Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The functions need to be evaluated quickly so the results I'm getting from mollification are to awkward to deal with...

Thanks in advance..

share|cite|improve this question
Wait... why did you remove the specific function? o_O – J. M. Oct 26 '10 at 7:04
Oh whoops... I was editing... that was a mistake... – alext87 Oct 26 '10 at 7:05
That's better... – alext87 Oct 26 '10 at 7:06
up vote 4 down vote accepted

You might want to try any of formulae 17 to 27 in the MathWorld page for the unit step function, since the function you're interested in is expressible as


share|cite|improve this answer

You can convolve your function with a bivariate Gaussian density kernel, $\Phi(x,y; \sigma_x, \sigma_y)$, and obtain a smooth representation. Then you can let the variance parameters $\sigma_x$ and $\sigma_y$ go to zero to obtain arbitrarily accurate approximation to $f(x,y)$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.