Pseudo-periodic function

with periods $ \omega _ {0} \dots \omega _ {r} $

A function $ f ( t , u _ {1} \dots u _ {r} ) $ of $ r + 1 $ variables satisfying:

$$ f ( t , u _ {1} \dots u _ {i} + \omega _ {i} \dots u _ {r} ) = $$

$$ = \ f ( t , u _ {1} \dots u _ {i} \dots u _ {r} ) ,\ i = 1 \dots r ; $$

$$ f ( t + \omega _ {0} , u _ {1} \dots u _ {r} ) = f ( t , u _ {1} + \omega _ {0} \dots u _ {r} + \omega _ {0} ) . $$

Example: if $ f _ {0} ( t) $ and $ f _ {1} ( t) $ are continuous periodic functions with periods $ \omega _ {0} $ and $ \omega _ {1} $, respectively, then $ f ( t , u _ {1} ) = f _ {0} ( t) + f _ {1} ( t + u _ {1} ) $ is a pseudo-periodic function.

A pseudo-periodic function is connected with a quasi-periodic function and is determined by it in a unique way: A function $ F ( t) $ is quasi-periodic with periods $ \omega _ {0} \dots \omega _ {r} $ if and only if there exists a continuous pseudo-periodic function $ f ( t , u _ {1} \dots u _ {r} ) $ with periods $ \omega _ {0} \dots \omega _ {r} $ such that $ F ( t) = f ( t , 0 \dots 0 ) $.

Comments

The expression "pseudo-periodic function" is also used to indicate a function with a pseudo- $ p $- period: $ g( t+ p) = e ^ {i \theta } w( t) $ for some $ \theta $ and all $ t $. For such a function $ g( t) $ the function $ h( t, u) = e ^ {ip ^ {-} 1 \theta u } g( t) $ is pseudo-periodic in the sense above.

References

[a1]	M. Urabe, "Green functions of pseudo-periodic differential operators" M. Urabe (ed.) , Japan-United States Sem. Ordinary Differential and Functional Eq. , Springer (1971) pp. 106–122
[a2]	J.A. Goldstein, "Asymptotics for bounded semigroups on Hilbert space" R. Nagel (ed.) et al. (ed.) , Aspects of Positivity in Funct. Anal. , North-Holland (1986) pp. 49–62