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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 ...
Fig 1. The top graph shows a non-periodic function () in blue defined only over the red interval from 0 to P. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function () is not.
The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon. [47] He introduced the notion of a superspace. Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more ...
2 Non-smooth functions. 3 Vector-valued functions. 4 Doubly periodic functions. ... This is a list of some well-known periodic functions. The constant function f (x) ...
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.
A periodic function, also called a periodic waveform ... is a non-zero real number such that is within the domain of , is periodic with period . For ...
The space of uniform almost periodic functions on G can be identified with the space of all continuous functions on the Bohr compactification of G. More generally the Bohr compactification can be defined for any topological group G , and the spaces of continuous or L p functions on the Bohr compactification can be considered as almost periodic ...
For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be ...