Approximation of Step function?Is every real function $L_1$-approximable by a step function?Poles of analytic approximation of step functionCompute the following integral of the Heaviside step functionConvolution of two unit step functions and one dirac delta functionWhich solver method for solving indicator function in a QP problemIs Heaviside step function or unit step function periodic?Fourier series of Heaviside step function?Laplace Transform of $ te^2t$ using unit step functionGraph Of Step FunctionsProperties of Heaviside Function

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Approximation of Step function?


Is every real function $L_1$-approximable by a step function?Poles of analytic approximation of step functionCompute the following integral of the Heaviside step functionConvolution of two unit step functions and one dirac delta functionWhich solver method for solving indicator function in a QP problemIs Heaviside step function or unit step function periodic?Fourier series of Heaviside step function?Laplace Transform of $ te^2t$ using unit step functionGraph Of Step FunctionsProperties of Heaviside Function













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.










share|cite|improve this question









$endgroup$











  • $begingroup$
    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
















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.










share|cite|improve this question









$endgroup$











  • $begingroup$
    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














1












1








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.










share|cite|improve this question









$endgroup$




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













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 21 at 18:30









malleamallea

32119




32119











  • $begingroup$
    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

















  • $begingroup$
    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
















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




$begingroup$
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





$begingroup$
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




$begingroup$
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




$begingroup$
@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





$begingroup$
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











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