Tube volumes via functional equations

Deniz, Ali; Koçak, Şahin; Özdemir, Yunus; Üreyen, A. Ersin

doi:10.1007/s00022-014-0241-3

Cited by 7 publications

(7 citation statements)

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“…One can easily check, by using the scaling property of the distance zeta function, that λA is strongly d-languid, for any λ ≥ 2 √ 3 and with κ d := −1. Hence, we can apply Theorem 3.1 in order to obtain the following exact pointwise tube formula, valid for all t ∈ (0, 1/2 √ 3), and which coincides with the one obtained in [27,28,29] and also, in [7]:…”

Section: Pointwise and Distributional Tube Formulas And A Criterion Fsupporting

confidence: 67%

“…Also, the exact tube formula stated in Theorem 3.1 has a counterpart with error term (much as in Theorem 3.2); see Remark 3.3(b). Finally, we refer to [38,37] and [41, §13.1] for many additional references on tube formulas (or related formulas) in various settings, including, [19,4,9,16,44,45,46,57,54,7,13,15,48,56,27,28,29,33,38,37,55,53,50,40].…”

Section: Pointwise and Distributional Tube Formulas And A Criterion F...mentioning

confidence: 99%

“…Furthermore, if A is only d-languid, we will have a distributional error term, with information about its asymptotic order given in the sense of [41, §5.4]. Namely, the distribution R ∈ D ′ (0, δ) is said to be of asymptotic order at most t α (resp., less than t α ) as t → 0 + if when applied to a test function ϕ ∈ D(0, δ), 7 we have that R, ϕ a = O(a α ) (resp., R, ϕ a = o(a α )), as a → 0 + , where ϕ a (t) := a −1 ϕ(t/a) (and the implicit constants may depend on ϕ). We then write that R(t) = O(t α ) (resp., R(t) = o(t α )) as t → 0 + .…”

Section: Pointwise and Distributional Tube Formulas And A Criterion F...mentioning

confidence: 99%

“…Here and in Theorem 3.2, we write ζA(s) instead of ζA(s; δ) since the residues in the formula do not depend on the parameter δ 7. Here, D(0, δ) := C ∞ c (0, δ) is the standard space of infinitely differentiable test functions with compact support.…”

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confidence: 99%

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Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

Lapidus

Radunović

Žubrinić

2019

Discrete &Amp; Continuous Dynamical Systems - S

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We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.1 The former statement is briefly justified at the end of this paper, while both statements are explained in detail in [37] and [33, Ch. 5]

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Section: Pointwise and Distributional Tube Formulas And A Criterion Fsupporting

confidence: 67%

Section: Pointwise and Distributional Tube Formulas And A Criterion F...mentioning

confidence: 99%

Section: Pointwise and Distributional Tube Formulas And A Criterion F...mentioning

confidence: 99%

mentioning

confidence: 99%

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Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

Lapidus

Radunović

Žubrinić

2019

Discrete &Amp; Continuous Dynamical Systems - S

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“…5].) Furthermore, we refer the interested reader to [LapRaŽu1,6] and §8.2 and §13.1] for additional references on tube formulas in various settings, including [DeKÖÜ,Fed1,Gra,HuLaWei,Schn,Zäh,Wey,LapLu,. (See also §13.1,§13.2 and §13.4].…”

Section: Fractal Tube Formulas and A Minkowski Measurability Criterionmentioning

confidence: 99%

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Lapidus

Radunović²,

Žubrinić³

2015

Fractal Geometry and Stochastics V

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In 2009, the first author introduced a class of zeta functions, called 'distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of 'tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of R N ).Keywords. Fractal set, fractal zeta functions, distance zeta function, tube zeta function, geometric zeta function of a fractal string, Minkowski content, Minkowski measurability, upper box (or Minkowski) dimension, complex dimensions of a fractal set, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, relative fractal drum, fractal tube formulas.

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Introduction

Lapidus¹,

Radunović²,

Žubrinić³

2017

Springer Monographs in Mathematics

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Tube volumes via functional equations

Abstract: We give a very simple approach to the computation of tube volumes of self-similar sprays using a functional equation satisfied by them.

Cited by 7 publications

References 7 publications

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

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