A simple unified method for the realization of generalized splines by using the matrix sweep algorithm (Q1921808)

Let \(L\) be a differential operator \[ L\equiv{d^n\over dx^n}+a_{n-1}{d^{n-1}\over dx^{n-1}}+\cdots+a_1{d\over dx}+a_0, \] where \(a_0\), \(a_1,\dots,a_{n-1}\) are constants. Let \(L^*\) be the conjugate operator to the operator \(L\). Let \(a=x_0<x_1<\cdots<x_N=b\) be the set of knots and let \(f_k\), \(k=0,1,\dots,N\), be a set of real numbers. The author proposes a method for the construction of a generalized spline \(S(x)\in C^{2(n-1)}[a,b]\) of order \(2n-1\), which satisfies the following conditions: 1) \(S(x_k)=f_k\), \(k=0,1,\dots,N\); 2) on each segment \([x_k,x_{k+1}]\), \(k=0,1,\dots,N-1\), the basis function of the spline \(S(x)\) is the solution of the differential equation \(L^*LS=0\). The author offers a special algorithm for the construction of the spline and investigates its stability.