On the existence and uniqueness of M-splines
type: Item
instance of: scholarly article
title: On the existence and uniqueness of M-splines
zbMATH Open document ID: 0562.41013
DOI: 10.1016/0021-9045(85)90146-7
author: Vimala Abraham
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 3895587
zbMATH Keywords: M-splines
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90146-7
OpenAlex ID: W2000620097
cites work: Fonctions 'spline' definies sur un ensemble convexe
cites work: M-splines
cites work: Konstruktion und algebraische Eigenschaften von M-Spline- Interpolierenden
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description: scientific article; zbMATH DE number 3895587