Does the Polygonal Confinement Theorem hold on the set of entire functions?
The Polygonal Confinement Theorem can be found in Section 2 of this paper by Rosenthal. I am interested in a generalization of Lemma 3.1 in the paper, which states:
$textbf{Lemma 3.1:} $ If $v_1,ldots,v_m in mathbb{R}^n$, and $left|displaystyle sum_{i=1}^m v_iright|<epsilon$, $|v_j|<epsilon$ for all $j$, then there is a constant $C$ which does not depend on $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that for each $1leq jleq m$,
$$ left|sum_{i=1}^j v_iright| leq Cepsilon.$$
I am wondering whether the following analogue for entire functions holds. In the question below, $|f|_R$ denotes the supremum of $f$ on the disk $|z|leq R$.
$textbf{Question:}$
Suppose $R>0$ and $f_i$ are entire functions on $mathbb{C}$. Let $epsilon>0$. If $displaystyle left| sum_{i=1}^m f_i right|_R< epsilon$ and $displaystyle |f_j|_R < epsilon$ for each $j$, does there exist a constant $C$ independent of $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that whenever $1leq jleq m$,
$$ left| sum_{i=1}^j f_{sigma(i)} right|_R< Cepsilon?$$
Lemma 3.1, called the rearrangement theorem by Rosenthal and the Steinitz lemma by others, was the crucial element for proving the Levy-Steinitz theorem (found in the Rosenthal paper). The Levy-Stenitz theorem was generalized to metrizable nuclear topological vector spaces by Banaszczyk in this paper. Since the space of entire functions is a Frechet space, and hence a nuclear space, I am hoping that the analgoue of Lemma 3.1 holds. In fact, it looks like the Corollary in the Banaszczyk paper on page 196 may be what I'm looking for. But the generality and technicality of the paper is well beyond my expertise.
Any insight or references would be most appreciated.
complex-analysis topological-vector-spaces
asked yesterday
user122916
576314
add a comment |
The Polygonal Confinement Theorem can be found in Section 2 of this paper by Rosenthal. I am interested in a generalization of Lemma 3.1 in the paper, which states:
$textbf{Lemma 3.1:} $ If $v_1,ldots,v_m in mathbb{R}^n$, and $left|displaystyle sum_{i=1}^m v_iright|<epsilon$, $|v_j|<epsilon$ for all $j$, then there is a constant $C$ which does not depend on $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that for each $1leq jleq m$,
$$ left|sum_{i=1}^j v_iright| leq Cepsilon.$$
I am wondering whether the following analogue for entire functions holds. In the question below, $|f|_R$ denotes the supremum of $f$ on the disk $|z|leq R$.
$textbf{Question:}$
Suppose $R>0$ and $f_i$ are entire functions on $mathbb{C}$. Let $epsilon>0$. If $displaystyle left| sum_{i=1}^m f_i right|_R< epsilon$ and $displaystyle |f_j|_R < epsilon$ for each $j$, does there exist a constant $C$ independent of $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that whenever $1leq jleq m$,
$$ left| sum_{i=1}^j f_{sigma(i)} right|_R< Cepsilon?$$
Lemma 3.1, called the rearrangement theorem by Rosenthal and the Steinitz lemma by others, was the crucial element for proving the Levy-Steinitz theorem (found in the Rosenthal paper). The Levy-Stenitz theorem was generalized to metrizable nuclear topological vector spaces by Banaszczyk in this paper. Since the space of entire functions is a Frechet space, and hence a nuclear space, I am hoping that the analgoue of Lemma 3.1 holds. In fact, it looks like the Corollary in the Banaszczyk paper on page 196 may be what I'm looking for. But the generality and technicality of the paper is well beyond my expertise.
