Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 41A / a

Expanding the square and using orthogonality of Fourier modes,Finite truncations justify the calculation directly, and passage to the ${\ell }^2$ limit proves Parseval identity for every square-summable sequence.

Fourier transformation in $m$ turns the recurrence intoAn amplification factor $z$ therefore satisfiesFor $\mu >0$, the quadratic Schur stability criterion giveswhere $a_1=-2\mu \cos \theta /(1+\mu )$ and $a_2=(\mu -1)/(\mu +1)$. Thus both roots lie in the closed unit disk; roots on it at $\theta =0,\pi$ are simple. At $\mu =0$, the roots are $\pm 1$. Parseval's identity then gives stability precisely forwhich is the full stated parameter range.