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This is a followup to my existing question: Question on Heaviside step function (distribution) identities

I was trying to show the equality between two Heaviside step functions using their derivatives.

The old question was:

For $c>0$, and in the sense of distributions:

First:

$$\frac {d}{dt} \theta (ct-z)=c\delta(ct-z)=\frac{1}{c} c\delta(t-\frac{z}{c})=\delta(t-\frac{z}{c})\tag{1}$$

Second:

$$\frac {d}{dt} \theta (t-\frac{z}{c})=\delta(t-\frac{z}{c})\tag{2}$$

So then

$$\theta (ct-z)=\theta (t-\frac{z}{c})\tag{3}$$

to within a constant of integration.

My question was: is that true?

The answer I got was YES.

MY QUESTION NOW IS: Does that mean that $\theta(f(x))$ is identical to $\theta(g(x))$ for any two functions as long as they have identical zeros? (Assuming the signs of their derivatives are the same at the zeros)

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user45664
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1 Answers1

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The primary necessity for $\theta \circ f$ to equal $\theta \circ g$ is $f(x)$ and $g(x)$ are negative for the same $x$ and positive for the same $x$ (modulo a null set).

If $f$ and $g$ are differentiable then this will occur if they have the same zeros and the same signs of the derivatives at the zeros. I here require the derivatives to be non-zero. The zero case is possible, but then we have to look at non-zero second derivatives. And so on.

The reason I write "non-zero derivatives" is that for example $\sin(x)^2$ and $-\sin(x)^2$ have the same zeros and the same derivatives at the zeros, but they have not the same sign: $\theta(\sin(x)^2)$ is a "square wave" while $\theta(-\sin(x)^2)$ vanishes.

md2perpe
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  • I think only the signs of the derivatives need be the same--not the derivatives themselves. Your criterion is too restrictive. – user45664 Oct 31 '19 at 21:27
  • @user45664. Are you referring to "and the same non-zero derivatives at the zeros"? If so, I agree. It should read something like "and the same signs of the non-zero derivatives at the zeros". This is still too restrictive; it excludes functions with zero derivatives at the zeros. – md2perpe Oct 31 '19 at 21:58
  • Please see my next question on this subject:https: //math.stackexchange.com/q/3417092/147776 – user45664 Oct 31 '19 at 22:06
  • update your answer here and i will accept it – user45664 Nov 01 '19 at 16:44
  • @user45664. Done. – md2perpe Nov 01 '19 at 16:51