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Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain. Recall that a Fourier series is only suitable to the analysis of periodic (or finite-domain) functions f(x) with period 2π. It can be expressed as an infinite series ...
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle . [ 1 ]
General mathematical techniques for analyzing non-periodic functions fall into the category of Fourier analysis. The Fourier transform of a function produces a frequency spectrum which contains all of the information about the original signal, but in a different form.
The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. In older texts, the phase of a periodic function is sometimes called the amplitude. [1]
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.
There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon. [47]
The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t, called the evolution parameter. X is called phase space or state space , while the variable x represents an initial state of the system.
The trapezoidal rule converges rapidly for periodic functions. This is an easy consequence of the Euler-Maclaurin summation formula , which says that if f {\displaystyle f} is p {\displaystyle p} times continuously differentiable with period T {\displaystyle T} ∑ k = 0 N − 1 f ( k h ) h = ∫ 0 T f ( x ) d x + ∑ k = 1 ⌊ p / 2 ⌋ B 2 k ...