Positivity 2: 369–377, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
A Method Which Generates Splines in H-Locally
Convex Spaces and Connections with Vectorial
Romanian Academy of Scientists, Bacãu State University, Department of Mathematical Sciences,
B-dul Traian nr. 11, bl. A1, sc. A, apt. 13 5600 - Piatra Neamt, Romania
(Received 1 October 1997; Accepted 19 May 1998)
Abstract. In this research paper we present a modality for generating splines in H-locally convex
spaces which allows us to solve some problems of best approximation by linear subspaces of spline
functions in these spaces. In this way one shows that the elements of best vectorial approximation
coincide with the spline functions introduced by us in a previous research work. These splines
are also the only elements of best simultaneous approximation by their generated linear subspaces
with respect to any family of seminorms which induces the H-locally convex topology and, conse-
quently, they are the only solutions for some frequent strong and vectorial optimization programs.
Moreover, as we shall see in the numerical examples, our construction leads to discover orthogonal
decompositions for H-locally convex spaces which, in general, are difﬁcult to be identiﬁed.
Mathematics Subject Classiﬁcation (1991): 41A15.
Key words:: H-locally convex space, spline function, best simultaneous (vectorial) approximation,
efﬁcient point, simultaneous strictly convex space.
It is known that the concept of H-locally convex space was introduced and studied
for the ﬁrst time by Th. Precupanu  and deﬁned as any Hausdorff locally convex
space with the seminorms satisfying the parallelogram law. On the other hand, we
introduced the notion of spline function in H-locally convex spaces  and we es-
tablished the basic properties of approximation and optimal interpolation for these
splines. Our splines are natural extensions in H-locally convex spaces of the usual
abstract splines which appear in a Hilbert space like the minimizing elements for
a seminorm subject to the restrictions given by set of linear continuous functionals
because they satisfy the corresponding best approximation property simultaneously
with respect to a family of seminorms (Section 1). We shall see in Section 2 that the
simultaneous minimal properties of the splines deﬁned by us persuade them to be
the only solutions for the vectorial optimization problems considered there. Section
3 is devoted to the examples which show that our method supplies a new way
to obtained the 2m-th order cardinal Hermite splines well-known from the spline