Brookes, Richard Gordon2015-07-272015-07-271987http://hdl.handle.net/10092/10705This report is concerned with the study of best uniform approximation to f E C[a,b) from the linear space generated by some finite subset U == {uo,u1, ... ,u} of C[a,b). n p* E span U such that By a best uniform approximation we mean max{jf(x) ~p*(x) j: x E [a,b]} = min{max{jf(x) -p(x) I x E [a,b]} : p E span u}. We explore, firstly, the case U = {l,x, ... ,xn}. It will be shown in Section 4 that in this situation each f E C[a,b) has a unique best approximation and for this best approximation there is a strong characterisation theorem. It is then natural to ask whether these results are true for a more general U = {u 0 ,u 1 , ••• ,u } . n If a strong type of linear independence known as the Haar condition is imposed on U then this will indeed turn out 1. to be the case. We will attempt to develop this condition using an approach more intuitively obvious than those found in many standard texts. When the Haar condition is not satisfied the problem rapidly becomes complicated and it appears that much work remains to be done in this area. A theorem concerning a particularly simple situation is given in Section 8.enCopyright Richard Gordon BrookesField of Research::01 - Mathematical Sciences::0101 - Pure Mathematics