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Symmetry in an almost periodic function

Fundamental period of two functionsInfinite sum of cosecant, $frac k{sin(a+ck)}$Can any periodic function be represented as a trigonometric series?Almost periodic functionsIf almost-periodic function is not identically zero, then it is not in L2Almost periodic functionAlmost periodic function with mean value zeroIs Heaviside step function or unit step function periodic?Besicovitch almost periodic functions with seminorm zeroMean value of an almost periodic functionQuasi-periodic sequence

6

0

$begingroup$

A comment under this answer suggests looking at the graph of $$f(t) = sin t + sin(sqrt 2 t) + sin(sqrt3 t),$$ and I did so, on the interval $0le tle 60.$ I was struck by a seeming near-symmetry, so I let $$g(t) = f(60-t)$$ and superimposed the graphs of $f$ and $g$ on each other and saw how close they are to each other. The correlation between $f$ and $g$ on that interval is more than $0.97.$ Is there some reason for that?

Remark: In one sense the answer is perfectly obvious and is that given by "Reese" below. Yet the fact that such small numbers of half-periods should be so close to each other, although it explains what we see, itself feels as if it calls for explanation.

trigonometric-series almost-periodic-functions

share|cite|improve this question

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  MichaelHardy: Since you are the tag creator, I though it might be useful to let you know that there is a post on meta suggesting the removal of the (coincindences) tag: math.meta.stackexchange.com/questions/27653/tag-management-2018/…
  $endgroup$
  – Martin Sleziak
  Jul 25 '18 at 18:38
  

  
  
  
  

add a comment | 

6

0

$begingroup$

A comment under this answer suggests looking at the graph of $$f(t) = sin t + sin(sqrt 2 t) + sin(sqrt3 t),$$ and I did so, on the interval $0le tle 60.$ I was struck by a seeming near-symmetry, so I let $$g(t) = f(60-t)$$ and superimposed the graphs of $f$ and $g$ on each other and saw how close they are to each other. The correlation between $f$ and $g$ on that interval is more than $0.97.$ Is there some reason for that?

Remark: In one sense the answer is perfectly obvious and is that given by "Reese" below. Yet the fact that such small numbers of half-periods should be so close to each other, although it explains what we see, itself feels as if it calls for explanation.

trigonometric-series almost-periodic-functions

share|cite|improve this question

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  MichaelHardy: Since you are the tag creator, I though it might be useful to let you know that there is a post on meta suggesting the removal of the (coincindences) tag: math.meta.stackexchange.com/questions/27653/tag-management-2018/…
  $endgroup$
  – Martin Sleziak
  Jul 25 '18 at 18:38
  

  
  
  
  

add a comment | 

$begingroup$

A comment under this answer suggests looking at the graph of $$f(t) = sin t + sin(sqrt 2 t) + sin(sqrt3 t),$$ and I did so, on the interval $0le tle 60.$ I was struck by a seeming near-symmetry, so I let $$g(t) = f(60-t)$$ and superimposed the graphs of $f$ and $g$ on each other and saw how close they are to each other. The correlation between $f$ and $g$ on that interval is more than $0.97.$ Is there some reason for that?

Remark: In one sense the answer is perfectly obvious and is that given by "Reese" below. Yet the fact that such small numbers of half-periods should be so close to each other, although it explains what we see, itself feels as if it calls for explanation.

trigonometric-series almost-periodic-functions

share|cite|improve this question

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

$endgroup$

share|cite|improve this question

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

share|cite|improve this question

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

edited Mar 13 at 4:47

Martin Sleziak

44.9k10121274

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

asked Feb 13 '17 at 1:16

Michael HardyMichael Hardy

1

1 Answer
1

active

oldest

votes

8

$begingroup$

$sqrt{2} cdot 60$ is almost exactly $27pi$; $sqrt{3} cdot 60$ is almost exactly $33pi$. And $60$ isn't far from $19pi$. And conveniently, $sin(npi - x) = sin(x)$ whenever $n$ is odd. So all three component functions come close to lining up under the transformation $t to 60 - t$, creating a cool little coincidence when you add them all together.

share|cite|improve this answer

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is all perfectly obvious, but still it's astonishing, partly because those multiples of the periods are so small.
  $endgroup$
  – Michael Hardy
  Feb 13 '17 at 2:11
  

  
  
  
  

add a comment | 

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8

$begingroup$

$sqrt{2} cdot 60$ is almost exactly $27pi$; $sqrt{3} cdot 60$ is almost exactly $33pi$. And $60$ isn't far from $19pi$. And conveniently, $sin(npi - x) = sin(x)$ whenever $n$ is odd. So all three component functions come close to lining up under the transformation $t to 60 - t$, creating a cool little coincidence when you add them all together.

share|cite|improve this answer

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is all perfectly obvious, but still it's astonishing, partly because those multiples of the periods are so small.
  $endgroup$
  – Michael Hardy
  Feb 13 '17 at 2:11
  

  
  
  
  

add a comment | 

8

$begingroup$

$sqrt{2} cdot 60$ is almost exactly $27pi$; $sqrt{3} cdot 60$ is almost exactly $33pi$. And $60$ isn't far from $19pi$. And conveniently, $sin(npi - x) = sin(x)$ whenever $n$ is odd. So all three component functions come close to lining up under the transformation $t to 60 - t$, creating a cool little coincidence when you add them all together.

share|cite|improve this answer

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is all perfectly obvious, but still it's astonishing, partly because those multiples of the periods are so small.
  $endgroup$
  – Michael Hardy
  Feb 13 '17 at 2:11
  

  
  
  
  

add a comment | 

$begingroup$

$sqrt{2} cdot 60$ is almost exactly $27pi$; $sqrt{3} cdot 60$ is almost exactly $33pi$. And $60$ isn't far from $19pi$. And conveniently, $sin(npi - x) = sin(x)$ whenever $n$ is odd. So all three component functions come close to lining up under the transformation $t to 60 - t$, creating a cool little coincidence when you add them all together.

share|cite|improve this answer

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338

$endgroup$

share|cite|improve this answer

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338

answered Feb 13 '17 at 1:22

ReeseReese

15.3k11338