Mixing
Convergence of Fourier series in $alpha$-Hölder norm.
$begingroup$
For $alphain(0,1]$, define the $alpha$-Hölder space on the 1-torus $mathbb{T}:=mathbb{R}/2pimathbb{Z}$ as the space:
$$C^{0,alpha}(mathbb{T}):={fin C(mathbb{T}) | sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}<+infty}.$$
Define:
$$|cdot|_{C^{0,alpha}(mathbb{T})}:C^{0,alpha}(mathbb{T})to[0,+infty), fmapsto |f|_infty+sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}.$$
Then $(C^{0,alpha}(mathbb{T}),|cdot|_{C^{0,alpha}(mathbb{T})})$ is a Banach space.
Define $$forall ninmathbb{Z}, forall tinmathbb{T},e_n(t):=e^{int}$$ and if $fin L^1(mathbb{T})$, the Fourier tranform of $f$ will be denoted by $hat{f}$, i.e. $$forall ninmathbb{Z},hat{f}(n)=int_{-pi}^pi f(t)e_{-n}(t)frac{operatorname{d}t}{2pi}.$$
I know from the answers to this question that:
$$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_infty to0, Nto +infty$$
While, since $C^{0,1}(mathbb{T})$ is essentially the Sobolev space $W^{1,infty}(mathbb{T})$, I know that: $$exists fin C^{0,1}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,1}(mathbb{T})}notto0, Nto+infty.$$
For which $alphain(0,1)$, is it true that $$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,alpha}(mathbb{T})}to0, Nto +infty?$$
fourier-series holder-spaces
asked Jan 28 at 22:27
BobBob
1,7101725
$endgroup$
add a comment |
$begingroup$
For $alphain(0,1]$, define the $alpha$-Hölder space on the 1-torus $mathbb{T}:=mathbb{R}/2pimathbb{Z}$ as the space:
$$C^{0,alpha}(mathbb{T}):={fin C(mathbb{T}) | sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}<+infty}.$$
Define:
$$|cdot|_{C^{0,alpha}(mathbb{T})}:C^{0,alpha}(mathbb{T})to[0,+infty), fmapsto |f|_infty+sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}.$$
Then $(C^{0,alpha}(mathbb{T}),|cdot|_{C^{0,alpha}(mathbb{T})})$ is a Banach space.
Define $$forall ninmathbb{Z}, forall tinmathbb{T},e_n(t):=e^{int}$$ and if $fin L^1(mathbb{T})$, the Fourier tranform of $f$ will be denoted by $hat{f}$, i.e. $$forall ninmathbb{Z},hat{f}(n)=int_{-pi}^pi f(t)e_{-n}(t)frac{operatorname{d}t}{2pi}.$$
I know from the answers to this question that:
$$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_infty to0, Nto +infty$$
While, since $C^{0,1}(mathbb{T})$ is essentially the Sobolev space $W^{1,infty}(mathbb{T})$, I know that: $$exists fin C^{0,1}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,1}(mathbb{T})}notto0, Nto+infty.$$
For which $alphain(0,1)$, is it true that $$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,alpha}(mathbb{T})}to0, Nto +infty?$$
fourier-series holder-spaces
asked Jan 28 at 22:27
BobBob
1,7101725
$endgroup$
$begingroup$
What is the explicit bound in the Dirichlet kernel + Holder continuity proof of the convergence of Fourier series ? If $f$ is $alpha$-Holder continuous $1$-periodic then $|f(x)-f ast D_N(x)| le h_{alpha,N}( |f|_infty,sup_{x,y} frac{|f(x)-f(y)|}{(x-y)^alpha})$ where $D_N(x) =sum_{|n| le N} e^{2i pi nx} $, $f ast D_N(x)= int_0^1 f(y) D_n(x-y)dy=sum_{|n| le N} hat{f}(n)e^{2i pi nx}$. I'm asking the exact form of that function $h_{alpha,N}$.
