Stability of trigonometric approximation in L^p and applications to prediction theory
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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 ∖{ 0 }, and [𝒯(S)]_μ_n,p the subspace of L^p(μ_n), p ∈ (0,∞), 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 [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.
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