International Tables for Crystallography (2010). Vol. B. ch. 1.3, pp. 24-113
https://doi.org/10.1107/97809553602060000760 |
Chapter 1.3. Fourier transforms in crystallography: theory, algorithms and applications
Contents
- 1.3. Fourier transforms in crystallography: theory, algorithms and applications (pp. 24-113) | html | pdf | chapter contents |
- 1.3.1. General introduction (p. 24) | html | pdf |
- 1.3.2. The mathematical theory of the Fourier transformation (pp. 24-52) | html | pdf |
- 1.3.2.1. Introduction (pp. 24-25) | html | pdf |
- 1.3.2.2. Preliminary notions and notation (pp. 25-28) | html | pdf |
- 1.3.2.2.1. Metric and topological notions in (p. 25) | html | pdf |
- 1.3.2.2.2. Functions over (pp. 25-26) | html | pdf |
- 1.3.2.2.3. Multi-index notation (p. 26) | html | pdf |
- 1.3.2.2.4. Integration, spaces (pp. 26-27) | html | pdf |
- 1.3.2.2.5. Tensor products. Fubini's theorem (p. 27) | html | pdf |
- 1.3.2.2.6. Topology in function spaces (pp. 27-28) | html | pdf |
- 1.3.2.3. Elements of the theory of distributions (pp. 28-35) | html | pdf |
- 1.3.2.3.1. Origins (pp. 28-29) | html | pdf |
- 1.3.2.3.2. Rationale (p. 29) | html | pdf |
- 1.3.2.3.3. Test-function spaces (pp. 29-30) | html | pdf |
- 1.3.2.3.4. Definition of distributions (p. 30) | html | pdf |
- 1.3.2.3.5. First examples of distributions (p. 30) | html | pdf |
- 1.3.2.3.6. Distributions associated to locally integrable functions (p. 30) | html | pdf |
- 1.3.2.3.7. Support of a distribution (p. 30) | html | pdf |
- 1.3.2.3.8. Convergence of distributions (pp. 30-31) | html | pdf |
- 1.3.2.3.9. Operations on distributions (pp. 31-35) | html | pdf |
- 1.3.2.3.9.1. Differentiation (pp. 31-32) | html | pdf |
- 1.3.2.3.9.2. Integration of distributions in dimension 1 (p. 32) | html | pdf |
- 1.3.2.3.9.3. Multiplication of distributions by functions (p. 32) | html | pdf |
- 1.3.2.3.9.4. Division of distributions by functions (pp. 32-33) | html | pdf |
- 1.3.2.3.9.5. Transformation of coordinates (p. 33) | html | pdf |
- 1.3.2.3.9.6. Tensor product of distributions (pp. 33-34) | html | pdf |
- 1.3.2.3.9.7. Convolution of distributions (pp. 34-35) | html | pdf |
- 1.3.2.4. Fourier transforms of functions (pp. 35-39) | html | pdf |
- 1.3.2.4.1. Introduction (p. 35) | html | pdf |
- 1.3.2.4.2. Fourier transforms in (pp. 35-37) | html | pdf |
- 1.3.2.4.2.1. Linearity (p. 35) | html | pdf |
- 1.3.2.4.2.2. Effect of affine coordinate transformations (p. 35) | html | pdf |
- 1.3.2.4.2.3. Conjugate symmetry (p. 35) | html | pdf |
- 1.3.2.4.2.4. Tensor product property (p. 35) | html | pdf |
- 1.3.2.4.2.5. Convolution property (pp. 35-36) | html | pdf |
- 1.3.2.4.2.6. Reciprocity property (p. 36) | html | pdf |
- 1.3.2.4.2.7. Riemann–Lebesgue lemma (p. 36) | html | pdf |
- 1.3.2.4.2.8. Differentiation (p. 36) | html | pdf |
- 1.3.2.4.2.9. Decrease at infinity (p. 36) | html | pdf |
- 1.3.2.4.2.10. The Paley–Wiener theorem (pp. 36-37) | html | pdf |
- 1.3.2.4.3. Fourier transforms in (p. 37) | html | pdf |
- 1.3.2.4.4. Fourier transforms in (pp. 37-39) | html | pdf |
