Electronic ISBN:  9781470401931 
Product Code:  MEMO/127/608.E 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $27.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 127; 1997; 97 ppMSC: Primary 35; 11; 28; 34; 58; Secondary 26; 47;
This memoir provides a detailed study of the effect of non powerlike 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.ReadershipGraduate students and research mathematicians interested in dynamical systems, fractal geometry, partial differential equations, analysis, measure theory, number theory or spectral geometry. Physicists interested in fractal geometry, condensed matter physics or wave propagation in random or fractal media.

Table of Contents

Chapters

1. Introduction

2. Statement of the main results

3. Sharp error estimates and their converse when $n$ = 1

4. Spectra of fractal strings and the Riemann zetafunction

5. The complex zeros of the Riemann zetafunction

6. Error estimates for $n \geq 2$

7. Examples

Appendix. Examples of gauge functions


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This memoir provides a detailed study of the effect of non powerlike 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.
Graduate students and research mathematicians interested in dynamical systems, fractal geometry, partial differential equations, analysis, measure theory, number theory or spectral geometry. Physicists interested in fractal geometry, condensed matter physics or wave propagation in random or fractal media.

Chapters

1. Introduction

2. Statement of the main results

3. Sharp error estimates and their converse when $n$ = 1

4. Spectra of fractal strings and the Riemann zetafunction

5. The complex zeros of the Riemann zetafunction

6. Error estimates for $n \geq 2$

7. Examples

Appendix. Examples of gauge functions