Interpolatory properties of best rational \(L_ 1\)-approximations
The problem concerns the distance between interpolation points of the best approximation to a continuous function. For example, for approximation by polynomials of degree n on [-1,1] the maximal distance between interpolating points is of the order O(1/n). This same question is asked of the interpolating points from \(R_{m,n}\)- the quotients of polynomials of degree m in the numerator and n in the denominator. They find for example, that if r is a best \(L_ 1\) approximation to f on [-1,1] and (a,b) is contained in [-1,1] and arccos(a)-arccos(b) is less than \(\pi /(n-m+2)\) then r agrees with f on some point in (a,b). Furthermore this bound is, in general, best possible. Their other results involve functions defined on subsets of \(R^ 2\), and they have several pertinent examples. In general terms, they find that the greater the quantity m-n, the more densely distributed are the interpolating points.
scientific article; zbMATH DE number 4107360