In simple harmonic motion, the average energy over one time period...

The total energy of a particle executing simple harmonic motion is given by $E = \dfrac{1}{2} m \omega^2 A^2$.

Since the total energy remains constant at all points during the motion, its average value over one complete time period is equal to the total energy itself.

Alternatively, the average kinetic energy over one time period is $\langle K \rangle = \dfrac{1}{4} m \omega^2 A^2$ and the average potential energy is $\langle U \rangle = \dfrac{1}{4} m \omega^2 A^2$.

The average total energy is $\langle E \rangle = \langle K \rangle + \langle U \rangle = \dfrac{1}{4} m \omega^2 A^2 + \dfrac{1}{4} m \omega^2 A^2 = \dfrac{1}{2} m \omega^2 A^2$.