## THE NONLINEAR RADON AND FOURIER TRANSFORMS

*Leon Ehrenpreis*

### in The Universality of the Radon Transform

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.003.0005
- Subject:
- Mathematics, Mathematical Physics

In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon ... More

## Parallel Scientific Computation: A Structured Approach using BSP and MPI

*Rob H. Bisseling*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529392
- eISBN:
- 9780191712869
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529392.001.0001
- Subject:
- Mathematics, Applied Mathematics

This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book ... More

## Fourier transforms and convolutions

*Dennis Sherwood and Jon Cooper*

### in Crystals, X-rays and Proteins: Comprehensive Protein Crystallography

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.003.0005
- Subject:
- Physics, Crystallography

This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the ... More

## Crystals, X-rays and Proteins: Comprehensive Protein Crystallography

*Dennis Sherwood and Jon Cooper*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.001.0001
- Subject:
- Physics, Crystallography

This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book ... More

## STRUCTURED MATRIX ANALYSIS

*D. A. Bini, G. Latouche, and B. Meini*

### in Numerical Methods for Structured Markov Chains

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0002
- Subject:
- Mathematics, Numerical Analysis

This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz ... More

## Fourier series and Fourier transforms

*Chun Wa Wong*

### in Introduction to Mathematical Physics: Methods & Concepts

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The Fourier expansion of an arbitrary piecewise-smooth function in the finite interval —π 〈 x 〈 π as an infinite sum of terms (a 0; a n ... More

## Diffraction

*Dennis Sherwood and Jon Cooper*

### in Crystals, X-rays and Proteins: Comprehensive Protein Crystallography

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.003.0006
- Subject:
- Physics, Crystallography

This chapter covers the principles of the interaction of waves with obstacles and the manner in which information on the nature of an obstacle is contained within the resulting wave forms. The ... More

## Simulating experimental data

*Reinhard B. Neder and Thomas Proffen*

### in Diffuse Scattering and Defect Structure Simulations: A cook book using the program DISCUS

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199233694
- eISBN:
- 9780191715563
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233694.003.0004
- Subject:
- Physics, Crystallography

In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental ... More

## Solving crystal structures from HRTEM images by crystallographic image processing

*Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov*

### in Electron Crystallography: Electron Microscopy and Electron Diffraction

- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199580200
- eISBN:
- 9780191731211
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199580200.003.0007
- Subject:
- Physics, Crystallography

A fundamentally important feature of electron crystallography is its power in solving crystal structures directly from HRTEM images, using crystallographic image processing. With the appropriate ... More

## Fourier Analysis

*B. D. Guenther*

### in Modern Optics

- Published in print:
- 2015
- Published Online:
- January 2016
- ISBN:
- 9780198738770
- eISBN:
- 9780191801983
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198738770.003.0006
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The techniques developed by Fourier and Dirichlet state that a Fourier series, a sum of sinusoidal functions, can be used to describe any periodic functions and that the Fourier transform, an ... More

## Fourier syntheses

*William Clegg*

### in Crystal Structure Analysis: Principles and Practice

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.003.0008
- Subject:
- Physics, Crystallography

A diffraction pattern is the (forward) Fourier transform of a crystal structure, obtained physically in an experiment and mathematically from a known or model structure (providing both amplitudes and ... More

## Postprocessing

*Gary A. Glatzmaier*

### in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and ... More

## The phase problem and the Patterson function

*Dennis Sherwood and Jon Cooper*

### in Crystals, X-rays and Proteins: Comprehensive Protein Crystallography

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.003.0012
- Subject:
- Physics, Crystallography

This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor ... More

## Introduction to diffraction

*Peter Main*

### in Crystal Structure Analysis: Principles and Practice

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.003.0001
- Subject:
- Physics, Crystallography

This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, ... More

## The Pseudospectral Method

*Heiner Igel*

### in Computational Seismology: A Practical Introduction

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198717409
- eISBN:
- 9780191835070
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198717409.003.0005
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

The concepts of Fourier series and the Fourier transforms are introduced as a way to approximate arbitrary functions. The interpolation properties on regular grids (collocation points, cardinal ... More

## From Abel-Poisson to Bochner-Riesz Summability

*Ali Taheri*

### in Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.003.0011
- Subject:
- Mathematics, Analysis

This chapter begins with an introduction of Fourier transform on ℝn and then covers various summability methods including Abel-Poisson, Gauss-Weierstrass and later Bochner-Riesz. It also discusses ... More

## Quantum algorithms

*Vlatko Vedral*

### in Introduction to Quantum Information Science

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780199215706
- eISBN:
- 9780191706783
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199215706.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

As computers get faster and faster, the size of the circuitry imprinted onto silicon chips decreases. The size of the circuitry becomes so small that its behavior is governed by the laws of quantum ... More

## Isotropic turbulence (in spectral space)

*P. A. Davidson*

### in Turbulence: An Introduction for Scientists and Engineers

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198722588
- eISBN:
- 9780191789298
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198722588.003.0008
- Subject:
- Mathematics, Applied Mathematics, Mathematical Physics

Chapter 8 recasts the dynamical properties of turbulence in spectral space. The chapter starts by redeveloping the kinematics of turbulence in Fourier space, introducing the ideas of the spectral ... More

## Linear-Gaussian systems and signal processing

*Max A. Little*

### in Machine Learning for Signal Processing: Data Science, Algorithms, and Computational Statistics

- Published in print:
- 2019
- Published Online:
- October 2019
- ISBN:
- 9780198714934
- eISBN:
- 9780191879180
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198714934.003.0007
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy, Mathematical Physics

Linear systems theory, based on the mathematics of vector spaces, is the backbone of all “classical” DSP and a large part of statistical machine learning. The basic idea -- that linear algebra ... More

## Molecular Scattering and Fourier Transforms

*Peter B. Moore*

### in Visualizing the Invisible: Imaging Techniques for the Structural Biologist

- Published in print:
- 2012
- Published Online:
- May 2015
- ISBN:
- 9780199767090
- eISBN:
- 9780190267841
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:osobl/9780199767090.003.0002
- Subject:
- Biology, Biochemistry / Molecular Biology

This chapter explores the mathematical relationships existing between known molecular structures from Chapter 1 and the structure factors, particularly dealing with the Fourier transforms and Fourier ... More