Sum over ghosts or use a diffrent formula for periodic cardinal sinus.

[jecampagne]

Hi, working for the variation of the Quintic kernel #1265 along the line of Gary & Daniel 2014 paper, I mention that the code below in Interpolant.cpp

    double Interpolant::xvalWrapped(double x, int N) const
    {
        // sum over all arguments x+jN that are within range.
        // Start by finding x+jN closest to zero
        double xdown = x - N*std::floor(x/N + 0.5);
        xassert(std::abs(xdown) <= N);
        if (xrange() <= N) {
            // This is the usual case.
            return xval(xdown);
        } else {
            double xup = xdown+N;
            double sum = 0.;
            while (std::abs(xdown) <= xrange()) {
                sum += xval(xdown);
                xdown -= N;
            }
            while (xup <= xrange()) {
                sum += xval(xup);
                xup += N;
            }
            return sum;
        }
    }

implements the following equation of the 2014 paper

But, discussing with Gary, I've mebtioned that in fact the true kernel (aka periodic cardinal sinus)

$$K^{\mathrm{ideal}}_u(\nu) = \frac{1}{N}\sum_{j=-N/2}^{N/2-1} e^{-2\pi i j \nu} = e^{i\pi \nu} \frac{\mathrm{sinc}(N \nu)}{\mathrm{sinc}(\nu)}$$$$

can be directly approximated by any "sinc" approximated kernel (noted ker) with the following generic python code
(of course this is not an optimized code)

def k_p(x,N,ker,xmax):
  x = np.abs(x)
  th=xmax/N
  return np.piecewise(x, [x<th,x>=th],[lambda x: ker(N*x)/ker(x), lambda x:0])

def rkWrapped(u,N,ker,xmax):
  return np.sign(0.5-np.floor(u+0.5)%2)*k_p(u-np.floor(u+0.5),N,ker,xmax)

def kernelWrapped(u,N,ker,umax):
  return np.exp(1j * np.pi * u) * kWrapped(u,N,ker,xmax)

and then one perform a single convolution (Eq. of 11 in the paper) with the wrapped kernel above

$$\tilde{F}(u) = K_x(u) \times \sum_{k=-N/2}^{N/2-1} \hat{a}_k\times \tilde{K}_u(u-k/N)$$

One can reproduce the figure of the paper using such convolution.

Gary says that : "it would be interesting to see numerically how accurate this is. I suppose it’s a speedup of a few since you don’t have to sum over the aliased frequencies any more."