The notion of quasi-periodic functions was generalised still further by Harald Bohr when he introduced almost-periodic functions.
32.
The totality of representations is governed by the group's " L " 2-space of periodic functions.
33.
This 2?-periodic function is the periodic summation of a Fourier transform, which makes it a widely used analysis tool.
34.
Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions.
35.
Notice that we consider here to be a periodic function, e . g . if and is negative then we define.
36.
In explicit form they are expressible in terms of Jacobian elliptic functions which are periodic functions ( effectively generalisations of trigonometrical functions ).
37.
For example, periodic functions such as sine produce segmented patterns with repetitions, while symmetric functions such as Gaussian produce symmetric patterns.
38.
The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves.
39.
Also shown is the frequency region containing one cycle of the discrete-time Fourier transform, which is always a periodic function.
40.
Stagnation zones theorems are closely related to pre-Liouville theorem about conversion of the entire doubly periodic function into the identical constant.
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