- 1.3.2.4.5. Various writings of Fourier transforms (p. 39) | html | pdf |
- 1.3.2.4.6. Tables of Fourier transforms (p. 39) | html | pdf |
- 1.3.2.5. Fourier transforms of tempered distributions (pp. 39-42) | html | pdf |
- 1.3.2.5.1. Introduction (p. 39) | html | pdf |
- 1.3.2.5.2. as a test-function space (p. 40) | html | pdf |
- 1.3.2.5.3. Definition and examples of tempered distributions (p. 40) | html | pdf |
- 1.3.2.5.4. Fourier transforms of tempered distributions (p. 40) | html | pdf |
- 1.3.2.5.5. Transposition of basic properties (p. 40) | html | pdf |
- 1.3.2.5.6. Transforms of δ-functions (pp. 40-41) | html | pdf |
- 1.3.2.5.7. Reciprocity theorem (p. 41) | html | pdf |
- 1.3.2.5.8. Multiplication and convolution (p. 41) | html | pdf |
- 1.3.2.5.9. aspects, Sobolev spaces (pp. 41-42) | html | pdf |
- 1.3.2.6. Periodic distributions and Fourier series (pp. 42-47) | html | pdf |
- 1.3.2.6.1. Terminology (p. 42) | html | pdf |
- 1.3.2.6.2. -periodic distributions in (p. 42) | html | pdf |
- 1.3.2.6.3. Identification with distributions over (p. 42) | html | pdf |
- 1.3.2.6.4. Fourier transforms of periodic distributions (pp. 42-43) | html | pdf |
- 1.3.2.6.5. The case of nonstandard period lattices (pp. 43-44) | html | pdf |
- 1.3.2.6.6. Duality between periodization and sampling (p. 44) | html | pdf |
- 1.3.2.6.7. The Poisson summation formula (p. 44) | html | pdf |
- 1.3.2.6.8. Convolution of Fourier series (p. 44) | html | pdf |
- 1.3.2.6.9. Toeplitz forms, Szegö's theorem (pp. 44-45) | html | pdf |
- 1.3.2.6.10. Convergence of Fourier series (pp. 45-47) | html | pdf |
- 1.3.2.7. The discrete Fourier transformation (pp. 47-52) | html | pdf |
- 1.3.2.7.1. Shannon's sampling theorem and interpolation formula (p. 47) | html | pdf |
- 1.3.2.7.2. Duality between subdivision and decimation of period lattices (pp. 48-49) | html | pdf |
- 1.3.2.7.2.1. Geometric description of sublattices (p. 48) | html | pdf |
- 1.3.2.7.2.2. Sublattice relations for reciprocal lattices (p. 48) | html | pdf |
- 1.3.2.7.2.3. Relation between lattice distributions (p. 48) | html | pdf |
- 1.3.2.7.2.4. Relation between Fourier transforms (pp. 48-49) | html | pdf |
- 1.3.2.7.2.5. Sublattice relations in terms of periodic distributions (p. 49) | html | pdf |
- 1.3.2.7.3. Discretization of the Fourier transformation (pp. 49-51) | html | pdf |
- 1.3.2.7.4. Matrix representation of the discrete Fourier transform (DFT) (p. 51) | html | pdf |
- 1.3.2.7.5. Properties of the discrete Fourier transform (pp. 51-52) | html | pdf |
- 1.3.3. Numerical computation of the discrete Fourier transform (pp. 52-62) | html | pdf |
- 1.3.3.1. Introduction (p. 52) | html | pdf |
- 1.3.3.2. One-dimensional algorithms (pp. 52-57) | html | pdf |
- 1.3.3.3. Multidimensional algorithms (pp. 58-62) | html | pdf |
- 1.3.3.3.1. The method of successive one-dimensional transforms (p. 58) | html | pdf |
- 1.3.3.3.2. Multidimensional factorization (pp. 58-61) | html | pdf |
- 1.3.3.3.2.1. Multidimensional Cooley–Tukey factorization (pp. 58-59) | html | pdf |
- 1.3.3.3.2.2. Multidimensional prime factor algorithm (pp. 59-60) | html | pdf |
