The degree of shape preserving weighted polynomial approximation

We analyze the degree of shape preserving weighted polynomial approximation for exponential weights on the whole real line. In particular, we establish a Jackson type estimate. Keywords: Shape Preserving Polynomials, k-Monotone, Exponential Weights, Jackson Theorem, Freud Weights. AMS Classi…cation: 41A29, 41A17 Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399 1. Introduction Shape preserving polynomial approximation has been an active research topic for decades. There are many interesting features, and a great many complex examples, and exceptional cases. Perhaps the oldest modern result is due to O. Shisha [14]. For continuous f : [ 1; 1]! R, let En [f ] = inf deg(P ) n kf PkL1[ 1;1] : In addition, let E n [f ] = inf deg(P ) n n kf PkL1[ 1;1] : P monotone in [ 1; 1] o : Shisha [14] essentially proved that when f 0 is non-negative and continuous, for n 1; (1.1) E n [f ] 2En 1 f 0 : This simple estimate is disappointing, in that one loses a factor of 1 n , when compared to Jackson-Favard estimates. However, it is best possible in the class of functions to which it applies [13]. Similar results hold for convex functions, and more generally, k-monotone functions. Recall that a function f is called k-monotone, if for any distinct x0; x1; :::; xk in the interval of de…nition, [x0; x1; :::; xk; f ] = k X i=0 f (xi) !0 (xi) 0; Date : August 2, 2011. 1 2 DANY LEVIATAN AND DORON S. LUBINSKY