Almost periodic functions

Cyclostationarity: half a century of research

Since the definition of quasiperiodicity is intimately connected to the indexing of a Fourier transform, for the case of an icosahedral solid, the step necessary to prove, using diffraction, that an object is quasiperiodic, is described. Various coordinate systems are discussed and reasons are given for choosing one aligned with a set of three orthogonal two-fold axes. Based on this coordinate system, the main crystallographic projections are presented and several analyzed single-crystal electron diffraction patterns are demonstrated. The extinction rules for three of the five icosahedral Bravais quasilattices are compared, and some simple relationships with the six-dimensional cut and projection crystallography are derived. This analysis leads to a simple application for indexing powder diffraction patterns.

/pdf/indexing-of-icosahedral-quasiperiodic-crystals-428tx01o2q.pdf

Indexing of icosahedral quasiperiodic crystals

Some trigonometrical inequalities with applications to the theory of series

Certain probability properties of $c_n(t)$, the empirical characteristic function $(\operatorname{ecf})$ are investigated. More specifically it is shown under some general restrictions that $c_n(t)$ converges uniformly almost surely to the population characteristic function $c(t).$ The weak convergence of $n^{\frac{1}{2}}(c_n(t) - c(t))$ to a Gaussian complex process is proved. It is suggested that the ecf may be a useful tool in numerous statistical problems. Application of these ideas is illustrated with reference to testing for symmetry about the origin: the statistic $\int\lbrack\mathbf{Im} c_n(t)\rbrack^2 dG(t)$ is proposed and its asymptotic distribution evaluated.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176343742

The Empirical Characteristic Function and Its Applications

/pdf/abstract-ergodic-theorems-and-weak-almost-periodic-functions-53hj8uktwt.pdf

Almost periodic functions

Citations

Cyclostationarity: half a century of research

Indexing of icosahedral quasiperiodic crystals

Some trigonometrical inequalities with applications to the theory of series

The Empirical Characteristic Function and Its Applications

Abstract ergodic theorems and weak almost periodic functions

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Abstract ergodic theorems and weak almost periodic functions