$endgroup$
– reuns
Jan 29 at 0:22
$begingroup$
I think that an explicit form for $h_{alpha,N}$ can be deduced from my answer in the linked question. It is hidden inside the constant $C$ in the last estimate: it is enough to unpack the other constants which $C$ is built from
$endgroup$
– Bob
Jan 29 at 0:31
add a comment |
$begingroup$
For $alphain(0,1]$, define the $alpha$-Hölder space on the 1-torus $mathbb{T}:=mathbb{R}/2pimathbb{Z}$ as the space:
$$C^{0,alpha}(mathbb{T}):={fin C(mathbb{T}) | sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}<+infty}.$$
Define:
$$|cdot|_{C^{0,alpha}(mathbb{T})}:C^{0,alpha}(mathbb{T})to[0,+infty), fmapsto |f|_infty+sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}.$$
Then $(C^{0,alpha}(mathbb{T}),|cdot|_{C^{0,alpha}(mathbb{T})})$ is a Banach space.
Define $$forall ninmathbb{Z}, forall tinmathbb{T},e_n(t):=e^{int}$$ and if $fin L^1(mathbb{T})$, the Fourier tranform of $f$ will be denoted by $hat{f}$, i.e. $$forall ninmathbb{Z},hat{f}(n)=int_{-pi}^pi f(t)e_{-n}(t)frac{operatorname{d}t}{2pi}.$$
I know from the answers to this question that:
$$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_infty to0, Nto +infty$$
While, since $C^{0,1}(mathbb{T})$ is essentially the Sobolev space $W^{1,infty}(mathbb{T})$, I know that: $$exists fin C^{0,1}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,1}(mathbb{T})}notto0, Nto+infty.$$
For which $alphain(0,1)$, is it true that $$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,alpha}(mathbb{T})}to0, Nto +infty?$$
fourier-series holder-spaces
asked Jan 28 at 22:27
BobBob
1,7101725
$endgroup$
For $alphain(0,1]$, define the $alpha$-Hölder space on the 1-torus $mathbb{T}:=mathbb{R}/2pimathbb{Z}$ as the space:
$$C^{0,alpha}(mathbb{T}):={fin C(mathbb{T}) | sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}<+infty}.$$
Define:
$$|cdot|_{C^{0,alpha}(mathbb{T})}:C^{0,alpha}(mathbb{T})to[0,+infty), fmapsto |f|_infty+sup_{s,tinmathbb{T}\sneq t}frac{|f(s)-f(t)|}{|s-t|^alpha}.$$
Then $(C^{0,alpha}(mathbb{T}),|cdot|_{C^{0,alpha}(mathbb{T})})$ is a Banach space.
Define $$forall ninmathbb{Z}, forall tinmathbb{T},e_n(t):=e^{int}$$ and if $fin L^1(mathbb{T})$, the Fourier tranform of $f$ will be denoted by $hat{f}$, i.e. $$forall ninmathbb{Z},hat{f}(n)=int_{-pi}^pi f(t)e_{-n}(t)frac{operatorname{d}t}{2pi}.$$
I know from the answers to this question that:
$$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_infty to0, Nto +infty$$
While, since $C^{0,1}(mathbb{T})$ is essentially the Sobolev space $W^{1,infty}(mathbb{T})$, I know that: $$exists fin C^{0,1}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,1}(mathbb{T})}notto0, Nto+infty.$$
For which $alphain(0,1)$, is it true that $$forall fin C^{0,alpha}(mathbb{T}), |sum_{n=-N}^N hat{f}(n)e_n-f|_{C^{0,alpha}(mathbb{T})}to0, Nto +infty?$$
fourier-series holder-spaces
fourier-series holder-spaces
asked Jan 28 at 22:27
BobBob
1,7101725
asked Jan 28 at 22:27
BobBob
1,7101725
edited Jan 29 at 21:20
Bob
asked Jan 28 at 22:27
BobBob
1,7101725
asked Jan 28 at 22:27
BobBob
1,7101725
asked Jan 28 at 22:27
BobBob
1,7101725
1,7101725
$begingroup$
What is the explicit bound in the Dirichlet kernel + Holder continuity proof of the convergence of Fourier series ? If $f$ is $alpha$-Holder continuous $1$-periodic then $|f(x)-f ast D_N(x)| le h_{alpha,N}( |f|_infty,sup_{x,y} frac{|f(x)-f(y)|}{(x-y)^alpha})$ where $D_N(x) =sum_{|n| le N} e^{2i pi nx} $, $f ast D_N(x)= int_0^1 f(y) D_n(x-y)dy=sum_{|n| le N} hat{f}(n)e^{2i pi nx}$. I'm asking the exact form of that function $h_{alpha,N}$.
