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On a criterion for a cutoff regularization in the coordinate representation

A. V. Ivanov

Abstract

The paper discusses an applicability criterion for a cutoff regularization in the coordinate representation in the Euclidean space with a dimension larger than two. It is shown that the set of functions satisfying the criterion is not empty. As an example, an explicit function is presented. It is proved by explicit construction that there are functions satisfying the criterion in a stronger formulation.

On a criterion for a cutoff regularization in the coordinate representation

Abstract

The paper discusses an applicability criterion for a cutoff regularization in the coordinate representation in the Euclidean space with a dimension larger than two. It is shown that the set of functions satisfying the criterion is not empty. As an example, an explicit function is presented. It is proved by explicit construction that there are functions satisfying the criterion in a stronger formulation.
Paper Structure (3 sections, 4 theorems, 35 equations, 1 figure)

This paper contains 3 sections, 4 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Conclusion

Key Result

Lemma 1

Taking into account all the above, the applicability criterion 31-5 can be equivalently represented by the condition where and $J_{n/2-1}(\cdot)$ is the Bessel function of the first kind.

Figures (1)

  • Figure 1: The function $\mathbf{f}_n(s)$ for $n\in\{3,4,5,6\}$ and $s\in[0,1]$. In the picture $\mathbf{f}_3(0)<\mathbf{f}_4(0)<\mathbf{f}_5(0)<\mathbf{f}_6(0)$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof