Stability of trigonometric approximation in $L^p$ and applications to prediction theory
Lutz Klotz, Michael Frank
TL;DR
This work investigates the stability of trigonometric approximation in $L^p$ spaces on locally compact abelian groups and applies the results to prediction theory for weakly stationary and harmonizable symmetric $p$-stable processes. It develops a unified framework based on the distance $d_p(\mu; S)$ to the trigonometric subspace $[\mathcal T(S)]_{\mu,p}$ and its metric projection $\phi_{\mu,p}$, and analyzes how perturbations of the spectral measure $\mu$ (via $\mu_n \to \mu_0$ in various senses) affect prediction errors and projections. The paper establishes general stability results under weak* convergence and identifies clear conditions (R1–R3) under which equality or convergence of prediction errors and projections holds, while highlighting limits through extensive examples. It then specializes to interpolation with one missing value, $m$-step ahead prediction, finite observation sets, and periodic observation patterns, deriving explicit formulas (e.g., via Szegö theory and Hardy spaces) and demonstrating both robust and fragile instability depending on $p$, convergence mode, and observation geometry. Overall, the results provide rigorous criteria for the robustness of spectral-based predictions and illuminate the roles of observation structure and $p$-norm in prediction stability with practical implications for time series and stochastic process forecasting.
Abstract
Let $Γ$ be an LCA group and $(μ_n)$ be a sequence of bounded regular Borel measures on $Γ$ tending to a measure $μ_0$. Let $G$ be the dual group of $Γ$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and $[{\mathcal T}(S)]_{μ_n,p}$ the subspace of $L^p(μ_n)$, $p \in (0,\infty)$, spanned by the characters of $Γ$ which are generated by the elements of $S$. The limit behaviour of the sequence of metric projections of the function $1$ onto $[{\mathcal T}(S)]_{μ_n,p}$ as well as of the sequence of the corresponding approximation errors are studied. The results are applied to obtain stability theorems for prediction of weakly stationary or harmonizable symmetric $p$-stable stochastic processes. Along with the general problem the particular cases of linear interpolation or extrapolation as well as of a finite or periodic observation set are studied in detail and compared to each other.