Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

Let X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q ∈ 〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the set W(X)q:=∩ S∈S(X,q)〈S〉, which we call the non-uniqueness set of q. We study the case dimX=1 and the case X a Veronese embedding of ℙn. We conclude the paper with a few remarks concerning this problem over the reals.

Poincaré type inequality for Dirichlet spaces and application to the uniqueness set

We give an extension of Poincaré's type capacitary inequality for Dirichlet spaces and provide an application to study the uniqueness sets on the unit circle for these spaces.

On the Recovery of Analytic Functions

In this paper we consider questions of recapturing an analytic function in a Banach space from its values on a uniqueness set. The principal method is to use reproducing kernels to construct a sequence in the Banach space which converges in norm to the given functions. The method works for several classical Banach spaces of analytic functions including some Hardy and Bergman spaces.

On Uniqueness Sets for Expansions in Sequences of Functions Arising from Singular Generating Functions

Let {pn(z)}; be a sequence of functions analytic in a region D. A problem in analysis which has received much attention is the following: describe the sets Z ⊂ D for which(1)implies hn is 0 for all n, (To make the problem interesting, only those situations are studied where finite subsets of the pn(z) are linearly independent in D.) Another way of phrasing this is: Characterize the uniqueness sets of pn(z), a uniqueness set Z being a set in D such that the restriction of {pn(z)}; to Z is linearly independent. If Z is not a uniqueness set then for some {hn}; not all 0, we have(2)This formula is called a non-trivial representation of 0 (on Z).

A uniqueness set for all Hp(Bn) with p>0

For n≥2, a hypersurface in the open unit ball Bn in is constructed which satisfies the generalized Blaschke condition and is a uniqueness set for all Hp(Bn) with p>0. If n≥3, the hypersurface can be chosen to have finite area.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 32 A 10.