On partial polynomial interpolation

On partial polynomial interpolation

The Alexander-Hirschowitz theorem says that a general collection of k double points in P imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says ...

On multivariate polynomial interpolation

We provide a map Θ 7→ ΠΘ which associates each finite set Θ of points in C with a polynomial space ΠΘ from which interpolation to arbitrary data given at the points in Θ is possible and uniquely so. Among all polynomial spaces Q from which interpolation at Θ is uniquely possible, our ΠΘ is of smallest degree. It is also Dand scale-invariant. Our map is monotone, thus providing a Newton form for...

Lectures on Multivariate Polynomial Interpolation

Polynomial Interpolation

Consider a family of functions of a single variable x: Φ(x; a0, a1, . . . , an), where a0, . . . , an are the parameters. The problem of interpolation for Φ can be stated as follows: Given n + 1 real or complex pairs of numbers (xi, fi), i = 0, . . . , n, with xi 6= xk for i 6= k, determine a0, . . . , an such that Φ(xi; a0, . . . , an) = fi, i = 0, . . . , n. The above is a linear interpolatio...

On Polynomial Interpolation on the Unit Ball

Polynomial interpolation on the unit ball of R d has a unique solution if the points are located on several spheres inside the ball and the points on each sphere solves the corresponding interpolation problem on the sphere. Furthermore, the solution can be computed in a recursive way.