Dear all,The context of my problem is the numerical solution of an elliptic boundary value problem on [-1,1]^2 with zero Neumann boundary conditions (if you're interested I'm solving the Helmholtz equation) using a meshfree method with radial basis functions. If you don't know much about meshfree methods that's fine, just think of it as FEM but instead we're using a finite dimensional subspace spanned by non-polynomial basis functions which have circular, compact support.Anyway my problem is the evaluation of integrals over circular (as well as lens-shaped) domains. I have developed a good numerical integration scheme for integrals over circles that are fully contained inside [-1,1]^2. However, these circles aren't always fully contained in [-1,1]^2 so I need to multiply the integrand by a characteristic function of the square. The characteristic function is defined as follows: it's equal to 1 if a point is in the square and it's 0 otherwise.Doing so ruins the accuracy of my numerical integration scheme and I blame it on the fact that the characteristic function is discontinuous. One way of getting rid of this problem is to approximate it by a continuous function I guess, which isn't hard to do in 1D where the domain is an interval but in 2D it's not obvious at all.Any input/relevant literature on the matter will be appreciated and I'll acknowledge you in my paper!

So if I understand well your problem it is just a problem of filter ?If you want a continuous function for a square like function you could use this window :Square(x,y)=1/(1+(x/c)^(2*n))*1/(1+(y/c)^(2n))with c the half size of the side and n the exposant. The most n will be high the most you square function will look to a square no need for acknowledgment for that if it was your problem here the figure with n=7;Don't know if it helps

Ah that may be the sort of thing I want! I'm not sure if this will actually make my quadrature rule any better but it's a continuous approximation to the characteristic function which is what I've been looking for.Well I may have overstated how important my work is in order to get more replies: it's part of a masters thesis so not quite publishable material! But thank your for your help!