International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 24-113   | 1 | 2 |
https://doi.org/10.1107/97809553602060000760

Chapter 1.3. Fourier transforms in crystallography: theory, algorithms and applications

Chapter index

`Almost everywhere' 1.3.2.2.4
Abelian groups 1.3.2.6.1, 1.3.4.3.4
Abel summation procedure 1.3.2.6.10.1
Absolutely integrable functions 1.3.2.2.4
Acentric reflections 1.3.4.2.2.7
Additive reindexing 1.3.3.3.3.1
Affine change of coordinates 1.3.2.4.2.2, 1.3.2.4.2.2
Affine change of variables 1.3.2.5.5
Agarwal's FFT implementation of the Fourier method 1.3.4.4.7.6
Algebraic integers 1.3.4.3.2, 1.3.4.3.4.3
Algebraic number theory 1.3.4.3.4.3
Algebra of functions 1.3.4.2.2.9
Analytical methods of probability theory 1.3.4.5.2.1
Anisotropic Gaussian atoms 1.3.4.2.1.2
Anisotropic temperature factors 1.3.4.2.2.6
Anomalous scatterers 1.3.4.2.1.4, 1.3.4.2.1.5, 1.3.4.2.2.7
Approximations
kinematical 1.3.4.1
saddlepoint 1.3.4.5.2, 1.3.4.5.2.1
Arithmetic classes 1.3.4.2.2.3
of representations 1.3.4.2.2.3
Artificial temperature factor 1.3.4.4.5, 1.3.4.4.7.10
Associated actions in function spaces 1.3.4.2.2.2
Associativity properties of convolution 1.3.4.4.7.10
Asymmetric unit 1.3.4.2.2.2, 1.3.4.2.2.4
Asymptotic distribution of eigenvalues of Toeplitz forms 1.3.2.6.9.3, 1.3.4.2.1.10
Asymptotic expansions
and limit theorems 1.3.4.5.2.1
of Gram–Charlier and Edgeworth 1.3.4.5.2.2
Atomic electron densities 1.3.4.2.2.10
Autocorrelation 1.3.4.2.1.6
Automorphism 1.3.4.2.2.2, 1.3.4.2.2.3
Back-shift correction 1.3.4.4.7.2
Backward convolution theorem 1.3.2.6.8, 1.3.4.2.2.9
Banach spaces 1.3.2.2.6.2
Band-limited function 1.3.2.7.3
Base-centred lattices 1.3.4.3.6.6
Basic crystallographic computations 1.3.4.4
Bayesian statistical approach to the phase problem 1.3.4.5.2.2
Beevers–Lipson factorization 1.3.3.3.1, 1.3.4.3.1, 1.3.4.3.1
Beevers–Lipson strips 1.3.4.3.1, 1.3.4.4.5
Bessel's inequality 1.3.2.6.10.2
Best Fourier 1.3.4.4.2
Bieberbach theorem 1.3.4.2.2.1
Body-centred lattices 1.3.4.3.6.6
Booth's differential Fourier syntheses 1.3.4.4.7.2
Booth's method of steepest descents 1.3.4.4.7.3
Bounded projections 1.3.4.2.1.8, 1.3.4.4.3.3
Bounded subset 1.3.2.2.1
Bragg–Lipson charts 1.3.4.4.4
Burg entropy 1.3.4.2.1.10
Burnside's theorem 1.3.4.2.2.3
Butterfly loop 1.3.3.2.1
Calculus
of asymmetric units 1.3.4.3.3
operational 1.3.2.3.1
Cartesian product 1.3.2.2, 1.3.2.6.1
Cauchy's theorem 1.3.4.5.2.1
Cauchy kernel 1.3.2.6.10.1
Cauchy–Schwarz inequality 1.3.2.2.4, 1.3.2.6.10.2
Cauchy sequence 1.3.2.2.4
Central-limit theorem 1.3.4.5.2.1
Centred lattices 1.3.4.2.2.5
Centric reflections 1.3.4.2.2.5
Cesàro sum 1.3.2.6.10.1
Chain rule 1.3.4.4.7.8
Characteristic functions 1.3.4.5.2.1
Chinese remainder theorem (CRT) 1.3.3.2.2.2, 1.3.3.3.3.1, 1.3.4.3.4.2
for polynomials 1.3.3.2.4, 1.3.4.3.4.3
reconstruction 1.3.3.2.2.2
reconstruction formula 1.3.3.2.4
Circular harmonic expansions 1.3.4.5.1.1
Classification of crystallographic groups 1.3.4.2.2.3
Closed subset 1.3.2.2.1
Cochran's Fourier method 1.3.4.4.7.4
Cocycle 1.3.4.3.5.1
Communication, statistical theory of 1.3.4.5.2.2
Commutative ring 1.3.3.2.2.1
