## Mathematical Preliminaries

*Partha P. Mitra and Hemant Bokil*

### in Observed Brain Dynamics

- Published in print:
- 2007
- Published Online:
- May 2009
- ISBN:
- 9780195178081
- eISBN:
- 9780199864829
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195178081.003.0005
- Subject:
- Neuroscience, Techniques, Molecular and Cellular Systems

This chapter reviews a broad range of mathematical topics relevant to the rest of the book. It begins with a brief discussion of real and complex numbers and elementary real and complex functions, ... More

## Linear Systems Analysis

*Russell L. De Valois and Karen K. De Valois*

### in Spatial Vision

- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195066579
- eISBN:
- 9780199872220
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195066579.003.0001
- Subject:
- Psychology, Cognitive Neuroscience

This chapter provides a background on linear systems analysis to allow the reader to follow the applications to visual problems discussed in this book. Specifically, it discusses the Fourier theorem ... More

## FOURIER ANALYSIS ON FINITE ABELIAN GROUPS: SOME GRAPHICAL APPLICATIONS

*Andrew Goodall*

### in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0007
- Subject:
- Mathematics, Probability / Statistics

This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in ... More

## Visual Pattern Analyzers

*Norma Van Surdam Graham*

- Published in print:
- 1989
- Published Online:
- January 2008
- ISBN:
- 9780195051544
- eISBN:
- 9780199872183
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195051544.001.0001
- Subject:
- Psychology, Cognitive Neuroscience

From the light that falls on the retina, the visual system must extract meaningful information about what is where in our environment. At an early stage in this process, it analyzes the incoming ... More

## LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS

*Sylvie Benzoni-Gavage and Denis Serre*

### in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199211234
- eISBN:
- 9780191705700
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211234.003.0001
- Subject:
- Mathematics, Applied Mathematics

For first-order systems, the well-posedness of the Cauchy problem is equivalent to an algebraic condition on the symbol of the operator: the hyperbolicity. The hyperbolic Cauchy problem has the ... More

## Historical Prelude

*Roger D. Roger and Miles A. Whittington*

### in Cortical Oscillations in Health and Disease

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780195342796
- eISBN:
- 9780199776276
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195342796.003.0002
- Subject:
- Neuroscience, Molecular and Cellular Systems, Development

One of the pioneers in the mathematical and conceptual study of oscillations was Norbert Wiener, whose book Cybernetics first appeared in 1948. The authors review the type of data, experimental ... More

## Selected Theorems by Eli Stein

*Charles Fefferman*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

### in Advances in Analysis: The Legacy of Elias M. Stein

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0001
- Subject:
- Mathematics, Numerical Analysis

This chapter is an overview of the scope and originality of Eli Stein's contributions to analysis. His work deals with representation theory, classical Fourier analysis, and partial differential ... More

## Fourier Analysis and Benfordʼs Law

*Steven J. Miller*

### in Benford's Law: Theory and Applications

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.003.0003
- Subject:
- Mathematics, Probability / Statistics

This chapter continues the development of the theory of Benford's law. It uses Fourier analysis (in particular, Poisson Summation) to prove many systems either satisfy or almost satisfy the ... More

## Advances in Analysis: The Legacy of Elias M. Stein

*Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.001.0001
- Subject:
- Mathematics, Numerical Analysis

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His ... More

## Fourier analysis in diffraction and image formation

*Christopher Hammond*

### in The Basics of Crystallography and Diffraction

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198738671
- eISBN:
- 9780191801938
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198738671.003.0013
- Subject:
- Physics, Crystallography, Condensed Matter Physics / Materials

This chapter starts with the Fourier series and Fourier transforms, and the representation of periodic functions. It then looks at Fourier analysis in crystallography. It details electron density, ... More