Best Approximation in Spaces of Continuously Differentiable Functions.

Let C('k){a,b} denote the space of continuously differentiable functions on the interval {a,b}, of the real line (//R), let (VBAR)(VBAR)q(VBAR)(VBAR) be any norm in C('k){a,b}. If M is a finite dimensional subspace of (C('k){a,b}, (VBAR)(VBAR)(,(.))(VBAR)(VBAR)(,k)), the local compactness of M guarantees the existence of at least one function g in M, such that (VBAR)(VBAR)f - g(VBAR)(VBAR)(,k) = d(f,M) = inf{(VBAR)(VBAR)f - h(VBAR)(VBAR)(,k): h (epsilon) M}, g is called a best approximation to f from M. The question of uniqueness of best approximation of functions in C('k){a,b}, from finite dimensional subspaces, has been investigated in several papers, with respect to several norms, for example,. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). In this thesis we introduce a new norm and derive results on the problem of best approximation with respect to this norm. It is defined as follows. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). With respect to (VBAR)(VBAR)(,(.))(VBAR)(VBAR)(,k), we will quickly notice that, finding the best approximation to f (epsilon) C('k){a,b} from (pi)(,n), the spaces of polynomials of degree (LESSTHEQ) n, reduces to finding the best approximation to f('(k)) from (pi)(,n -k); when n > k, with respect to (VBAR)(VBAR)(,(.))(VBAR)(VBAR)(,(INFIN)). And the subspaces (pi)(,n) are Chebyshev with respect to (VBAR)(VBAR)(,(.))(VBAR)(VBAR)(,k), for m (epsilon) N. Also, we will find conditions on the n-dimensional subspace M of C('k){a,b}, and on the point of f (epsilon) C('k){a,b}, that reduces the computation of the distance. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). to the distance of a point in (//R)('k+1) to a subspace of dimension (LESSTHEQ) n in (//R)('k+1), with respect to. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). We will also study subspaces where one of the following conditions is satisfied: h (epsilon) P(,m)(x) if and only if h('(k)) (epsilon) P(,m('(k)))(x('(k))). or h (epsilon) P(,m)(x) if and only if h('(k)) (epsilon) P(,m('(k)))(x('(k))). Then we will characterize finite dimensional Chebyshev subspaces of (C('k){a,b}, (VBAR)(VBAR)(,(.))(VBAR)(VBAR)(,k)), and we will find a property similar to the Haar property that we will call condition (H). We will also define a property derived from condition (H), that we will call condition (A). And we will study subspaces satisfying either condition H or condition (A).