how to initialize function values in Fourier space

I'm trying to learn classical density functional theory and learning to numerical implementation FMT functional which invovled convolutions of weight functions and density.

Some weight functions have the following forms: (R is constant,0.5.)

w2(r) = 2*R*(R^2-r^2)^-0.5*heaviside(R-r)

I am not quite sure about how to calculate the weight function values in Fourier space to avoid singularities in real space. My codes are here, thanks for your precious help.

clear
syms r y2(r) 
R = 0.5;
y2(r) =  2*R/sqrt(R^2-r^2)*heaviside(R-r);

fy2 = fourier(y2);
dx = 0.05;dy = 0.05;
N = 500; M = 500;
fw2 = zeros(500,500);
Norm = 1/(N*M);
for i = 1:N
    for j = 1:M
        if i <= N/2 
            kx = 2*pi*i/(N*dx);
        else
            kx = -2*pi*(N-i)/(N*dx);
        end
        if j <= M/2 
            ky = 2*pi*j/(M*dy);
        else
            ky = -2*pi*(M-j)/(M*dy);
        end
        
        k_abs = sqrt(kx^2+ky^2);
        if k_abs == 0
            fw2(i,j) = Norm*4*pi*R^2;
        else
            fw2(i,j) = fy2(k_abs);
        end
        
    end
end

Answer 1

[haohao gu]

to ensure the periodic boundary, weight function or other functions may used in later convolution as response singnal should be initiated from 4 directions instead of 2.

2d weight function in fourier space can be calculated as shown in fweight.m