Interpolation by quadratic spline with periodic derivative on uniform meshes
Let \(0=x_ 0<x_ 1<...<x_{N+1}=1\) be an equidistant subdivision and \(\{f_ 0,f'_ 1,...,f'_ N,f_{N+1}\}\) be an arbitrary set of real numbers. The authors construct a quadratic spline interpolant s with periodic derivatives: \(s\in C^ 1[0,1]\), s is a quadratic polynomial in each subinterval \([x_ i,x_{i+1}]\), \(s'(0)=s'(1)\), such that \(s'(x_ i)=f'_ i\) for \(i=1,...,N\), \(s(0)=f_ 0\) and \(s(x_{N+1})=f_{N+1}.\) The existence and uniqueness of such a spline is proved. Error estimations in the uniform norm for the function and its derivatives in the case of a smooth function f, \(f\in C^ k[0,1]\), \(k=2,3\) are given (in this case, \(f_ i=f(x_ i)\), \(f'_ i=f'(x_ i))\). Two numerical test examples are given, too. Periodic splines using the nodal values \(f_ i\) \((i=0,1,...,N+1)\) have already been studied by \textit{F. Dubeau} and \textit{J. Savoie} [J. Approx. Theory 39, 77-88 (1983; Zbl 0516.41005) and 44, 43-54 (1985; Zbl 0622.41002)].
scientific article; zbMATH DE number 4180628