Compact subset 1.3.2.2.1
Complete normed space 1.3.2.2.6.2
Complete vector spaces 1.3.2.2.4
Complex antisymmetric transforms 1.3.4.3.5.3
Complex symmetric transforms 1.3.4.3.5.3
Computer architecture 1.3.3.1, 1.3.3.3.3.2
Conjugacy classes of subgroups 1.3.4.2.2.2
Conjugate and parity-related symmetry 1.3.4.3.5
Conjugate distribution 1.3.4.5.2.1, 1.3.4.5.2.2
Conjugate families of distributions 1.3.4.5.2.2
Conjugate symmetry 1.3.2.4.2.3, 1.3.2.5.5
Conjugation 1.3.4.2.2.2
Consistency condition 1.3.2.4.5
Contragredient 1.3.2.5.5
of a matrix 1.3.2.4.2.2
Convergence
of distributions 1.3.2.3.8
of Fourier series 1.3.2.6.10
Conversion of translations to phase shifts 1.3.2.4.2.2
Convolution 1.3.4.2.1.6
associativity properties of 1.3.4.4.7.10
of distributions 1.3.2.3.9.7
of Fourier series 1.3.2.6.8
of probability densities 1.3.4.5.2.1
of two distributions 1.3.2.3.9.7
Convolution property 1.3.2.4.2.5, 1.3.2.7.5
Convolution theorems with crystallographic symmetry 1.3.4.2.2.9
Cooley–Tukey algorithm 1.3.3.1, 1.3.3.2.1, 1.3.3.3.3.2, 1.3.4.3.1
vector-radix version 1.3.3.3.2.1
Cooley–Tukey factorization, multidimensional 1.3.3.3.2.1, 1.3.3.3.2.2, 1.3.4.3.4.1
Coordinates
affine change of 1.3.2.4.2.2, 1.3.2.4.2.2
nonstandard 1.3.2.6.1
transformation of 1.3.2.3.9.5
Core of discrete Fourier transform matrix 1.3.4.3.4.3
Correlation 1.3.4.2.1.6
Correlation functions 1.3.4.2.2.10, 1.3.4.4.8
Coset averaging 1.3.2.7.2.3, 1.3.2.7.2.5
Coset decomposition 1.3.2.7.2.1, 1.3.3.3.2.1
Coset reversal 1.3.3.3.2.1
Cosine strips 1.3.4.3.1
Cross correlation 1.3.4.2.2.10
Cross-rotation function 1.3.4.4.8
CRT (Chinese remainder theorem) 1.3.3.2.2.2, 1.3.3.3.3.1, 1.3.4.3.4.2
for polynomials 1.3.3.2.4, 1.3.4.3.4.3
reconstruction 1.3.3.2.2.2
reconstruction formula 1.3.3.2.4
Cruickshank's modified Fourier method 1.3.4.4.7.5
Crystallographic applications of Fourier transforms 1.3.4
Crystallographic discrete Fourier transform 1.3.4.3.2
algorithms 1.3.4.3.1
Crystallographic extension of the Rader/Winograd factorization 1.3.4.3.4.3
Crystallographic Fourier transform theory 1.3.4.2.1.1
Crystallographic group action 1.3.4.3.4
in real space 1.3.4.2.2.4
in reciprocal space 1.3.4.2.2.5
Crystallographic groups 1.3.4.2.2.1
classification of 1.3.4.2.2.3
Crystal periodicity 1.3.4.2.1.1
Crystal symmetry 1.3.4.2.2.1
Crystal systems 1.3.4.2.2.3
Cubic groups 1.3.4.3.6.5
Cumulant-generating functions 1.3.4.5.2.1
Cyclic convolution 1.3.2.7.5, 1.3.3.2.3.1
Cyclic groups 1.3.4.2.2.3
Cyclic symmetry 1.3.4.3.4.3
Cyclotomic polynomials 1.3.3.2.4
Data flow 1.3.3.3.3.2
and subdivision of period lattices, duality between 1.3.2.7.2
de la Vallée Poussin kernel 1.3.2.6.10.1
Delta functions 1.3.2.1
Dirac 1.3.2.3.1
transforms of 1.3.2.5.6
Density modification 1.3.4.4.3.2
Derivatives
for model refinement 1.3.4.4.7
for variational phasing techniques 1.3.4.4.6
Determinantal inequalities 1.3.4.2.1.10
Diamond's real-space refinement method 1.3.4.4.7.9
Differential syntheses 1.3.2.4.2.8, 1.3.4.2.1.9, 1.3.4.4.7.5
Differentiation 1.3.2.1, 1.3.2.4.2.8
and multiplication by a monomial 1.3.2.5.5
of distributions 1.3.2.3.9.1
under the duality bracket 1.3.2.3.9.1
