On Montel and Montel–Popoviciu theorems in several variables

Abstract

Abstract. We present an elementary proof of a general version of Montel's theorem in several variables which is based on the use of tensor product polynomial interpolation. We also prove a Montel-Popoviciu's type theorem for functions f : R d → R for d > 1. Furthermore, our proof of this result is also valid for the case d = 1, differing in several points from Popoviciu's original proof. Finally, we demonstrate that our results are optimal.

On certain generalizations of the Levi-Civita and Wilson functional equations

“…The equation (1) can be formulated also for distributions, since the shift operator τ y : f pxq Þ Ñ f px`yq and dilation operator σ b : f pxq Þ Ñ f pbxq can be extended to the space DpR d q 1 of Schwartz complex-valued distributions (as the adjoint of the corresponding operators on DpR d q). Our results in this setting extend the Anselone-Korevaar [9] theorem on finite-dimensional shift-invariant subspaces of DpR d q 1 and the results of [22], [25] (see also [1] - [8]). They show in particular that the continuity restrictions on solutions and coefficients of (1) can be weakened at least to local integrability.…”

Characterization of functions whose forward differences are exponential polynomials

“…Let us state two technical results, which are important for our arguments in this section. These results were, indeed, recently introduced by the author, and have proved their usefulness for the study of several Montel-type theorems for polynomial and exponential polynomial functions (see, e.g., [1]- [5]). We include the proofs for the sake of completeness.…”

Characterization of polynomials as solutions of certain functional equations

“…Qpyqq, which implies that ∆ hm¨¨¨∆h2 ∆ h1 P pxq and ∆´c´1 m hm¨¨¨∆´c´1 2 h2 ∆´c´1 1 h1 Qpyq are both constant functions. In particular, taking h m`1 P R d and applying the operator ∆ phm`1,0q to both sides of the equation, we get ∆ hm`1 ∆ hm¨¨¨∆h2 ∆ h1 P pxq¨1pyq " 0 and, analogously, if we apply ∆ p0,hm`1q to both sides of the equation, we get 1pxq¨∆ hm`1 ∆´c´1 m hm¨¨¨∆´c´1 2 h2 ∆´c´1 1 h1 Qpyq " 0 Thus ∆ hm`1 ∆ hm¨¨¨∆h2 ∆ h1 P pxq " ∆ hm`1 ∆´c´1 m hm¨¨¨∆´c´1 2 h2 ∆´c´1 1 h1 Qpyq " 0 for all h 1 ,¨¨¨, h m`1 in R d (with equality in the sense of DpR d q 1 ), and the result follows from the corresponding Fréchet's type theorem for distributions, which is known (see again [1], [2], [5] or [7]). Note that, if we assume Q " 0 in the hypotheses of the theorem, then equation (16) takes the form 0 " p∆ hm¨¨¨∆h2 ∆ h1 P pxqq¨1pyq, which leads to ∆ hm¨¨¨∆h2 ∆ h1 P pxq " 0 and henceforth, in that case, P is a polynomial of degree ď m´1.…”