Periodic continuation

In mathematics , especially in Fourier analysis , periodic continuation or periodization is an operation with which a function that is only defined in a certain interval becomes periodic .

One application are Fourier series that are only defined for periodic functions. In order to be able to use them for non-periodic functions as well, one must periodize them .

definition

Be a function with . ${\displaystyle f\colon \mathbb {[} a,b]\rightarrow \mathbb {R} }$ ${\displaystyle f(a)=f(b)}$

Then the periodization of is defined as: ${\displaystyle f_{p}\colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle f}$

${\displaystyle f_{p}(x)=f(x+a-\left\lfloor {\frac {x}{T}}\right\rfloor \cdot T)}$.

${\displaystyle T=b-a}$ is called the period of and denotes the rounding function . ${\displaystyle f}$ ${\displaystyle \lfloor \cdot \rfloor }$

Individual evidence

1. ↑ Michael Knorrenschild: Mathematics for Engineers 2. Applied Analysis in the Bachelor's degree . Carl Hanser Verlag, Munich 2014, ISBN 978-3-446-41347-4 , pp. 178 .