Necessary Conditions for Uniqueness in L ’ - Approximation

Let K be a compact subset of R " with K=~nt K. Necessary conditions on an n-dimensional subspace U, of C(K) are given so that for each f~ C(K) there exists a unique best L'(w)-approximation from U,,, for every fixed positive weight function w. Let K be a compact subset of R ". For convenience we assume that K= int K. W will denote the set of bounded, integrable functions on K for which inf(w(x): XE K} > 0, and @ the set of strictly positive continuous functions on K. By C(K) we mean the set of real-valued continuous functions with domain of definition K. U, will always denote an n-dimensional subspace of C(K). For w E W, the L'(w)-norm offE C(K) is defined by Ilf'll, = s, If(x)I w(x) dx. DEFINITION 1. We say that U, is a unicity space for w, w E W, if to each f~ C(K) there exists a unique best approximation to f from U, in the L'(w)-norm. Similarly we say that U,, is a unicity space jtir W (@) if U,, is a unicity space for w for all w E W (w E i?l). DEFINITION 2. For each f E C(K), we set Z(f) = {x: f(x) = O}. Similarly, for a set Fs C(K), we set Z(F) = {x: f(x) = 0 for all f~ J'}. DEFINITION 3. For a relatively open subset D of K, we denote by IDI the number (possibly infinite but countable) of the connected components of D. For given u E U,, we set M(u) = I K/Z(u)l. We fix an order on the connected components Aj= Ai of K/Z(u), and set K/Z(u) = UFf;) Ai.