Differentiation identities 1.3.2.7.5
Differentiation property 1.3.4.5.2.2
Diffraction
by helical structures 1.3.4.5.1
Diffraction conditions 1.3.4.2.1.1
Digital electronic computation of Fourier series 1.3.4.3.1
Dihedral symmetry 1.3.4.3.4.3
Dirac delta function 1.3.2.3.1
Direct lattice 1.3.2.6.2
Direct methods 1.3.4.5.2
Direct phase determination 1.3.2.4.2.10
Dirichlet kernel 1.3.2.6.10.1
Discrete Fourier transformation 1.3.2.7.1
Discrete Fourier transform matrix
core of 1.3.4.3.4.3
Discrete Fourier transforms 1.3.2.1, 1.3.4.3.2
algorithms 1.3.4.3.1
matrix representation of 1.3.2.7.4
numerical computation of 1.3.3.1
properties of 1.3.2.7.5
Distance function 1.3.2.2.6.1
Distributions
associated with locally integrable functions 1.3.2.3.6
conjugate families of 1.3.4.5.2.2
convergence of 1.3.2.3.8
convolution of 1.3.2.3.9.7, 1.3.2.3.9.7
definition of 1.3.2.3.4
differentiation of 1.3.2.3.9.1
division of 1.3.2.3.9.4
Fourier transforms of 1.3.2.5.1
integration of 1.3.2.3.9.2
maximum-entropy 1.3.2.4.2.10, 1.3.4.5.2.2
moments of 1.3.4.5.2.1
multiplication of 1.3.2.3.9.3
of finite order 1.3.2.3.4
of random atoms 1.3.4.5.2.2
operations on 1.3.2.3.9
support of 1.3.2.3.7, 1.3.2.3.7
tensor products of 1.3.2.3.9.6
theory of 1.3.2.1, 1.3.2.3.1
T on Ω 1.3.2.3.4
Division of distributions 1.3.2.3.9.4
Division problem 1.3.2.3.9.4
Dual, topological 1.3.2.3.4, 1.3.2.3.7, 1.3.2.5.1, 1.3.2.5.3
Duality
between differentiation and multiplication by a monomial 1.3.4.2.1.9
between periodization and sampling 1.3.2.6.6
between sections and projections 1.3.2.5.8
between subdivision and decimation of period lattices 1.3.2.7.2
Duality bracket 1.3.2.4.4.4
Duality product 1.3.2.3.9
Edgeworth series 1.3.4.5.2.1
Eigenspace decomposition of L2 1.3.2.4.3.4
Electron-density calculations 1.3.4.2.2.7
Electron-density maps, Fourier synthesis of 1.3.4.4.2
Electronic analogue computer X-RAC 1.3.4.3.1
Entire functions 1.3.2.4.2.10
Entropy 1.3.4.4.6
Equal distribution 1.3.2.6.9.3
Essential bounds 1.3.2.6.9.3
Essentially bounded function 1.3.2.2.4
Euclidean algorithm 1.3.2.7.2.1, 1.3.3.2.4, 1.3.4.2.1.8
Euclidean norm 1.3.2.2.1
Euclidean space 1.3.2.2.1
Exchange between differentiation and multiplication by monomials 1.3.4.5.2
Exchange between multiplication and convolution 1.3.2.1
Face-centred lattices 1.3.4.3.6.6
Factor group 1.3.4.2.2.2, 1.3.4.2.2.3
Factorization 1.3.4.3.2
Fast Fourier transform (FFT) 1.3.4.3.1
Fejér kernel 1.3.2.6.10.1
spherical 1.3.4.2.1.3
FFT (fast Fourier transform) 1.3.4.3.1
Fibre diffraction 1.3.2.5.8
Fibres
axially periodic, transform of 1.3.4.5.1.3
Finite field 1.3.3.2.3.1
Form factor 1.3.4.2.1.2
Forward convolution theorem 1.3.2.6.8, 1.3.4.2.1.6, 1.3.4.2.1.6, 1.3.4.2.2.9
Fourier analysis 1.3.4.2.1.1
Fourier coefficient 1.3.2.6.10.1
Fourier cotransform 1.3.2.4.1
Fourier cotransformation 1.3.2.5.7
Fourier method
Agarwal's FFT implementation of 1.3.4.4.7.6
Cochran's 1.3.4.4.7.4
Cruickshank's modified 1.3.4.4.7.5
Fourier series 1.3.2.1
convergence of 1.3.2.6.10
convolution of 1.3.2.6.8
digital electronic computation of 1.3.4.3.1
electron density and its summation 1.3.4.2.1.3
partial sum of 1.3.2.6.10.1
