What are the almost periodic functions on the complex plane?











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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.










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    up vote
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    down vote

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    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.










    share|cite|improve this question









    New contributor




    Merry is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      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.










      share|cite|improve this question









      New contributor




      Merry is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      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






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      edited 4 hours ago









      Arun Debray

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          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.






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            up vote
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            down vote













            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.






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              up vote
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              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.






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                up vote
                3
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                up vote
                3
                down vote









                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.






                share|cite|improve this answer














                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.







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                edited 2 hours ago

























                answered 3 hours ago









                Nik Weaver

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