Multivariate Lebesgue bounds for R-Leja lower grids

Abstract

We establish an algebraic upper bound for the Lebesgue constant of multivariate polynomial interpolation on $\mathfrak{R}$-Leja lower grids. For each fixed dimension $d\ge 2$ and every nonempty finite lower set $A\subset {\mathbb{N}}_0^d$, let $P_A$ denote the corresponding grid with $N=|A|$ nodes. We prove $\Lambda (P_A)\le C_d{N}^{{\beta }_d}$, where $q_d=d/(d-1)$ and ${\beta }_d=1+q_d^{-1}{\log }_2(1+2^{q_d})$. The exponent satisfies ${\beta }_2=2.160964\dots$ and increases to ${\log }_{2}6=2.584962\dots$ as $d\to \infty$, improving both the cubic estimate and the exponent $3-1/d$ obtained from quadratic prefix bounds. Our proof exploits the binary structure of known univariate estimates to control sums of powers of Lebesgue constants over interpolation prefixes. A weighted Hölder inequality and a disjoint-box packing argument then transfer these prefix-moment bounds to arbitrary lower sets. The resulting estimate is uniform over all lower-set shapes and choices of disk Leja maximizers, with constants depending only on $d$.