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Statement

For $p>0$ the weighted norm of a sequence $\xi:\mathbb Z\to\mathbb C$ is given by (p. 8)

$$\|\xi\|_p^2=\sum_{m\in\mathbb Z}\left(\frac{|\xi(m)|}{1+|m|^p}\right)^2 .$$

$\sigma(\xi)$ is the spectrum of a sequence, as on the [[analysis/borichev_et_al_2017_spectra_stationary_processes_z/theorem_1|Theorem 1]] page.

Lemma 3 (p. 8). Let $\delta,p,M>0$, and let $J$ be an open arc with $\bar J\subsetneq\mathbb T$. Then there are $n\in\mathbb N$ and $q_0,\ldots,q_{n-1}\in\mathbb C$ such that every sequence $\xi:\mathbb Z\to\mathbb C$ with $\|\xi\|_p\le M$ and $\sigma(\xi)\subset J$ satisfies

$$\Bigl|\xi(n)+\sum_{k=0}^{n-1}q_k\xi(k)\Bigr|<\delta .$$

The paper traces the lemma to Szegő and calls it the main ingredient of the proof of [[analysis/borichev_et_al_2017_spectra_stationary_processes_z/theorem_4|Theorem 4]] (p. 8).

The two lemmas it uses

Lemma 1 (p. 7). For a closed arc $J\subsetneq\mathbb T$ and $\delta>0$ there is a polynomial $P$ with $P(0)=1$ and $\|P\|_{C(J)}<\delta$. The paper proves it in two lines from Runge's theorem.

Lemma 2 (p. 7). For a closed arc $J\subsetneq\mathbb T$ there is a constant $K(J)$ such that every polynomial $P$ of degree $n$ satisfies $\|P'\|_{C(J)}\le K(J)n^2\|P\|_{C(J)}$. This is V. S. Videnskii's Bernstein inequality on an arc, cited from Borwein and Erdélyi, Polynomials and polynomial inequalities (Section 5.1.E19.c); the paper remarks that any bound polynomial, or even subexponential, in $n$ would do.

Proof pointer

Pp. 8--9. After a rotation the arc is $\{e^{i\theta}:|\theta|<\pi-\varepsilon\}$; a smooth cutoff $\varphi$ equals $1$ on $J$ and $0$ off a slightly larger arc $J'$. Lemma 1 gives $P$ with $P(0)=1$ and $\|P\|_{C(J')}\le\frac12$, and $Q(t)=t^{-n}P(t)^\ell$ with $n=\ell\deg P$ has the form $t^{-n}+\sum_{k<n}q_kt^{-k}$ and is exponentially small on $J'$, with its derivatives controlled by Lemma 2. Since $1-\varphi$ vanishes near $\sigma(\xi)$, the prediction error equals $F_\xi(Q\varphi)$, and the Cauchy--Schwarz inequality against the weighted norm bounds it by a constant depending on $p$ and $J$ times $Mn^{2p}e^{-cn}$, which is below $\delta$ once $n\ge n_0(p,J,M,\delta)$.

Used by. [[analysis/borichev_et_al_2017_spectra_stationary_processes_z/theorem_4|Theorem 4]].

Source. Alexander Borichev, Mikhail Sodin, Benjamin Weiss, Spectra of stationary processes on $\mathbb Z$, arXiv:1701.03407v1 (12 January 2017), identified on the [[analysis/borichev_et_al_2017_spectra_stationary_processes_z/_index|source card]]; labels and pages are that version's.

Read depth. Claims checked: Lemmas 1--3 and the proof of Lemma 3 were read clause by clause on pp. 7--9; Lemma 2 is recorded as the paper cites it and was not checked against its source. Nothing here is independently reviewed.

Bears on. Problem 1150 (context only): the source card notes that this lemma, with the proof of Theorem 4, would make a stationary-limit argument quantitative when a fixed arc contains the spectrum. The paper does not relate the prediction length $n$ to the maximum modulus of a $\pm1$ polynomial, and it does not mention the problem.