Fourier synthesis 1.3.4.2.1.1
of electron-density maps 1.3.4.4.2
Fourier transformation
discrete 1.3.2.7.1
mathematical theory of 1.3.2
Fourier transforms 1.3.2.1, 1.3.2.4.1, 1.3.2.4.1
crystallographic, discrete 1.3.4.3.2
crystallographic, theory of 1.3.4.2.1.1
crystallographic applications 1.3.4
discrete 1.3.2.1
discrete, core of matrix 1.3.4.3.4.3
discrete, matrix representation of 1.3.2.7.4
discrete, numerical computation of 1.3.3.1
discrete, properties of 1.3.2.7.5
exchange of subdivision and decimation 1.3.2.7.2.4
in {\scr S} 1.3.2.4.4.1
in L2 1.3.2.4.3
in polar coordinates 1.3.4.5.1.2
kernels of 1.3.2.4.2.3
of a distribution 1.3.2.5.1
of periodic distributions 1.3.2.6.4
of tempered distributions 1.3.2.5.1, 1.3.2.5.4
tables of 1.3.2.4.6
tensor product property of 1.3.4.3.1
various writings of 1.3.2.4.5
Fractional coordinates 1.3.2.6.1, 1.3.4.2.1.1
Fréchet space 1.3.2.2.6.2
Frobenius congruences 1.3.4.2.2.3, 1.3.4.2.2.5
Fubini's theorem 1.3.2.2.5, 1.3.2.4.2.4, 1.3.2.4.4.4
Functional derivative 1.3.4.4.7.8
Functions of polynomial growth 1.3.2.5.8
Function spaces
associated actions in 1.3.4.2.2.2
topology in 1.3.2.2.6
Gaussian atomic densities 1.3.2.4.4.2
anisotropic 1.3.4.2.1.2
Gaussian function 1.3.2.4.4.3
Gaussians 1.3.4.4.7.10
Generalized multiplexing 1.3.4.3.5.6
Generalized Rader/Winograd algorithms 1.3.4.3.6.4
Generalized support condition 1.3.2.3.9.7
General linear change of variable 1.3.2.4.2.2
General multivariate Gaussians 1.3.2.4.4.2
General topology 1.3.2.2.6.1
Geometric redundancies 1.3.4.2.1.7
Gibbs phenomenon 1.3.2.6.10.1, 1.3.4.2.1.3
G-invariant function 1.3.4.2.2.2
Global crystallographic algorithms 1.3.4.3.6
Good algorithm 1.3.3.2.2
Good factorization, multidimensional 1.3.4.3.4.2
Gram–Charlier series 1.3.4.5.2.1
Green's theorem 1.3.2.3.9.1, 1.3.4.4.3.5
crystallographic 1.3.4.3.4
crystallographic, real space 1.3.4.2.2.4
crystallographic, reciprocal space 1.3.4.2.2.5
Group characters 1.3.4.3.5.6
Group cohomology 1.3.4.3.4.1
Group extensions 1.3.4.2.2.3
Group of units 1.3.3.1
Group ring
integral 1.3.4.3.4
module over 1.3.1
Hankel transform 1.3.4.5.1.2
Hardy's theorem 1.3.2.4.4.3
Harker peaks 1.3.4.2.2.10
Heisenberg's inequality 1.3.2.4.4.3, 1.3.4.4.3.2
Helical structures
diffraction by 1.3.4.5.1
Helical symmetry 1.3.4.5.1.4
Hermite function 1.3.2.4.4.2, 1.3.4.5.2.1
Hermitian-antisymmetric transforms 1.3.4.3.5.2
Hermitian form 1.3.2.6.9.1
Hermitian symmetry 1.3.4.2.1.4, 1.3.4.2.2.7, 1.3.4.3.5
Hexagonal groups 1.3.4.3.6.4
Hypothetical atoms 1.3.4.4.5
Idempotents 1.3.3.2.2.2
Image of a function by a geometric operation 1.3.2.2.2
Implicit function theorem 1.3.2.3.9.5
Induction formula 1.3.4.5.2.2
Inductive limit 1.3.2.3.3.3
Integral group ring 1.3.4.3.4
Integral representation 1.3.4.2.2.1
Integrals
Lebesgue 1.3.2.2.4
Riemann 1.3.2.2.4
Integration
by parts 1.3.2.3.9.1
Lebesgue's theory of 1.3.2.3.1
of distributions 1.3.2.3.9.2
Intensity statistics 1.3.4.5.2.2
Interaction between symmetry and decomposition 1.3.4.3.3
Interaction between symmetry and factorization 1.3.4.3.4
Interatomic vectors 1.3.4.2.1.6
Interference function 1.3.4.2.1.7
Interpolation formula 1.3.2.7.1
