greg@endor.harvard.edu (Greg) (10/19/86)
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.
----
gregregreggreg@endor.harvard.edu (Greg) (10/21/86)
In article <471@husc6.HARVARD.EDU> greg@endor.UUCP (Greg) writes: >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 from P to R 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)? Here are the essentials of an article that I stupidly posted to net.math rather than sci.math. I also wish to make a minor correction: I'm convinced that if f has degree k+1, f usually doesn't have enough degrees of freedom to satisfy all of the above conditions. So I'll change the question some: For what integer d (which may depend on k, n, and T) can we always find an f with the above properties such that f restricted to S is a polynomial of degree d? ---- Greg