On the other hand, every $p \in \mathcal{P}(X)$ satisfies an apparently stronger condition (cf. [DR2; Prop. 1]):
(2.21) Condition. For every subspace $M$ of $\mathbb{R}^s$ which is spanned by elements from $X$, and for every $v \in \mathbb{R}^s$,
Hence we conclude the following from (2.7) Theorem:
(2.22) COROLLARY. The space $\mathcal{P}(X)$ is characterized either by (2.20) Condition or by (2.21) Condition. In particular, these two conditions are equivalent on $\Pi$.
(2.22) Corollary verifies the claim made in [DR2; Remark 2] that (2.21) Condition characterizes $\mathcal{P}(X)$. In case $X$ consists of $N$ repetitions of $s+1$ vectors $Y \subset \mathbb{R}^s$ in general position, the characterization of $\mathcal{P}(X)$ by (2.21) Condition has been proved in [G] by other methods.
3. An associated polynomial interpolation problem. Here we identify certain exponential spaces $H$ whose corresponding “limit at the origin” $H_\downarrow$ coincides with $\mathcal{P}(X)$ (for an appropriate choice of $X$), and use this identification in the solution of an associated interpolation problem. The map
which associates with every finite-dimensional space of entire functions a homogeneous space of polynomials of the same dimension, has been introduced and studied in [BR1] in the context of a multivariate polynomial interpolation problem, and has been discussed as well in [BR2] in the context of kernels of polynomial ideals. To begin with, we recall the definition of $H_\downarrow$ and review some of the results of [BR1, 2] needed here. Then we discuss a certain interpolation problem and its relation to $\mathcal{P}(X)$.
Given a function $f \neq 0$ analytic at the origin, we write its power series expansion at the origin in the form
where, for each $j$, $f_j$ is a homogeneous polynomial of degree $j$, and define $f_\downarrow := f_k$ with $k = \min{j: f_j \neq 0}$; i.e., $f_\downarrow$ is the first non-trivial homogeneous polynomial in the power expansion of $f$. Using