Interpolation kernel 1.3.4.4.3.4
Invariance of L2 1.3.2.4.3.1
Inverse Fourier transformation 1.3.2.4.2.6, 1.3.2.5.7
Ising model 1.3.4.2.1.10
Isometry 1.3.2.4.3.3
Isometry property 1.3.2.4.3.5
Isotropic temperature factors 1.3.4.2.2.6
Isotropy subgroups 1.3.4.2.2.2, 1.3.4.2.2.4, 1.3.4.2.2.4
Joint probability distribution of structure factors 1.3.4.5.2.2
Kernels 1.3.3.3.2.1
de la Vallée Poussin 1.3.2.6.10.1
Dirichlet 1.3.2.6.10.1
interpolation 1.3.4.4.3.4
of Fourier transformations 1.3.2.4.2.3
Poisson 1.3.2.6.10.1
spherical Dirichlet 1.3.4.2.1.3, 1.3.4.4.2
spherical Fejér 1.3.4.2.1.3
Kinematical approximation 1.3.4.1
Lagrange's theorem 1.3.4.2.2.2
Lagrange multiplier 1.3.4.4.6, 1.3.4.5.2.2
Lattice 1.3.2.6.1
base-centred 1.3.4.3.6.6
body-centred 1.3.4.3.6.6
centred 1.3.4.2.2.5
direct 1.3.2.6.2
face-centred 1.3.4.3.6.6
nonprimitive 1.3.4.2.2.3, 1.3.4.2.2.4
nonstandard 1.3.2.6.1
nonstandard period 1.3.2.6.5
primitive 1.3.4.2.2.3
residual 1.3.2.7.2.1
rhombohedral 1.3.4.3.6.6
standard 1.3.2.6.1
Lattice distributions 1.3.2.6.5, 1.3.2.6.7, 1.3.2.7.1, 1.3.2.7.2.3
Lattice mode 1.3.4.2.2.3
Lattice sum 1.3.2.6.7
Least-squares method, multivariate 1.3.4.4.7.1
Lebesgue's theory of integration 1.3.2.3.1
Lebesgue integral 1.3.2.2.4
Left cosets 1.3.4.2.2.2
Left representation 1.3.4.2.2.5
Length
of a function 1.3.2.2.3
Lifchitz's reformulation 1.3.4.4.7.7
Linear change of variable, general 1.3.2.4.2.2
Linear forms 1.3.2.3.4
Linear functionals 1.3.2.2.6.2
Linearization formulae 1.3.4.2.2.9
Liouville's theorem 1.3.2.4.2.10
Lissajous curve 1.3.4.5.2.2
Locally integrable functions 1.3.2.3.6
distributions associated with 1.3.2.3.6
Locally summable function of polynomial growth 1.3.2.5.3
Lp spaces 1.3.2.2.4
Macromolecular refinement techniques 1.3.4.4.7.9
Mapping 1.3.2.2
Mathematical theory of Fourier transformation 1.3.2
Matrix representation
of discrete Fourier transform 1.3.2.7.4
Maximum entropy 1.3.4.5.2.2
Maximum-entropy distributions 1.3.2.4.2.10
of atoms 1.3.4.5.2.2
Maximum-entropy methods 1.3.4.5.2
Maximum-entropy theory 1.3.4.5.2.2
Metric space 1.3.2.2.1, 1.3.2.2.6.1
Metrizability 1.3.2.1
Metrizable topology 1.3.2.2.6.1
Module 1.3.4.3.4
over a group ring 1.3.1
Molecular averaging by noncrystallographic symmetry 1.3.4.4.3.4
Moment-generating functions 1.3.2.4.2.8, 1.3.4.5.2.1
Moment-generating properties 1.3.4.5.2
of \bar{\scr F} 1.3.4.2.1.9
Moments of a distribution 1.3.4.5.2.1
Monoclinic groups 1.3.4.3.6.2
Motif distribution 1.3.4.2.1.1
Multidimensional algorithms 1.3.3.3
Multidimensional Cooley–Tukey factorization 1.3.3.3.2.1, 1.3.3.3.2.2, 1.3.4.3.4.1
Multidimensional factorization 1.3.3.3.2
Multidimensional Good factorization 1.3.4.3.4.2
Multidimensional prime factor algorithm 1.3.3.3.2.2
Multi-index notation 1.3.2.2.3
Multiplexing, generalized 1.3.4.3.5.6
Multiplexing–demultiplexing 1.3.4.3.5.1
Multiplication by a monomial 1.3.2.1
Multiplication of distributions 1.3.2.3.9.3
Multiplicative group of units 1.3.3.2.3.2
Multiplicative reindexing 1.3.3.3.3.1
Multiplier functions 1.3.2.5.8
Multipliers 1.3.2.6.6
Multivariate Gaussian 1.3.2.6.7
Multivariate Hermite functions 1.3.2.4.4.2, 1.3.4.4.7.10
