Effect of rescaling on the Laplacian

Jun 2013
Hello all,

My question is on Fourier series and the Laplacian. I'm working in cylinderical coordinates.

Imagine a function depending on the radius and azimuth: A(r,phi). That function is periodic over 2*pi/N (phi in [0,2*pi/N]).
Now I want to rescale that function so that it becomes periodic over delta (< 2*pi/N) and thus: phi in [0,delta]. Note that the function is not compressed, we're just getting rid of whatever information is between delta and 2*pi/N. I use Fouries series to do so, the Fourier constants are thus defined by an integration of A(r,phi)cos(k*2*pi/delta*phi) and A(r,phi)sin(k*2*pi/delta*phi).

Assume that the laplacian of the original function A(r,phi) is zero, will this also be true for the "rescaled" function?

Thank you very much,
Bert Hannon