Shape-preserving multivariate polynomial approximation inC[−1,1]m

Shape-preserving multivariate polynomial approximation in C[-1, 1]m

where C > 0 is independent of f , n1, and n2, satisfying, moreover, the following shapepreserving properties: (i) if f is convex of order (0,0), then so is Pn1,n1(f ); (ii) if f is simultaneously convex of orders (−1,0), (0,−1), and (0,0) (i.e., totally upper monotone), then so is Pn1,n2(f ); (iii) if f is convex of order (1,1), then so is Pn1,n2(f ); (iv) if f is simultaneously convex of order...

Shape - Preserving Multivariate Polynomial Approximation in C [ − 1 , 1 ]

where C > 0 is independent of f , n1, and n2, satisfying, moreover, the following shapepreserving properties: (i) if f is convex of order (0,0), then so is Pn1,n1(f ); (ii) if f is simultaneously convex of orders (−1,0), (0,−1), and (0,0) (i.e., totally upper monotone), then so is Pn1,n2(f ); (iii) if f is convex of order (1,1), then so is Pn1,n2(f ); (iv) if f is simultaneously convex of order...

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. Intro...

Shape preserving histogram approximation

We present a new method for reconstructing the density function underlying a given histogram. First we analyze the univariate case taking the approximating function in a class of quadratic-like splines with variable degrees. For the analogous bivariate problem we introduce a new scheme based on the Boolean sum of univariate B-splines and show that for a proper choice of the degrees, the splines...

Lectures on multivariate polynomial approximation