Multivariate least-squares method 1.3.4.4.7.1
Nested algorithms 1.3.3.3.3.3
Nesting 1.3.3.3.3.1
of Winograd small FFTs 1.3.3.3.2.3
Noncrystallographic symmetry 1.3.4.2.1.7
molecular averaging by 1.3.4.4.3.4
Nonprimitive lattice 1.3.4.2.2.3, 1.3.4.2.2.4
Nonstandard coordinates 1.3.2.6.1
Nonstandard lattice 1.3.2.6.1
Nonstandard n-torus 1.3.2.6.1
Nonstandard period lattice 1.3.2.6.5
Normal equations 1.3.4.4.7.1
Normalizer 1.3.4.2.2.2
Normal matrix 1.3.4.4.7.5
Normed space 1.3.2.2.6.2
complete 1.3.2.2.6.2
Norm
Euclidean 1.3.2.2.1
on a vector space 1.3.2.2.6.2
Notation, multi-index 1.3.2.2.3
n-shift rule 1.3.4.4.7.5
n-torus
nonstandard 1.3.2.6.1
standard 1.3.2.6.1
Numerical computation of discrete Fourier transform 1.3.3.1
Nussbaumer–Quandalle algorithm 1.3.3.3.2.4
Observational equations 1.3.4.4.7.1
Offset 1.3.3.2.1
Operational calculus 1.3.2.3.1
Operations on distributions 1.3.2.3.9
Orthorhombic groups 1.3.4.3.6.3
Paley–Wiener theorem 1.3.2.4.2.10, 1.3.4.5.2.2
Parallel processing 1.3.3.3.3.2
Parseval's identity 1.3.4.2.1.5, 1.3.4.2.1.10
Parseval's theorem 1.3.2.4.1, 1.3.4.4.6
with crystallographic symmetry 1.3.4.2.2.8
Parseval–Plancherel property 1.3.2.7.5
Parseval–Plancherel theorem 1.3.2.4.3.3, 1.3.2.6.10.2
Partial sum of Fourier series 1.3.2.6.10.1
Period decimation 1.3.2.7.2.3
Periodic distributions 1.3.2.6.2, 1.3.2.6.8, 1.3.4.1
and Fourier series 1.3.2.6
Fourier transforms of 1.3.2.6.4
Periodicity
crystal 1.3.4.2.1.1
Periodization 1.3.2.1, 1.3.2.6.6, 1.3.3.2.1
and sampling, duality between 1.3.2.6.6
nonstandard 1.3.2.6.5
Period matrix 1.3.2.6.5
Period subdivision 1.3.2.7.2.3
Phase determination
statistical theory of 1.3.4.5.2.2
Phase problem
Bayesian statistical approach 1.3.4.5.2.2
Phase restriction 1.3.4.2.2.5
Phase shift 1.3.2.1, 1.3.3.2.1
Pipelining 1.3.3.3.3.2
Plancherel's theorem 1.3.2.5.9
Poisson kernel 1.3.2.6.10.1
Poisson summation formula 1.3.2.6.7
Polynomial growth
functions of 1.3.2.5.8
locally summable function of 1.3.2.5.3
Polynomials
Chinese remainder theorem for 1.3.3.2.4, 1.3.4.3.4.3
cyclotomic 1.3.3.2.4
Polynomial transforms 1.3.3.3.2.4
Prime factor algorithm 1.3.3.1, 1.3.3.2.2
multidimensional 1.3.3.3.2.2
Primitive lattice 1.3.4.2.2.3
Primitive root mod p 1.3.3.2.3.1
Principal central projections and sections 1.3.4.2.1.8
Principal projections 1.3.4.3.1
Principal sections and projections 1.3.4.2.1.8
Probability densities, convolution of 1.3.4.5.2.1
Probability theory 1.3.4.5.2
analytical methods of 1.3.4.5.2.1
Projection(s) 1.3.2.1
and sections, principal central 1.3.4.2.1.8
principal 1.3.4.3.1
Projector 1.3.4.2.2.2
Prolate spheroidal wavefunctions 1.3.2.4.4.3
Pseudo-distances 1.3.2.2.6.1
Punched-card machines 1.3.4.3.1
Pure imaginary transforms 1.3.4.3.5.2
Rader/Winograd algorithms, generalized 1.3.4.3.6.4
Rader/Winograd factorization, crystallographic extension of 1.3.4.3.4.3
Rader algorithm 1.3.3.1
Radon measure 1.3.2.3.4
Random-walk problem 1.3.4.5.2.2
Rapidly decreasing functions 1.3.2.4.4.1, 1.3.2.5.1
Real antisymmetric transforms 1.3.4.3.5.5
Real symmetric transforms 1.3.4.3.5.4
Real-valued transforms 1.3.4.3.5.1
Reciprocity 1.3.2.4.3.2
