The Parabolic Wave

Consider a continuous and differentiable periodic function built with segments of quadratic functions, given by the sine series

$$S_{\rm s} = \frac{32}{\pi^{3}} \sum_{j=0}^{\infty} \frac{1}{k^{3}}\, \sin(k\theta),$$

where $k=2j+1$. The corresponding FC series is then

$$S_{\rm c} = \frac{32}{\pi^{3}} \sum_{j=0}^{\infty} \frac{1}{k^{3}}\, \cos(k\theta),$$

where $k=2j+1$. Note that due to the factors of $1/k^{3}$ these series are already absolutely and uniformly convergent. The complex power series $S_{z}$ is given by

$$S_{z} = \frac{32}{\pi^{3}} \sum_{j=0}^{\infty} \frac{1}{k^{3}}\, z^{k},$$

where $k=2j+1$, of which the two DP Fourier series above are the real and imaginary parts on the unit circle.