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Approximation of Step function?

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1

$begingroup$

Is it possible to approximate the step function?

f(x)=begincases
0 quadtext if xle 0\
1quad textotherwise
endcases

I want to implement it by polynomial approximation.

step-function

share|cite|improve this question

asked Mar 21 at 18:30

malleamallea

32119

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you want to approximate it on the whole line? Or a subset of the line?
  $endgroup$
  – Umberto P.
  Mar 21 at 18:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm considering the approximation for the whole line. It would be appreciated if you know subset version of it.
  $endgroup$
  – mallea
  Mar 21 at 18:43
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You cannot approximate a bounded function on the whole line with a polynomial, as a polynomial has the form $p(x)=a_nx^n+cdots$, and will go to $pminfty$ for $xrightarrowinfty$.
  $endgroup$
  – Alex R.
  Mar 21 at 18:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @AlexR. Thank you for your response!! Ok, then what about fixing the domain? Say, $-r leq x leq r$ for some integer $r$.
  $endgroup$
  – mallea
  Mar 21 at 18:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On a bounded domain, you can uniformly approximate any continuous function with polynomials to whatever accuracy you like. Of course, your function is not continuous at $0$, but it is the limit of a sequence of continuous functions (ones that rise linearly from $0$ up to $1$, for ever inceasing slopes). And thus you can approximate the function with sequence, and the sequence with polynomials. The convergence of the polynomials to your function will not be uniform at $0$, but it will converge.
  $endgroup$
  – Paul Sinclair
  Mar 22 at 2:35
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Is it possible to approximate the step function?

f(x)=begincases
0 quadtext if xle 0\
1quad textotherwise
endcases

I want to implement it by polynomial approximation.

step-function

share|cite|improve this question

asked Mar 21 at 18:30

malleamallea

32119

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you want to approximate it on the whole line? Or a subset of the line?
  $endgroup$
  – Umberto P.
  Mar 21 at 18:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm considering the approximation for the whole line. It would be appreciated if you know subset version of it.
  $endgroup$
  – mallea
  Mar 21 at 18:43
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You cannot approximate a bounded function on the whole line with a polynomial, as a polynomial has the form $p(x)=a_nx^n+cdots$, and will go to $pminfty$ for $xrightarrowinfty$.
  $endgroup$
  – Alex R.
  Mar 21 at 18:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @AlexR. Thank you for your response!! Ok, then what about fixing the domain? Say, $-r leq x leq r$ for some integer $r$.
  $endgroup$
  – mallea
  Mar 21 at 18:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On a bounded domain, you can uniformly approximate any continuous function with polynomials to whatever accuracy you like. Of course, your function is not continuous at $0$, but it is the limit of a sequence of continuous functions (ones that rise linearly from $0$ up to $1$, for ever inceasing slopes). And thus you can approximate the function with sequence, and the sequence with polynomials. The convergence of the polynomials to your function will not be uniform at $0$, but it will converge.
  $endgroup$
  – Paul Sinclair
  Mar 22 at 2:35
  
  

  
  
  
  

add a comment | 

$begingroup$

Is it possible to approximate the step function?

f(x)=begincases
0 quadtext if xle 0\
1quad textotherwise
endcases

I want to implement it by polynomial approximation.

step-function

share|cite|improve this question

asked Mar 21 at 18:30

malleamallea

32119

$endgroup$

share|cite|improve this question

asked Mar 21 at 18:30

malleamallea

32119

share|cite|improve this question

asked Mar 21 at 18:30

malleamallea

32119

asked Mar 21 at 18:30

malleamallea

32119

asked Mar 21 at 18:30

malleamallea

32119