p}(mathbb{T}), |sum_{n=-N}^N hat f(n)e_n-f|_{W^{-1,p}(mathbb{T})}to 0, Nto+infty?$
I know from the fact that the Fourier transform is an isometry between $W^{-1,2}(mathbb{T})$ and ${ainmathbb{C}^{mathbb{Z}} | sum_{ninmathbb{Z}backslash{0}}frac{|a_n|^2}{|n|^2}<+infty}$ that the result is true for $p=2$, but what if $pin(1,2)cup(2,+infty)$?
fourier-series sobolev-spaces
asked Feb 2 at 23:18
BobBob
1,7051725
$endgroup$
If $mathbb{T}$ is the 1-torus and $1<p<infty$, then for every $f$ in the Sobolev space $W^{1,p}(mathbb{T})$ we have that the Fourier series of $f$ converges in the $W^{1,p}(mathbb{T})$ to $f$.
From this, we can deduce that the Fourier series of every $fin W^{-1,p}(mathbb{T}):=(W^{1,p}(mathbb{T}))'$ converges in the weak* topology of $W^{-1,p}(mathbb{T})$ to $f$. In fact, denoting $e_n(t):=e^{int}$ so that $hat f(n):=langle f,e_{-n}rangle$, we have that:
$$forall fin W^{-1,p}(mathbb{T}),forall varphi in W^{1,p}(mathbb{T}), langlesum_{n=-N}^Nhat{f}(n)e_n,varphirangle = sum_{n=-N}^Nhat{f}(n)hatvarphi(-n) \= langle f,sum_{n=-N}^Nhat{varphi}(-n)e_{-n}rangle = langle f,sum_{n=-N}^Nhat{varphi}(n)e_{n}rangle to langle f,varphirangle, Nto+infty.$$
What about the convergence in the norm topology of $W^{-1,p}(mathbb{T})$? I.e.
Is it true that $forall fin W^{-1,p}(mathbb{T}), |sum_{n=-N}^N hat f(n)e_n-f|_{W^{-1,p}(mathbb{T})}to 0, Nto+infty?$
I know from the fact that the Fourier transform is an isometry between $W^{-1,2}(mathbb{T})$ and ${ainmathbb{C}^{mathbb{Z}} | sum_{ninmathbb{Z}backslash{0}}frac{|a_n|^2}{|n|^2}<+infty}$ that the result is true for $p=2$, but what if $pin(1,2)cup(2,+infty)$?
fourier-series sobolev-spaces
fourier-series sobolev-spaces
asked Feb 2 at 23:18
BobBob
1,7051725
asked Feb 2 at 23:18
BobBob
1,7051725
edited Feb 3 at 17:05
Bob
asked Feb 2 at 23:18
BobBob
1,7051725
asked Feb 2 at 23:18
BobBob
1,7051725
asked Feb 2 at 23:18
BobBob
1,7051725
1,7051725
add a comment |
add a comment |
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