- 1.3.3.3.2.3. Nesting of Winograd small FFTs (p. 60) | html | pdf |
- 1.3.3.3.2.4. The Nussbaumer–Quandalle algorithm (pp. 60-61) | html | pdf |
- 1.3.3.3.3. Global algorithm design (pp. 61-62) | html | pdf |
- 1.3.4. Crystallographic applications of Fourier transforms (pp. 62-106) | html | pdf |
- 1.3.4.1. Introduction (p. 62) | html | pdf |
- 1.3.4.2. Crystallographic Fourier transform theory (pp. 62-76) | html | pdf |
- 1.3.4.2.1. Crystal periodicity (pp. 62-68) | html | pdf |
- 1.3.4.2.1.1. Period lattice, reciprocal lattice and structure factors (pp. 62-63) | html | pdf |
- 1.3.4.2.1.2. Structure factors in terms of form factors (p. 63) | html | pdf |
- 1.3.4.2.1.3. Fourier series for the electron density and its summation (pp. 63-64) | html | pdf |
- 1.3.4.2.1.4. Friedel's law, anomalous scatterers (p. 64) | html | pdf |
- 1.3.4.2.1.5. Parseval's identity and other theorems (p. 64) | html | pdf |
- 1.3.4.2.1.6. Convolution, correlation and Patterson function (pp. 64-65) | html | pdf |
- 1.3.4.2.1.7. Sampling theorems, continuous transforms, interpolation (pp. 65-66) | html | pdf |
- 1.3.4.2.1.8. Sections and projections (pp. 66-67) | html | pdf |
- 1.3.4.2.1.9. Differential syntheses (p. 67) | html | pdf |
- 1.3.4.2.1.10. Toeplitz forms, determinantal inequalities and Szegö's theorem (pp. 67-68) | html | pdf |
- 1.3.4.2.2. Crystal symmetry (pp. 68-76) | html | pdf |
- 1.3.4.2.2.1. Crystallographic groups (p. 68) | html | pdf |
- 1.3.4.2.2.2. Groups and group actions (pp. 68-70) | html | pdf |
- 1.3.4.2.2.3. Classification of crystallographic groups (pp. 70-71) | html | pdf |
- 1.3.4.2.2.4. Crystallographic group action in real space (pp. 71-72) | html | pdf |
- 1.3.4.2.2.5. Crystallographic group action in reciprocal space (pp. 72-73) | html | pdf |
- 1.3.4.2.2.6. Structure-factor calculation (pp. 73-74) | html | pdf |
- 1.3.4.2.2.7. Electron-density calculations (p. 74) | html | pdf |
- 1.3.4.2.2.8. Parseval's theorem with crystallographic symmetry (p. 74) | html | pdf |
- 1.3.4.2.2.9. Convolution theorems with crystallographic symmetry (p. 75) | html | pdf |
- 1.3.4.2.2.10. Correlation and Patterson functions (pp. 75-76) | html | pdf |
- 1.3.4.2.1. Crystal periodicity (pp. 62-68) | html | pdf |
- 1.3.4.3. Crystallographic discrete Fourier transform algorithms (pp. 76-91) | html | pdf |
- 1.3.4.3.1. Historical introduction (pp. 76-77) | html | pdf |
- 1.3.4.3.2. Defining relations and symmetry considerations (pp. 77-78) | html | pdf |
- 1.3.4.3.3. Interaction between symmetry and decomposition (pp. 78-79) | html | pdf |
- 1.3.4.3.4. Interaction between symmetry and factorization (pp. 79-85) | html | pdf |
- 1.3.4.3.5. Treatment of conjugate and parity-related symmetry properties (pp. 85-89) | html | pdf |
- 1.3.4.3.5.1. Hermitian-symmetric or real-valued transforms (pp. 85-87) | html | pdf |
- 1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms (p. 87) | html | pdf |
- 1.3.4.3.5.3. Complex symmetric and antisymmetric transforms (pp. 87-88) | html | pdf |
- 1.3.4.3.5.4. Real symmetric transforms (p. 88) | html | pdf |
