Bernoulli monosplines

Question

Please help me with Bernoulli monosplines. Let's consider $2\pi$-periodic cubic spline, which is consist from $N$ ranges $0<x_1<x_2<\cdots<x_N<2\pi$. We can introduce a periodic function on the interval $[0, 2\pi]$ by the formula: $$ D(u)=\sum\limits_{k=1}^{\infty}\frac{\cos(ku)}{k^4}. $$ For any spline $s \in S^{N}$ we can be unambiguously calculate breaks of the third derivative in the nodes of our grid. Spline is a polynomial of the third degree between nodes of grid, and the third derivative between nodes is constant. We can introduce the following notation: $$ \beta_i = s'''(u_i+0)-s'''(u_i-0); i = 1,\ldots,N; \beta_0 = 0 $$ According [Korneychuk, N.P., Splines in the Theory of Approximation], breaks of the third derivative at the nodes define the spline accurate to the integral over the interval $[0, 2 \pi]$: $$ s(u) = \beta_0 + \sum\limits_{k=1}^{N} \beta_i D(u-u_i) \tag{$*$} $$ I want to recostruct any $2\pi$-periodic function by representing expression. I programmed this algorithm at Matlab, but the problem is that the function $s(u) = \sin(u)$ is perfectly reconstructed with the help of (*), but other functions, such as $s(u) = \cos(u)$, does not reconstruct correctly. Thank you.