Twice continuously differentiable semilocal smoothing spline / D. A. Silaev. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2009. № 5. P. 11-19
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 187-194].

Twice continuously differentiable periodic local and semilocal smoothing splines, or
S-splines from the class C^2 are considered.
These splines consist of polynomials of 5th degree, first three coefficients
of each polynomial are determined by values of the previous polynomial
and two its derivatives at the point of splice, coefficients at higher
terms of the polynomial are determined by the least squares method.
These conditions are supplemented by the periodicity condition for the
spline function on the whole segment of definition or by initial conditions.
Uniqueness and existence theorems are proved. Stability and convergence conditions
for these splines are established.