Any insight or references would be most appreciated.
complex-analysis topological-vector-spaces
asked yesterday
user122916
576314
add a comment |
The Polygonal Confinement Theorem can be found in Section 2 of this paper by Rosenthal. I am interested in a generalization of Lemma 3.1 in the paper, which states:
$textbf{Lemma 3.1:} $ If $v_1,ldots,v_m in mathbb{R}^n$, and $left|displaystyle sum_{i=1}^m v_iright|<epsilon$, $|v_j|<epsilon$ for all $j$, then there is a constant $C$ which does not depend on $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that for each $1leq jleq m$,
$$ left|sum_{i=1}^j v_iright| leq Cepsilon.$$
I am wondering whether the following analogue for entire functions holds. In the question below, $|f|_R$ denotes the supremum of $f$ on the disk $|z|leq R$.
$textbf{Question:}$
Suppose $R>0$ and $f_i$ are entire functions on $mathbb{C}$. Let $epsilon>0$. If $displaystyle left| sum_{i=1}^m f_i right|_R< epsilon$ and $displaystyle |f_j|_R < epsilon$ for each $j$, does there exist a constant $C$ independent of $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that whenever $1leq jleq m$,
$$ left| sum_{i=1}^j f_{sigma(i)} right|_R< Cepsilon?$$
Lemma 3.1, called the rearrangement theorem by Rosenthal and the Steinitz lemma by others, was the crucial element for proving the Levy-Steinitz theorem (found in the Rosenthal paper). The Levy-Stenitz theorem was generalized to metrizable nuclear topological vector spaces by Banaszczyk in this paper. Since the space of entire functions is a Frechet space, and hence a nuclear space, I am hoping that the analgoue of Lemma 3.1 holds. In fact, it looks like the Corollary in the Banaszczyk paper on page 196 may be what I'm looking for. But the generality and technicality of the paper is well beyond my expertise.
Any insight or references would be most appreciated.
complex-analysis topological-vector-spaces
asked yesterday
user122916
576314
The Polygonal Confinement Theorem can be found in Section 2 of this paper by Rosenthal. I am interested in a generalization of Lemma 3.1 in the paper, which states:
$textbf{Lemma 3.1:} $ If $v_1,ldots,v_m in mathbb{R}^n$, and $left|displaystyle sum_{i=1}^m v_iright|<epsilon$, $|v_j|<epsilon$ for all $j$, then there is a constant $C$ which does not depend on $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that for each $1leq jleq m$,
$$ left|sum_{i=1}^j v_iright| leq Cepsilon.$$
I am wondering whether the following analogue for entire functions holds. In the question below, $|f|_R$ denotes the supremum of $f$ on the disk $|z|leq R$.
$textbf{Question:}$
Suppose $R>0$ and $f_i$ are entire functions on $mathbb{C}$. Let $epsilon>0$. If $displaystyle left| sum_{i=1}^m f_i right|_R< epsilon$ and $displaystyle |f_j|_R < epsilon$ for each $j$, does there exist a constant $C$ independent of $m$ and a permutation $sigma$ of ${1,ldots,m}$ such that whenever $1leq jleq m$,
$$ left| sum_{i=1}^j f_{sigma(i)} right|_R< Cepsilon?$$
Lemma 3.1, called the rearrangement theorem by Rosenthal and the Steinitz lemma by others, was the crucial element for proving the Levy-Steinitz theorem (found in the Rosenthal paper). The Levy-Stenitz theorem was generalized to metrizable nuclear topological vector spaces by Banaszczyk in this paper. Since the space of entire functions is a Frechet space, and hence a nuclear space, I am hoping that the analgoue of Lemma 3.1 holds. In fact, it looks like the Corollary in the Banaszczyk paper on page 196 may be what I'm looking for. But the generality and technicality of the paper is well beyond my expertise.
Any insight or references would be most appreciated.
complex-analysis topological-vector-spaces
complex-analysis topological-vector-spaces
asked yesterday
user122916
576314
asked yesterday
user122916
576314
asked yesterday
user122916
576314
asked yesterday
user122916
576314
asked yesterday
user122916
576314
576314
add a comment |
add a comment |
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