What are the almost periodic functions on the complex plane?
$begingroup$
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally compact group under addition).
In particular, I am trying to figure out if there exists a non-constant almost periodic function $f$ on $mathbb{C}$ such that $f$ is invariant under rotations i.e. $f(tz) = f(z)$ for all $tin mathbb{T}$, $zin mathbb{C}$.
Any help is much appreciated.
fa.functional-analysis fourier-analysis topological-groups abelian-groups almost-periodic-function
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
asked Nov 24 '18 at 0:13
MerryMerry
1161
$endgroup$
add a comment |
$begingroup$
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally compact group under addition).
In particular, I am trying to figure out if there exists a non-constant almost periodic function $f$ on $mathbb{C}$ such that $f$ is invariant under rotations i.e. $f(tz) = f(z)$ for all $tin mathbb{T}$, $zin mathbb{C}$.
Any help is much appreciated.
fa.functional-analysis fourier-analysis topological-groups abelian-groups almost-periodic-function
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
asked Nov 24 '18 at 0:13
MerryMerry
1161
$endgroup$
add a comment |
$begingroup$
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally compact group under addition).
In particular, I am trying to figure out if there exists a non-constant almost periodic function $f$ on $mathbb{C}$ such that $f$ is invariant under rotations i.e. $f(tz) = f(z)$ for all $tin mathbb{T}$, $zin mathbb{C}$.
Any help is much appreciated.
fa.functional-analysis fourier-analysis topological-groups abelian-groups almost-periodic-function
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
asked Nov 24 '18 at 0:13
MerryMerry
1161
$endgroup$
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally compact group under addition).
In particular, I am trying to figure out if there exists a non-constant almost periodic function $f$ on $mathbb{C}$ such that $f$ is invariant under rotations i.e. $f(tz) = f(z)$ for all $tin mathbb{T}$, $zin mathbb{C}$.
Any help is much appreciated.
fa.functional-analysis fourier-analysis topological-groups abelian-groups almost-periodic-function
fa.functional-analysis fourier-analysis topological-groups abelian-groups almost-periodic-function
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
asked Nov 24 '18 at 0:13
MerryMerry
1161
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
asked Nov 24 '18 at 0:13
MerryMerry
1161
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
edited Nov 24 '18 at 0:23
Arun Debray
3,15611442
3,15611442
asked Nov 24 '18 at 0:13
MerryMerry
1161
asked Nov 24 '18 at 0:13
MerryMerry
1161
asked Nov 24 '18 at 0:13
MerryMerry
1161
1161
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $mathbb{R}^2$ these are the functions $e^{i(ax + by)}$.
I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $f$ is a rotationally invariant nonconstant almost periodic function. WLOG $f(0,0) =0$ and $f(1,0) = 1$. So $f$ is constantly $1$ on the unit circle. Now find $t > 1$ such that $f$ and its shift by $(t, 0)$ are uniformly at most $1/3$ apart. Then $f(t,0)$ is within $1/3$ of $0$, so the same must be true at any point on the circle of radius $t$ about the origin. But at the same time, $f$ must be within $1/3$ of $1$ on the circle of radius $1$ about $(t,0)$, and that is contradictory.
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $mathbb{R}^2$ these are the functions $e^{i(ax + by)}$.
I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $f$ is a rotationally invariant nonconstant almost periodic function. WLOG $f(0,0) =0$ and $f(1,0) = 1$. So $f$ is constantly $1$ on the unit circle. Now find $t > 1$ such that $f$ and its shift by $(t, 0)$ are uniformly at most $1/3$ apart. Then $f(t,0)$ is within $1/3$ of $0$, so the same must be true at any point on the circle of radius $t$ about the origin. But at the same time, $f$ must be within $1/3$ of $1$ on the circle of radius $1$ about $(t,0)$, and that is contradictory.
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
$endgroup$
add a comment |
$begingroup$
Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $mathbb{R}^2$ these are the functions $e^{i(ax + by)}$.
I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $f$ is a rotationally invariant nonconstant almost periodic function. WLOG $f(0,0) =0$ and $f(1,0) = 1$. So $f$ is constantly $1$ on the unit circle. Now find $t > 1$ such that $f$ and its shift by $(t, 0)$ are uniformly at most $1/3$ apart. Then $f(t,0)$ is within $1/3$ of $0$, so the same must be true at any point on the circle of radius $t$ about the origin. But at the same time, $f$ must be within $1/3$ of $1$ on the circle of radius $1$ about $(t,0)$, and that is contradictory.
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
$endgroup$
add a comment |
$begingroup$
Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $mathbb{R}^2$ these are the functions $e^{i(ax + by)}$.
I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $f$ is a rotationally invariant nonconstant almost periodic function. WLOG $f(0,0) =0$ and $f(1,0) = 1$. So $f$ is constantly $1$ on the unit circle. Now find $t > 1$ such that $f$ and its shift by $(t, 0)$ are uniformly at most $1/3$ apart. Then $f(t,0)$ is within $1/3$ of $0$, so the same must be true at any point on the circle of radius $t$ about the origin. But at the same time, $f$ must be within $1/3$ of $1$ on the circle of radius $1$ about $(t,0)$, and that is contradictory.
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
$endgroup$
Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $mathbb{R}^2$ these are the functions $e^{i(ax + by)}$.
I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $f$ is a rotationally invariant nonconstant almost periodic function. WLOG $f(0,0) =0$ and $f(1,0) = 1$. So $f$ is constantly $1$ on the unit circle. Now find $t > 1$ such that $f$ and its shift by $(t, 0)$ are uniformly at most $1/3$ apart. Then $f(t,0)$ is within $1/3$ of $0$, so the same must be true at any point on the circle of radius $t$ about the origin. But at the same time, $f$ must be within $1/3$ of $1$ on the circle of radius $1$ about $(t,0)$, and that is contradictory.
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
edited Nov 24 '18 at 2:09
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
answered Nov 24 '18 at 1:30
Nik WeaverNik Weaver
21.1k147128
21.1k147128
add a comment |
add a comment |
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