On the existence and uniqueness of M-splines
This paper gives necessary and sufficient conditions for the existence and uniqueness of M-splines defined by \textit{T. R. Lucas} [ibid. 5, 1-14 (1972; Zbl 0227.41002)] who gave two sets of sufficient conditions for the existence of M-splines. Let X be a real Hilbert space and let M(x,y) be a continuous bilinear functional on \(X\times X\). Suppose that A is a closed subspace of X such that \(M(a,a)\geq 0\), \(\forall a\in A\). There exists a continuous linear transformation T from X to A such that \(M(x,a)=<Tx,a>\), \(\forall x\in X\), \(\forall a\in A\). Let \(A^{\perp}\) be the orthogonal complement of A in X. In this paper it is proved that for each \(a^{\perp}\in A^{\perp}\) there exists an M-spline in \(\Phi (a^{\perp};A)=a^{\perp}+A\) if and only if \(T(A^{\perp})\subset TA\). A necessary and sufficient condition for the M-spline in \(\Phi(a^{\perp};A)\) to be unique is \(\hat A=\{\hat a\in A\mid M(\hat a,a)=0,\) \(\forall a\in A\}=0.\) A necessary condition for the M-spline interpolation problem to have a solution is \(M(x,\hat a)=0\), \(\forall x\in X\), \(\forall \hat a\in \hat A\). A number of sufficient conditions for the existence and uniqueness of M-splines are derived.
scientific article; zbMATH DE number 3895587
scientific article