Multivariate polynomial approximation in the hypercube

Lectures on multivariate polynomial approximation

Obtaining exact interpolation multivariate polynomial by approximation

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In order to improve the situation, exactly interpolating methods are often proposed for the exact results and approximate interpolating methods for the approximate...

Locating good points for multivariate polynomial approximation

Locating good points for multivariate polynomial approximation, in particular interpolation, is an open challenging problem, even in standard domains. One set of points that is always good, in theory, is the so-called Fekete points. They are defined to be those points that maximize the (absolute value of the) Vandermonde determinant on the given compact set. However, these are known analyticall...

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...