Prove that if (f(t)) is a characteristic function equal to zero when (|t| geqslant a), then the

Question:

Prove that if $f(t)$ is a characteristic function equal to zero when $|t| \geqslant a$, then the function $\varphi(t)$ defined by the equations

\[
\varphi(t)= \begin{cases}f(t) & \text { when }|t| \leqslant a \\ f(t+2 a) & \text { when }-\infty\]

is also a characteristic function.

Make use of the Bochner-Khinchin theorem.

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