# Periodic Extension

Definition 5: Periodic Extension

The periodic extension of a function $f$ defined on $[a,b]$ is the $b−a$ periodic function $f_{p}$ such that

- $f_{p}(t)=f(t)$ for $t∈(a,b)$ where $f(t)$ is continuous.
- $f_{p}(t)=2f(t_{−})+f(t_{+}) $ for $t∈(a,b)$ where $f(t)$ is not continuous.
- $f_{p}(a)=2f(a)+f(b) =fp(b)$