'"vi* Dl-82-0538 MEAN-SQUARE APPROXIMATION BY GENERALIZED RATIONAL FUNCTIONS

The problem of numerical analysis to which this study is directed is that of determining an optimum approximation (in the least squares sense) to a given function f by a function of the form p/q, where p and q are confined to certain prescribed linear spaces. The analogous approximation problem employing the uniform norm has received much recent attention. See, for example, [1, 2, 3, 4, 5, 9]. To our knowledge no investigation of the present problem has appeared. ~^mgpr*mtmm ■ jm —n mmm iJI — The problem of numerical analysis to which this study is directed is that of determining an optimum approximation (in the least squares sense) to a given function f by a function of the form p/q, where p and q are confined to certain prescribed linear spaces. The analogous approximation problem employing the uniform norm has received much recent attention. See, for example, [1, 2, 3, 4, 5, 9]. To our knowledge no investigation of the present problem has appeared. The exact setting of the problem will be as follows. We consider the linear space, C[a,b], of all continuous real-valued functions defined on a fixed, closed, interval [a,b]. In many of the results, the reader will observe that the domain [a,b] can be replaced by an arbitrary compact measure space. In C[a,b] we consider two norms fll^ = max lf(x) a = /f(x)g(x)w(x)dx.