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From: greg@endor.harvard.edu (Greg)
Newsgroups: sci.math
Subject: Geometry/Algebraic Geometry query.
Message-ID: <471@husc6.HARVARD.EDU>
Date: Sun, 19-Oct-86 11:05:39 EDT
Article-I.D.: husc6.471
Posted: Sun Oct 19 11:05:39 1986
Date-Received: Tue, 21-Oct-86 21:30:18 EDT
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Reply-To: greg@endor.UUCP (Greg)
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Does anyone have references, knowledge, or ideas about the following class
of problems:

Let k be a non-negative integer.  Suppose you have n points in the plane
{(x_i,y_i)} such that if i>j, x_i>x_j.  There is any easy theorem that states
that there is a function f such that:

1) The k'th derivative of f is defined and continuous.
2) For all i f(x) restricted to [x_i,x_i+1] is a polynomial of degree k+1.
3) f(x_i) = y_i for all i.

My question is about possible generalizations (they would come in handy for
some computer graphics stuff that I'm doing).  Let T be a triangulation of the
interior of a polyhedron P in R^n.  Let g be a function from the vertices of T
to the reals, and let k be a non-negative integer.  Is there always a function
f such that:

1) The k'th derivative of f is defined and continuous.
2) For any n-dimensional simplex S of T, f restricted to S is a polynomial of
degree k+1.
3) For any vertex v of T, f(v) = y(v)?

For an even broader generalization, we may allow the faces of the simplices of
T to be algebraic surfaces of some degree (of course, we would need to allow
f to have degree >k+1 as well) rather than flat, and we could allow T to be
embedded in an algebraic surface of some degree rather than R^n.
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gregregreg