On the divergence of polynomial interpolation

On the divergence of polynomial interpolation

On the Divergence of Polynomial Interpolation in the Complex Plane

We extend the results in [1] and [2] from the divergence of Hermite–Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, we prove the following theorem: Let 1n = −1 ≤ t (n) 1 < · · · < t (n) n < 1. Let φ k be polynomials of arbitrary degree such that φ k (t (n) j ) = δk j . Then the Lebesgue function3n(x) = ∑n j=1 ...

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...

extensions of some polynomial inequalities to the polar derivative

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