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What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?
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Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $mathrm D:=<1$ for which there exist constants $a_0$, $a_1$, ... in $Bbb C$ and $n$ in $Bbb N$ such that $$f(z)=sum_k=0^infty a_kz^kquad(zinmathrm D)quadtextwithquad|a_k|<k^n$$for all sufficiently large $k$ in $Bbb N$. This set $R$ is a ring, commutative and without zero-divisors, under the standard arithmetic operations $f+g:zmapsto f(z)+g(z)$ and $fg:zmapsto f(z)g(z)$. What is the usual name for this ring?
Clearly there is an exact real analogue of this, with $mathrm D$ replaced by the open real interval $(-1,pmb,,1)$. If it has a name, that would also be of interest.
real-analysis complex-analysis ring-theory taylor-expansion analytic-functions
asked Jul 27 at 11:54
John Bentin
10.8k22350
add a comment |Â
up vote
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down vote
favorite
Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $mathrm D:=<1$ for which there exist constants $a_0$, $a_1$, ... in $Bbb C$ and $n$ in $Bbb N$ such that $$f(z)=sum_k=0^infty a_kz^kquad(zinmathrm D)quadtextwithquad|a_k|<k^n$$for all sufficiently large $k$ in $Bbb N$. This set $R$ is a ring, commutative and without zero-divisors, under the standard arithmetic operations $f+g:zmapsto f(z)+g(z)$ and $fg:zmapsto f(z)g(z)$. What is the usual name for this ring?
Clearly there is an exact real analogue of this, with $mathrm D$ replaced by the open real interval $(-1,pmb,,1)$. If it has a name, that would also be of interest.
real-analysis complex-analysis ring-theory taylor-expansion analytic-functions
asked Jul 27 at 11:54
John Bentin
10.8k22350
1
There are many interesting constructs and phenomena that have no established name.
– Lubin
Jul 27 at 14:34
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $mathrm D:=<1$ for which there exist constants $a_0$, $a_1$, ... in $Bbb C$ and $n$ in $Bbb N$ such that $$f(z)=sum_k=0^infty a_kz^kquad(zinmathrm D)quadtextwithquad|a_k|<k^n$$for all sufficiently large $k$ in $Bbb N$. This set $R$ is a ring, commutative and without zero-divisors, under the standard arithmetic operations $f+g:zmapsto f(z)+g(z)$ and $fg:zmapsto f(z)g(z)$. What is the usual name for this ring?
Clearly there is an exact real analogue of this, with $mathrm D$ replaced by the open real interval $(-1,pmb,,1)$. If it has a name, that would also be of interest.
real-analysis complex-analysis ring-theory taylor-expansion analytic-functions
asked Jul 27 at 11:54
John Bentin
10.8k22350
Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $mathrm D:=<1$ for which there exist constants $a_0$, $a_1$, ... in $Bbb C$ and $n$ in $Bbb N$ such that $$f(z)=sum_k=0^infty a_kz^kquad(zinmathrm D)quadtextwithquad|a_k|<k^n$$for all sufficiently large $k$ in $Bbb N$. This set $R$ is a ring, commutative and without zero-divisors, under the standard arithmetic operations $f+g:zmapsto f(z)+g(z)$ and $fg:zmapsto f(z)g(z)$. What is the usual name for this ring?
Clearly there is an exact real analogue of this, with $mathrm D$ replaced by the open real interval $(-1,pmb,,1)$. If it has a name, that would also be of interest.
real-analysis complex-analysis ring-theory taylor-expansion analytic-functions
asked Jul 27 at 11:54
John Bentin
10.8k22350
asked Jul 27 at 11:54
John Bentin
10.8k22350
asked Jul 27 at 11:54
John Bentin
10.8k22350
asked Jul 27 at 11:54
John Bentin
10.8k22350
10.8k22350
1
There are many interesting constructs and phenomena that have no established name.
– Lubin
Jul 27 at 14:34
add a comment |Â
1
There are many interesting constructs and phenomena that have no established name.
– Lubin
Jul 27 at 14:34
1
1
There are many interesting constructs and phenomena that have no established name.
– Lubin
Jul 27 at 14:34
There are many interesting constructs and phenomena that have no established name.
– Lubin
Jul 27 at 14:34
add a comment |Â
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1
There are many interesting constructs and phenomena that have no established name.
– Lubin
Jul 27 at 14:34