Interpolation in normed spaces with applications to the approximate solution of nonlinear equations
by
Adrian Diaconu
"Babes-Bolyai" University, Faculty of Mathematics, Cluj-Napoca, Romania

In this talk we shall study some ways of extending the model of interpolating the real functions of a real variable with simple nodes to the case of the functions defined on and taking values in linear normed spaces.

A first model is based in a consequence of the well-known theorem of Hahn-Banach which ensures for any element a in X \{\theta_X}, where X is a linear normed space and \theta_Y is the null element of this space, the existence of a linear and continuous functional u:X --> R so that || u|| = 1 and u( a) = || a|| _X .

So if D subset or equal X,   f:D --> Y,    n in N   and x₀, x₁, ..., x_n in D   for any 

i, j in { 0, 1, .., n}   with i =/= j.  The linear and continuous functional

U_ij:X --> R   exists so that ||U_ij|| = 1   and U_ij( x_i-x_j) = ||x_i-x_j|| _X. Thus the abstract interpolation polynomial will be L( x₀, x₁, ...x_n) :X --> Y  with

L( x0, x1, ...xn) ( x) = \dsumi=0nli( x) f( xi)

and

li( x) = \dprod\Sb j=0 j =/= i \endSb n\dfracUij( xi-xj) || xi-xj|| X.

In order to keep as many characteristics as possible from the case of the interpolation of real functions, in next paragraph we present a model of construction of the abstract interpolation polynomials, based on the properties of multilinear mappings.

So that let us consider U in L( X, Y) and B in L₂( Y, Y) . Using these we will build the sequence ( A_n) _{n in N} where A_n in L_n(X, Y) through

A₁( u) = U( u)   for every u in X, and

An( u1, ..., un) = B( An-1(u1, ..., un-1) , U( un) )

 for every ( u₁, ..., u_n) in X  and n in N,   n >= 2.  Here for every n in N the space L_n(X, Y)  represent the set of linear and continous mappings from X to Y.   

We suppose that the mapping U in L( X, Y) and B in L₂( Y, Y) verifies certain conditions.

Given ( x_n) _{n in N} subset or equal D subset or equal X  for k, n in N we introduce the non-linear mappings w_{n, k}:X --> Y, defined by:

wk, n( x) = An+1( x-xk, x-xk+1, ..., xk+n)

and for any x in X, the mapping w_{k, n}^'( x) in L( X, Y) define by:

wk, n'( x) h=\dsumi=kk+nAn+1(x-xk, ..., x-xi-1, x-xi+1, ..., x-xk+n, h)

We introduce the mapping L( x₀, x₁, ..., x_n;f):X --> Y,

L( x0, x1, ..., xn;f) ( x) = =\dsumi=0nAn+1(x-x0, ..., x-xi-1, x-xi+1, ..., x-xn, [ w0, n'(xi) ] *-1  f( xi) ) ,

which is called ( U-B)   interpolation abstract polynomial of the function f  on the nodes  x₀, x₁, ..., x_n.

Here [ w_{0, n}^'( x_i) ] _*^-1 represent the invers,  in certain sense of w_{k, n}^'( x) in L( X, Y) .

The mapping [ x₀, x₁, ..., x_n;f] in L_n( X, Y)    defined by:

[ x0, x1, ..., xn;f]h1...hn=\dsumi=0nAn+1( h1, ..., hn;[w0, n'( xi) ] *-1f( xi))

is called the divided difference of the order n of the function f on the nodes x₀, x₁, ..., x_n.

We will establish interpolation formulas with the rest expressed as a divided difference. We give the example the type of interpolation polynomial built for non-linear mappings between spaces of functions defined on a certain interval.

Date received: February 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caen-15.