Difference between revisions of "Conditionally-periodic function"
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+ | $#C+1 = 12 : ~/encyclopedia/old_files/data/C024/C.0204540 Conditionally\AAhperiodic function | ||
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+ | A function $ A \circ \phi $ | ||
+ | which is the composition of a $ 2 \pi $- | ||
+ | periodic function $ A: T ^ { n } \rightarrow \mathbf C $, | ||
+ | where $ T ^ { n } $ | ||
+ | is the $ n $- | ||
+ | dimensional torus, and a function $ \phi : \mathbf R \rightarrow \mathbf R ^ {n} $ | ||
+ | such that $ \dot \phi = \omega $, | ||
+ | where $ \omega = ( \omega _ {1} \dots \omega _ {n} ) $ | ||
+ | is a constant vector whose components are linearly independent over the rational numbers. Examples of conditionally-periodic functions are given by partial sums of Fourier series | ||
+ | |||
+ | $$ | ||
+ | \sum _ {i = 1 } ^ { n } | ||
+ | [ A _ {i} \sin ( \omega _ {i} t + \psi _ {i} ) + | ||
+ | B _ {i} \cos ( \omega _ {i} t + \psi _ {i} )], | ||
+ | $$ | ||
where | where | ||
− | + | $$ | |
+ | A = A ( \phi _ {1} \dots \phi _ {n} ) = \ | ||
+ | \sum _ {i = 1 } ^ { n } | ||
+ | [ A _ {i} \sin \phi _ {i} + | ||
+ | B _ {i} \cos \phi _ {i} ], | ||
+ | $$ | ||
− | + | $$ | |
+ | \phi = ( \phi _ {1} ( t) \dots \phi _ {n} ( t)) = ( \omega _ {1} t + \psi _ {1} \dots \omega _ {n} t + \psi _ {n} ). | ||
+ | $$ | ||
− | If a conditionally-periodic function is continuous, then it coincides with a [[Quasi-periodic function|quasi-periodic function]] with periods | + | If a conditionally-periodic function is continuous, then it coincides with a [[Quasi-periodic function|quasi-periodic function]] with periods $ \omega _ {1} \dots \omega _ {n} $. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
A conditionally-periodic function is almost periodic, cf. [[Almost-periodic function|Almost-periodic function]]. | A conditionally-periodic function is almost periodic, cf. [[Almost-periodic function|Almost-periodic function]]. |
Latest revision as of 17:46, 4 June 2020
A function $ A \circ \phi $
which is the composition of a $ 2 \pi $-
periodic function $ A: T ^ { n } \rightarrow \mathbf C $,
where $ T ^ { n } $
is the $ n $-
dimensional torus, and a function $ \phi : \mathbf R \rightarrow \mathbf R ^ {n} $
such that $ \dot \phi = \omega $,
where $ \omega = ( \omega _ {1} \dots \omega _ {n} ) $
is a constant vector whose components are linearly independent over the rational numbers. Examples of conditionally-periodic functions are given by partial sums of Fourier series
$$ \sum _ {i = 1 } ^ { n } [ A _ {i} \sin ( \omega _ {i} t + \psi _ {i} ) + B _ {i} \cos ( \omega _ {i} t + \psi _ {i} )], $$
where
$$ A = A ( \phi _ {1} \dots \phi _ {n} ) = \ \sum _ {i = 1 } ^ { n } [ A _ {i} \sin \phi _ {i} + B _ {i} \cos \phi _ {i} ], $$
$$ \phi = ( \phi _ {1} ( t) \dots \phi _ {n} ( t)) = ( \omega _ {1} t + \psi _ {1} \dots \omega _ {n} t + \psi _ {n} ). $$
If a conditionally-periodic function is continuous, then it coincides with a quasi-periodic function with periods $ \omega _ {1} \dots \omega _ {n} $.
References
[1] | V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian) |
Comments
A conditionally-periodic function is almost periodic, cf. Almost-periodic function.
Conditionally-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditionally-periodic_function&oldid=11212