Mixed‐Norm Spaces and Prediction of S<i>α</i>S Moving Averages

Cheng, Raymond; Harris, Charles B.

doi:10.1111/jtsa.12134

Cited by 8 publications

(1 citation statement)

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“…These processes include α-stable processes with α ∈ (1, 2], L p -harmonizable processes, and strictly stationary L p processes. The orthogonality condition is connected to associated prediction problems, Wold-type decompositions, and moving-average representations [7,10,11,12,13,14,27].…”

Section: Banach Space Geometrymentioning

confidence: 99%

Inner functions and zero sets for ℓ^{𝑝}_{𝐴}

Cheng

Mashreghi

Ross

2019

Trans. Amer. Math. Soc.

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In this paper we characterize the zero sets of functions from ℓ p A (the analytic functions on the open unit disk D whose Taylor coefficients form an ℓ p sequence) by developing a concept of an "inner function" modeled by Beurling's discussion of the Hilbert space ℓ 2 A , the classical Hardy space. The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, we give an alternative proof of a result of Vinogradov [34] which says that when p > 2, there are zero sets for ℓ p A which are not Blaschke sequences.This work was supported by NSERC (Canada). 1 (4.1), provides many examples. Definition 5.3. For f ∈ ℓ p A , let f be the metric projection of f onto [Sf ] from Definition 3.6. Proposition 5.4. For f ∈ ℓ p A , let J = f − f be the co-projection of f . Then J is p-inner and any p-inner function arises in this way.Proof. First observe that f is the unique vector in [Sf ] which satisfieswhere P denotes the set of analytic polynomials. Then for any t ∈ C and n ∈ N we see that sinceFrom the definition of Birkhoff-James orthogonality from (4.1), J ⊥ p S n J for all n ∈ N and thus J is p-inner.Conversely, suppose that J is p-inner. Then J ⊥ p S n J for all n ∈ N. But since the criterion for Birkhoff-James orthogonality in the ℓ p A setting (4.2) is linear in the second argument, we see that J ⊥ p QJ for all Q ∈ P, Q(0) = 0. This means that J − J = inf J + QJ p : Q ∈ P, Q(0) = 0} J p .By the definition of J, we see that J = 0 and so the p-inner function J = J − J is of the desired form.

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