Reciprocity property 1.3.2.4.2.6
Reduced orbit 1.3.4.2.2.7
Reducibility of the representation 1.3.4.2.2.4
Reflection conditions 1.3.4.2.2.5
Reflections
acentric 1.3.4.2.2.7
Regularization 1.3.2.3.9.7
by convolution 1.3.2.6.2
Reindexing
additive 1.3.3.3.3.1
multiplicative 1.3.3.3.3.1
Representation operators 1.3.4.2.2.4, 1.3.4.3.3
Representation property 1.3.4.2.2.2
Residual lattice 1.3.2.7.2.1
Rhombohedral lattice 1.3.4.3.6.6
Riemann integral 1.3.2.2.4
Riemann–Lebesgue lemma 1.3.2.4.2.7
Right cosets 1.3.4.2.2.2
Right representation 1.3.4.2.2.2
Robertson's sorting board 1.3.4.3.1
Row–column method 1.3.3.3.1
Saddlepoint approximation 1.3.4.5.2, 1.3.4.5.2.1
Saddlepoint equation 1.3.4.5.2.2
Saddlepoint expansion 1.3.4.5.2.1
Saddlepoint method 1.3.2.4.2.10, 1.3.4.5.2.2
Sampling 1.3.2.1, 1.3.2.6.6
and periodization, duality between 1.3.2.6.6
considerations 1.3.4.4.7.10
theorems 1.3.4.2.1.7
Sayre's equation 1.3.4.4.3.1
Sayre's squaring method 1.3.4.4.6
Scattering
X-ray 1.3.4.1
Schur's lemma 1.3.4.2.2.4, 1.3.4.3.3
Scrambling 1.3.3.2.1
Search directions 1.3.4.4.6
Section 1.3.2.1
Sections and projections 1.3.2.1, 1.3.4.2.1.8
duality betweenProjection(s)and sections, duality between 1.3.2.5.8
duality betweenProjection(s)and sections, duality between 1.3.2.5.8
principal 1.3.4.2.1.8
Selection rules 1.3.4.5.1.4
Self-Patterson 1.3.4.4.8
Self-rotation function 1.3.4.4.8
Semi-direct product 1.3.4.2.2.2, 1.3.4.2.2.2
Semi-norm on a vector space 1.3.2.2.6.2
Series-termination errors 1.3.4.2.1.3, 1.3.4.4.2, 1.3.4.4.7.9
Shannon interpolation 1.3.2.1, 1.3.2.7.3
Shannon interpolation formula 1.3.2.7.1, 1.3.4.4.3.3
Shannon interpolation theorem 1.3.4.2.1.7
Shannon sampling criterion 1.3.2.7.1, 1.3.4.2.1.10, 1.3.4.4.3.4
Shannon sampling theorem 1.3.2.7.1, 1.3.4.2.1.7
Shift property 1.3.2.7.5, 1.3.4.2.1.6
Short cyclic convolutions 1.3.3.2.4
Sine strips 1.3.4.3.1
Skew-circulant matrix 1.3.3.2.3.2
Sobolev space 1.3.2.5.9
Solvable space groups 1.3.4.2.2.3
Solvent flattening 1.3.4.4.3.3
Solvent regions 1.3.4.2.1.7
Space groups 1.3.4.2.2.3
solvable 1.3.4.2.2.3
symmorphic 1.3.4.2.2.3
Space-group types 1.3.4.2.2.3
Special position 1.3.4.2.2.4
condition 1.3.4.2.2.4
Special reflection 1.3.4.2.2.5
Spherical Dirichlet kernel 1.3.4.2.1.3, 1.3.4.4.2
Spherical Fejér kernel 1.3.4.2.1.3
Square-integrable functions 1.3.2.5.9
Square-summable sequences 1.3.2.6.10.2
Squaring method equation 1.3.4.4.3.1
Standard basis of {\bb R}sup{n} 1.3.2.6.1
Standard Gaussian function 1.3.2.4.4.2, 1.3.2.5.6
Standard lattice 1.3.2.6.1
Standard n-torus 1.3.2.6.1
Statistical theory of communication 1.3.4.5.2.2
Statistical theory of phase determination 1.3.4.5.2.2
Steepest descents, Booth's method 1.3.4.4.7.3
Stirling's formula 1.3.4.5.2.2
Structure-factor algebra 1.3.4.2.2.9, 1.3.4.5.2.2, 1.3.4.5.2.2
Structure factors 1.3.4.2.1.1
calculation of 1.3.4.2.2.6
from model atomic parameters 1.3.4.4.4
in terms of form factors 1.3.4.2.1.2
joint probability distribution of 1.3.4.5.2.2
via model electron-density maps 1.3.4.4.5
Structure theorem 1.3.2.3.9.7
for distributions with compact support 1.3.2.6.4, 1.3.2.6.10.3
Subdivision and decimation of period lattices, duality between 1.3.2.7.2
