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Approximations of the Identity

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2084)

Abstract

In this chapter, we study approximations of the identity on \({\mathbb{R}}^{D}\) with the measures μ satisfying (0.0.1). To this end, we first introduce an important notion of coefficients δ(Q, R) for cubes Q and R in \({\mathbb{R}}^{D}\). It turns out that δ(Q, R) characterizes the geometric relationship between Q and R. Using this notion, we further study cubes of different generations in terms of δ(Q, R), which are versions of dyadic cubes in the setting \(({\mathbb{R}}^{D},\vert \cdot \vert,\mu )\). Then we construct the functions, \(\{f_{y,\,k}\}_{y\in \mathrm{\,supp\,}\mu,\ k\in \mathbb{Z}}\), which originate the kernels of approximations of the identity. Via these functions, we introduce and establish some important properties of the approximations of the identity on \({\mathbb{R}}^{D}\).

Keywords

  • Dyadic Cubes
  • Geometric Relationship
  • Double Cube
  • Initial Cube
  • Sense Modulo

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    See, for example, [40].

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© 2013 Springer International Publishing Switzerland

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Yang, D., Yang, D., Hu, G. (2013). Approximations of the Identity. In: The Hardy Space H1 with Non-doubling Measures and Their Applications. Lecture Notes in Mathematics, vol 2084. Springer, Cham. https://doi.org/10.1007/978-3-319-00825-7_2

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