Unconditionality for $m$-homogeneous polynomials on $\ell _{\infty }^{n}$

UNCONDITIONALITY IN SPACES OF m-HOMOGENEOUS POLYNOMIALS

Let E be a Banach space with an unconditional basis. We prove that for m 2 the Banach space P(m E) of all m-homogeneous polynomials on E has an unconditional basis if and only if E is finite dimensional. This answers a problem of S. Dineen.

Two Results on Homogeneous Hessian Nilpotent Polynomials

Let z = (z1, · · · , zn) and ∆ = ∑n i=1 ∂ 2 ∂z i the Laplace operator. A formal power series P (z) is said to be Hessian Nilpotent(HN) if its Hessian matrix HesP (z) = ( ∂ 2 P ∂zi∂zj ) is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture(VC) of HN polynomials: for any homogeneous HN polynomial P (z) ...

Sampling of Homogeneous Polynomials

Conditions for reconstruction of multivariate homogeneous polynomials from sets of sample values are introduced, together with a frame-based method for explicitly obtaining the polynomial coefficients from the sample data.

Approximation by homogeneous polynomials

A new, elementary proof is given for the fact that on a centrally symmetric convex curve on the plane every continuous even function can be uniformly approximated by homogeneous polynomials. The theorem has been proven before by Benko and Kroó, and independently by Varjú using the theory of weighted potentials. In higher dimension the new method recaptures a theorem of Kroó and Szabados, which ...

Right gw-Majorization on M_{n,m}