Sublattice 1.3.2.7.2.1
Summable functions 1.3.2.2.4
Summation problem in crystallography 1.3.2.6.10.1
Support 1.3.2.2.2
of a distribution 1.3.2.3.7, 1.3.2.3.7
of a tensor product 1.3.2.3.9.6
Support condition 1.3.2.3.9.7, 1.3.2.6.8
generalized 1.3.2.3.9.7
Symmetry
conjugate and parity-related 1.3.4.3.5
crystal 1.3.4.2.2.1
dihedral 1.3.4.3.4.3
helical 1.3.4.5.1.4
noncrystallographic 1.3.4.2.1.7
noncrystallographic, molecular averaging by 1.3.4.4.3.4
Symmetry property 1.3.2.4.4.4
Symmorphic space groups 1.3.4.2.2.3
Systematic absences 1.3.4.2.2.5
Temperature factors 1.3.4.2.2.6
anisotropic 1.3.4.2.2.6
isotropic 1.3.4.2.2.6
definition and examples of 1.3.2.5.3
Fourier transforms of 1.3.2.5.1, 1.3.2.5.4
Tensor product 1.3.2.2.5, 1.3.2.3.9.6, 1.3.3.1
of distributions 1.3.2.3.9.6
structure of 1.3.4.3.2
support of 1.3.2.3.9.6
Tensor product property 1.3.2.4.2.4, 1.3.4.2.1.8, 1.3.4.5.1.3
of a Fourier transform 1.3.4.3.1
Test functions 1.3.2.5.1
Test-function spaces 1.3.2.3.3
Tetragonal groups 1.3.4.3.6.4
Theory of distributions 1.3.2.1, 1.3.2.3.1
Toeplitz–Carathéodory–Herglotz theorem 1.3.2.6.9.2
Toeplitz determinants 1.3.2.6.9.2, 1.3.4.2.1.10
Toeplitz forms 1.3.2.6.9, 1.3.4.2.1.10
asymptotic distribution of eigenvalues of 1.3.2.6.9.3, 1.3.4.2.1.10
Toeplitz matrices 1.3.2.6.9.3
Topological vector spaces 1.3.2.2.6.2
general 1.3.2.2.6.1
in function spaces 1.3.2.2.6
metrizable 1.3.2.2.6.1
not metrizable 1.3.2.3.3.3
on {\scr D}_{k}(\Omega) 1.3.2.3.3.2
on {\scr D}(\Omega) 1.3.2.3.3.3
on {\scr E}(\Omega) 1.3.2.3.3.1
Transfer function 1.3.3.1
Transformations
of coordinates 1.3.2.3.9.5
unitary 1.3.2.4.3.3
Transforms
complex antisymmetric 1.3.4.3.5.3
complex symmetric 1.3.4.3.5.3
Hermitian-antisymmetric 1.3.4.3.5.2
of an axially periodic fibre 1.3.4.5.1.3
of delta functions 1.3.2.5.6
polynomial 1.3.3.3.2.4
pure imaginary 1.3.4.3.5.2
real antisymmetric 1.3.4.3.5.5
real symmetric 1.3.4.3.5.4
real-valued 1.3.4.3.5.1
Translate 1.3.2.2.2
Translation 1.3.2.1
Translation functions 1.3.4.4.8
Translations
conversion to phase shifts 1.3.2.4.2.2
Transposition formula 1.3.4.3.4.1, 1.3.4.3.4.1
for intermediate results 1.3.4.3.1
Triangular inequality 1.3.2.2.6.1
Triclinic groups 1.3.4.3.6.1
Trigonal groups 1.3.4.3.6.4
Trigonometric moment problem 1.3.2.6.9
Trigonometric structure-factor expressions, vectors of 1.3.4.5.2.2
Uniformizable space 1.3.2.2.6.1
Unitary transformations 1.3.2.4.3.3
Unit cube 1.3.2.6.1
Unscrambling 1.3.4.3.4.1
Vector processing 1.3.3.3.3.2
Vector radix Cooley–Tukey algorithm 1.3.3.3.2.1
Vector radix FFT algorithms 1.3.3.3.2.1
Vectors
interatomic 1.3.4.2.1.6
of trigonometric structure-factor expressions 1.3.4.5.2.2
Vector space
complete 1.3.2.2.4
norm on 1.3.2.2.6.2
semi-norm on 1.3.2.2.6.2
topological 1.3.2.2.6.2
Wavefunctions, prolate spheroidal 1.3.2.4.4.3
Weighted difference map 1.3.4.4.7.4, 1.3.4.4.7.8
Weighted lattice distribution 1.3.2.6.6
Weighted reciprocal-lattice distribution 1.3.4.2.1.1
Winograd algorithms 1.3.3.1, 1.3.3.2.4
Winograd small FFT(s)
algorithms 1.3.3.2.4
nesting of 1.3.3.3.2.3
Wyckoff symbols 1.3.4.2.2.4
X-ray scattering 1.3.4.1