- 1.3.4.3.5.5. Real antisymmetric transforms (p. 88) | html | pdf |
- 1.3.4.3.5.6. Generalized multiplexing (pp. 88-89) | html | pdf |
- 1.3.4.3.6. Global crystallographic algorithms (pp. 89-91) | html | pdf |
- 1.3.4.3.6.1. Triclinic groups (p. 89) | html | pdf |
- 1.3.4.3.6.2. Monoclinic groups (p. 89) | html | pdf |
- 1.3.4.3.6.3. Orthorhombic groups (pp. 89-90) | html | pdf |
- 1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups (p. 90) | html | pdf |
- 1.3.4.3.6.5. Cubic groups (p. 90) | html | pdf |
- 1.3.4.3.6.6. Treatment of centred lattices (p. 90) | html | pdf |
- 1.3.4.3.6.7. Programming considerations (pp. 90-91) | html | pdf |
- 1.3.4.4. Basic crystallographic computations (pp. 91-100) | html | pdf |
- 1.3.4.4.1. Introduction (p. 91) | html | pdf |
- 1.3.4.4.2. Fourier synthesis of electron-density maps (p. 91) | html | pdf |
- 1.3.4.4.3. Fourier analysis of modified electron-density maps (pp. 91-93) | html | pdf |
- 1.3.4.4.3.1. Squaring (p. 91) | html | pdf |
- 1.3.4.4.3.2. Other nonlinear operations (pp. 91-92) | html | pdf |
- 1.3.4.4.3.3. Solvent flattening (p. 92) | html | pdf |
- 1.3.4.4.3.4. Molecular averaging by noncrystallographic symmetries (pp. 92-93) | html | pdf |
- 1.3.4.4.3.5. Molecular-envelope transforms via Green's theorem (p. 93) | html | pdf |
- 1.3.4.4.4. Structure factors from model atomic parameters (p. 93) | html | pdf |
- 1.3.4.4.5. Structure factors via model electron-density maps (pp. 93-94) | html | pdf |
- 1.3.4.4.6. Derivatives for variational phasing techniques (pp. 94-95) | html | pdf |
- 1.3.4.4.7. Derivatives for model refinement (pp. 95-100) | html | pdf |
- 1.3.4.4.7.1. The method of least squares (pp. 95-96) | html | pdf |
- 1.3.4.4.7.2. Booth's differential Fourier syntheses (p. 96) | html | pdf |
- 1.3.4.4.7.3. Booth's method of steepest descents (p. 96) | html | pdf |
- 1.3.4.4.7.4. Cochran's Fourier method (pp. 96-97) | html | pdf |
- 1.3.4.4.7.5. Cruickshank's modified Fourier method (pp. 97-98) | html | pdf |
- 1.3.4.4.7.6. Agarwal's FFT implementation of the Fourier method (p. 98) | html | pdf |
- 1.3.4.4.7.7. Lifchitz's reformulation (p. 98) | html | pdf |
- 1.3.4.4.7.8. A simplified derivation (pp. 98-99) | html | pdf |
- 1.3.4.4.7.9. Discussion of macromolecular refinement techniques (p. 99) | html | pdf |
- 1.3.4.4.7.10. Sampling considerations (pp. 99-100) | html | pdf |
- 1.3.4.4.8. Miscellaneous correlation functions (p. 100) | html | pdf |
- 1.3.4.5. Related applications (pp. 100-106) | html | pdf |
- 1.3.4.5.1. Helical diffraction (pp. 100-102) | html | pdf |
- 1.3.4.5.1.1. Circular harmonic expansions in polar coordinates (p. 100) | html | pdf |
- 1.3.4.5.1.2. The Fourier transform in polar coordinates (p. 101) | html | pdf |
- 1.3.4.5.1.3. The transform of an axially periodic fibre (p. 101) | html | pdf |
- 1.3.4.5.1.4. Helical symmetry and associated selection rules (pp. 101-102) | html | pdf |
- 1.3.4.5.2. Application to probability theory and direct methods (pp. 102-106) | html | pdf |
- 1.3.4.5.1. Helical diffraction (pp. 100-102) | html | pdf |
- References | html | pdf |
- Figures