$endgroup$
– reuns
Jan 29 at 0:22
$begingroup$
I think that an explicit form for $h_{alpha,N}$ can be deduced from my answer in the linked question. It is hidden inside the constant $C$ in the last estimate: it is enough to unpack the other constants which $C$ is built from
$endgroup$
– Bob
Jan 29 at 0:31
add a comment |
$begingroup$
What is the explicit bound in the Dirichlet kernel + Holder continuity proof of the convergence of Fourier series ? If $f$ is $alpha$-Holder continuous $1$-periodic then $|f(x)-f ast D_N(x)| le h_{alpha,N}( |f|_infty,sup_{x,y} frac{|f(x)-f(y)|}{(x-y)^alpha})$ where $D_N(x) =sum_{|n| le N} e^{2i pi nx} $, $f ast D_N(x)= int_0^1 f(y) D_n(x-y)dy=sum_{|n| le N} hat{f}(n)e^{2i pi nx}$. I'm asking the exact form of that function $h_{alpha,N}$.
$endgroup$
– reuns
Jan 29 at 0:22
$begingroup$
I think that an explicit form for $h_{alpha,N}$ can be deduced from my answer in the linked question. It is hidden inside the constant $C$ in the last estimate: it is enough to unpack the other constants which $C$ is built from
$endgroup$
– Bob
Jan 29 at 0:31
$begingroup$
What is the explicit bound in the Dirichlet kernel + Holder continuity proof of the convergence of Fourier series ? If $f$ is $alpha$-Holder continuous $1$-periodic then $|f(x)-f ast D_N(x)| le h_{alpha,N}( |f|_infty,sup_{x,y} frac{|f(x)-f(y)|}{(x-y)^alpha})$ where $D_N(x) =sum_{|n| le N} e^{2i pi nx} $, $f ast D_N(x)= int_0^1 f(y) D_n(x-y)dy=sum_{|n| le N} hat{f}(n)e^{2i pi nx}$. I'm asking the exact form of that function $h_{alpha,N}$.
$endgroup$
– reuns
Jan 29 at 0:22
$begingroup$
What is the explicit bound in the Dirichlet kernel + Holder continuity proof of the convergence of Fourier series ? If $f$ is $alpha$-Holder continuous $1$-periodic then $|f(x)-f ast D_N(x)| le h_{alpha,N}( |f|_infty,sup_{x,y} frac{|f(x)-f(y)|}{(x-y)^alpha})$ where $D_N(x) =sum_{|n| le N} e^{2i pi nx} $, $f ast D_N(x)= int_0^1 f(y) D_n(x-y)dy=sum_{|n| le N} hat{f}(n)e^{2i pi nx}$. I'm asking the exact form of that function $h_{alpha,N}$.
$endgroup$
– reuns
Jan 29 at 0:22
$begingroup$
I think that an explicit form for $h_{alpha,N}$ can be deduced from my answer in the linked question. It is hidden inside the constant $C$ in the last estimate: it is enough to unpack the other constants which $C$ is built from
$endgroup$
– Bob
Jan 29 at 0:31
$begingroup$
I think that an explicit form for $h_{alpha,N}$ can be deduced from my answer in the linked question. It is hidden inside the constant $C$ in the last estimate: it is enough to unpack the other constants which $C$ is built from
$endgroup$
– Bob
Jan 29 at 0:31
add a comment |
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$begingroup$
What is the explicit bound in the Dirichlet kernel + Holder continuity proof of the convergence of Fourier series ? If $f$ is $alpha$-Holder continuous $1$-periodic then $|f(x)-f ast D_N(x)| le h_{alpha,N}( |f|_infty,sup_{x,y} frac{|f(x)-f(y)|}{(x-y)^alpha})$ where $D_N(x) =sum_{|n| le N} e^{2i pi nx} $, $f ast D_N(x)= int_0^1 f(y) D_n(x-y)dy=sum_{|n| le N} hat{f}(n)e^{2i pi nx}$. I'm asking the exact form of that function $h_{alpha,N}$.
$endgroup$
– reuns
Jan 29 at 0:22
$begingroup$
I think that an explicit form for $h_{alpha,N}$ can be deduced from my answer in the linked question. It is hidden inside the constant $C$ in the last estimate: it is enough to unpack the other constants which $C$ is built from
$endgroup$
– Bob
Jan 29 at 0:31