International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 25-98
https://doi.org/10.1107/97809553602060000551 |
Chapter 1.3. Fourier transforms in crystallography: theory, algorithms and applications
Chapter index
`Almost everywhere' 1.3.2.2.4
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
Action 1.3.4.2.2.2
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
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
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
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
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
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
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
cyclic 1.3.2.7.5, 1.3.3.2.3.1
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 theorem 1.3.2.4.3.5, 1.3.2.4.4.1, 1.3.2.6.4, 1.3.2.6.6, 1.3.2.6.10.1, 1.3.4.2.1.10, 1.3.4.5.2, 1.3.4.5.2.2
backward version 1.3.2.6.8, 1.3.4.2.2.9
Convolution theorems with crystallographic symmetry 1.3.4.2.2.9
vector-radix version 1.3.3.3.2.1
Coordinates
affine change of 1.3.2.4.2.2, 1.3.2.4.2.2
fractional 1.3.2.6.1, 1.3.4.2.1.1
non-standard 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
Cruickshank's modified Fourier method 1.3.4.4.7.5
Crystallographic applications of Fourier transforms 1.3.4
Crystallographic extension of the Rader/Winograd factorization 1.3.4.3.4.3
Crystallographic Fourier transform theory 1.3.4.2.1.1
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
in time 1.3.3.2.1, 1.3.4.3.5.1
period 1.3.2.7.2.3
Decomposition 1.3.4.3.2
coset 1.3.2.7.2.1, 1.3.3.3.2.1
de la Vallée Poussin kernel 1.3.2.6.10.1
Density modification 1.3.4.4.3.2
Determinantal inequalities 1.3.4.2.1.10
Diamond's real-space refinement method 1.3.4.4.7.9
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
spherical 1.3.4.2.1.3, 1.3.4.4.2
Discrete Fourier transformation 1.3.2.7.1
Discrete Fourier transform matrix
core of 1.3.4.3.4.3
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 1.3.4.5.2.1, 1.3.4.5.2.2
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
motif 1.3.4.2.1.1
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
tensor products of 1.3.2.3.9.6
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
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
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 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
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
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
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
inverse 1.3.2.4.2.6, 1.3.2.5.7
mathematical theory of 1.3.2
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 1.3.2.4.4.1
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
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
Functional derivative 1.3.4.4.7.8
Functions of polynomial growth 1.3.2.5.8
Gaussian atomic densities 1.3.2.4.4.2
anisotropic 1.3.4.2.1.2
Gaussian function 1.3.2.4.4.3
standard 1.3.2.4.4.2, 1.3.2.5.6
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
Groups 1.3.4.2.2.2
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
multivariate 1.3.2.4.4.2, 1.3.4.4.7.10
Hermitian-antisymmetric transforms 1.3.4.3.5.2
Hermitian form 1.3.2.6.9.1
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
Index 1.3.4.2.2.2
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
theory 1.3.4.2.2.3
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
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
Jacobians 1.3.2.3.9.5, 1.3.2.7.5
Joint probability distribution of structure factors 1.3.4.5.2.2
Kernels 1.3.3.3.2.1
of Fourier transformations 1.3.2.4.2.3
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
non-primitive 1.3.4.2.2.4
non-standard 1.3.2.6.1
non-standard 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 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
Linearity 1.3.2.4.2.1, 1.3.2.7.5
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 summable function of polynomial growth 1.3.2.5.3
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
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 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 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
Lagrange 1.3.4.4.6, 1.3.4.5.2.2
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
Non-primitive lattice 1.3.4.2.2.3, 1.3.4.2.2.4
Non-standard coordinates 1.3.2.6.1
Non-standard lattice 1.3.2.6.1
Non-standard n-torus 1.3.2.6.1
Non-standard 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
Notation, multi-index 1.3.2.2.3
n-shift rule 1.3.4.4.7.5
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
formula 1.3.4.2.2.2, 1.3.4.2.2.6
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
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
Periodicity
crystal 1.3.4.2.1.1
non-standard 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
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 transforms 1.3.3.3.2.4
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
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
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
Regularization 1.3.2.3.9.7
by convolution 1.3.2.6.2
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
equation 1.3.4.5.2.2
expansion 1.3.4.5.2.1
method 1.3.2.4.2.10, 1.3.4.5.2.2
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
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
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 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-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 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 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 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 1.3.2.4.2.3, 1.3.2.5.5
conjugate and parity-related 1.3.4.3.5
crystal 1.3.4.2.2.1
cyclic 1.3.4.3.4.3
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
artificial 1.3.4.4.5, 1.3.4.4.7.10
isotropic 1.3.4.2.2.6
definition and examples of 1.3.2.5.3
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
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
Topology 1.3.2.2.6.1, 1.3.2.5.1
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 ) 1.3.2.3.3.2
Transfer function 1.3.3.1
Transformations
of coordinates 1.3.2.3.9.5
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
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
Wyckoff symbols 1.3.4.2.2.4