Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings, and the Riemann Zeta-Function

Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings, and the Riemann Zeta-Function

by Christina Q. He, Michel L. Lapidus
     
 

ISBN-10: 0821805975

ISBN-13: 9780821805978

Pub. Date: 05/12/1997

Publisher: American Mathematical Society

This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of ''fractal drums'' (and especially of ''fractal strings''). In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable ''gauge

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Overview

This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of ''fractal drums'' (and especially of ''fractal strings''). In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable ''gauge functions'' other than power functions. (This content is used to measure the irregularity (or ''fractality'') of the boundary of an open set in $R^n$ by evaluating the volume of its small tubular neighborhoods.) In the situation when the power function is not the natural ''gauge function'', this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators.

Product Details

ISBN-13:
9780821805978
Publisher:
American Mathematical Society
Publication date:
05/12/1997
Series:
Memoirs of the American Mathematical Society Series, #127
Pages:
97

Table of Contents

1Introduction1
2Statement of the Main Results6
2.1One-dimensional case (n = 1)6
2.2Higher dimensional case10
3Sharp Error Estimates and their Converse when n = 114
3.1Preliminaries14
3.2Two-sided estimates22
3.3One-sided estimates35
4Spectra of Fractal Strings and the Riemann Zeta-Function38
4.1Characterization of h-Minkowski measurability38
4.2Existence of a monotonic second term: the Riemann zetafunction51
5The Complex Zeros of the Riemann Zeta-Function57
6Error Estimates for n [greater than or equal] 269
7Examples80
AppendixExamples of Gauge Functions88
References94

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