Boris Shekhtman

Ideal Interpolation: Translations to and from Algebraic Geometry

In this survey I will discuss four themes that surfaced in multivariate interpolation and seem to have analogues in algebraic geometry. The hope is that mixing these two areas together will benefit both.

Bivariate ideal projectors and their perturbations

In this paper we present a complete description of ideal projectors from the space of bivariate polynomials

Norming points and unique minimality of orthogonal projections

\mathbb{F}[x,y]

On discrete norms of polynomials

onto its subspace

Extension constants of unconditional two-dimensional operators

\mathbb{F }_{<n}[x,y]

On minimal interpolating projections and trace duality

of polynomials of degree less than n. Several applications are given. In particular, we study small perturbations of ideal projectors as well as the limits of Lagrange projectors. The latter results verify one particular case of a conjecture of Carl de Boor.

Covering by complements of subspaces

We study the norming points and norming functionals of symmetric operators on Lp spaces for p=2m or p=2m/(2m−1). We prove some general result relating uniqueness of minimal projection to the set of norming functionals. As a main application, we obtain that the Fourier projection onto span [1,sin⁡x,cos⁡x] is a unique minimal projection in Lp.

On the norms of interpolating operators

For a polynomial p of degree n < N we compare two norms: ||p||:= sup{|p(z)|: z ∈ C; |z| = 1} and ||p||N:= sup {|p(zj)|: j=0,..., N - 1}; zj = e2πi j/N . We show that there exist universal constants C1 and C2 such that 1 + C1 log (N/N - n) ≤ sup{||p||/||p||N: p ∈ Pn}, ≤ C2 log (N/N-n) + 1.

MINIMAL PROJECTIONS AND ABSOLUTE PROJECTION CONSTANTS FOR REGULAR POLYHEDRAL SPACES

It is shown that the (absolute) extension constant e(T) of an operator T such that Tvk = λkvk, k = 1, 2, for some unconditional basis (v1, v2) of a two-dimensional real normed space is less than or equal to λ1| + |λ2| + 2√λ21 − |λ1λ2| + λ22)3. In fact, it is demonstrated that e(T) is attained by exactly one unconditional two-dimensional space (up to an isometry).

Do the chain rules for matrix functions hold without commutativity

Abstract We construct a two-dimensional subspace V ⊂ C ( K ) such that an interpolating projection on V is a minimal projection with the norm >1. That answers a question posed by B. L. Chalmers. It also answers a question posed implicitly by a theorem of P. Morris and E. W. Cheney. We also give a quantitative generalization of the above mentioned theorem. As is suggested by the title, we use trace duality to obtain these results.