The Remez Exchange Algorithm for Approximation with Linear Restrictions

This paper demonstrates a Remez exchange algorithm applicable to approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions. As special cases are included the notion of restricted derivatives approximation (examples of which are monotone and convex approximation and restricted range approximation) and the notion of approximation with restrictions at poised Birkhoff data (examples of which are bounded coefficients approximation, e-interpolator approximation, and polynomial approximation with restrictions at Her mite and "FergusonAt kinson-Sharma" data and pyramid matrix data). Furthermore the exchange procedure is completely simplified in all the cases of approximation with restrictions at poised Birkhoff data. Also results are obtained in the cases of general linear restrictions where the Haar condition prevails. In the other cases (e.g., monotone approximation) the exchange in general requires essentially a matrix inversion, although insight into the exchange is provided and partial alternation results are obtained which lead to simplifications.