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Finite difference calculus in the continuum

Dmitri Finkelshtein, Yuri Kondratiev, Eugene Lytvynov, Maria Joao Oliveira

Abstract

The paper describes known and new results about finite difference calculus on configuration spaces. We describe finite difference geometry on configuration spaces, connect finite difference operators with cannonical commutation relations, find explicit form for certain finite difference Markov generators on configuration spaces, and describe spaces of Newton series defined over the configuration spaces.

Finite difference calculus in the continuum

Abstract

The paper describes known and new results about finite difference calculus on configuration spaces. We describe finite difference geometry on configuration spaces, connect finite difference operators with cannonical commutation relations, find explicit form for certain finite difference Markov generators on configuration spaces, and describe spaces of Newton series defined over the configuration spaces.

Paper Structure

This paper contains 8 sections, 13 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Elements of spatial combinatorics
  3. Finite difference geometry in the continuum
  4. Finite difference operators and the canonical commutation relations
  5. Finite difference Markov generators
  6. Spaces of Newton series
  7. Banach spaces
  8. Nuclear spaces

Key Result

Theorem 2.1

(i) For $\omega \in \mathcal{D}' ({\mathbb{R}^d})$, In the special case $\omega=\gamma =\sum_{i\in\mathbb N}\delta_{x_i}\in\Gamma({\mathbb{R}^d})$, In particular, $\binom \gamma n$ is a symmetric (discrete) Radon measure on $({\mathbb{R}^d})^n$. (ii) For any $\omega_1,\omega_2\in\mathcal{D}'({\mathbb{R}^d})$,

Theorems & Definitions (29)

  • Theorem 2.1: Umbral
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • ...and 19 more