Reproducing Pairs and Gabor Systems

This thesis can be summarized as follows. Chapter 1 establishes the foundational concepts and definitions, ensuring the work is largely self-contained. It introduces normed linear spaces, including Banach and Hilbert spaces, with examples like ℓ2(ℤ) and L2(E) for measurable sets E. It covers key notions such as convergence, Cauchy sequences, and orthogonality, with the use of an inner product. The chapter also covers operator theory, including linear and antilinear operators, boundedness, continuity, and the extension of bounded operators from dense subsets. The dual and anti-dual spaces are introduced, culminating in an antilinear version of the Riesz representation theorem. Chapter 2 explores reproducing pairs in Hilbert spaces, focusing on the discrete case. Reproducing pairs are a generalization of frames. They consist of two sequences ψ and Φ and an induced bounded invertible operator Sψ,Φ: ℋ ⟶ ℋ, defined weakly by ⟨S,f , h⟩ = iJ⟨f , ψᵢ⟩ ⟨φᵢ , h⟩ , f, h ∈ ℋ In the context of reproducing pairs, we study sequences that are overcomplete by one element—those that become exact upon removal of a single element. Theorem 2.2.2 shows that if such a sequence has a reproducing partner, the resulting subsequence must be a Schauder basis. This is applied to weighted exponential systems and the Gaussian Gabor system at critical density, both of which lack Schauder bases and hence cannot admit reproducing partners. Theorem 2.5.3 generalizes this result to sequences overcomplete by finitely many elements, with applications to weighted exponential systems. The chapter concludes by introducing exponential reproducing pairs, where the sequences are weighted exponentials. The weak operator Sgɣ is defined by ⟨Sgf , h⟩ = nℤ⟨f ,gen⟩ ⟨en , h⟩ , f, h ∈ ℋ where en = e2πint and the inner product is on L2([0,1]). Theorem2.6.4 shows that Sgɣ acts as a multiplication operator: Sgɣf = f g̅ɣ for f ∈ L2([0,1]). Theorem 2.6.6 gives necessary and sufficient conditions for (g,ɣ) to form an exponential reproducing pair, with further refinements in Section 2.7 under weakened assumptions. In Chapter~3, we extend a 2012 result of Heil and Yoon, on exact weighted exponential sequences to two dimensions. We first characterize when the weighted double exponential system ℰ(d_{x₀,ξ₀}(x, ξ)ⁿ, ℤ² \ F), with F ⊆ ℤ² finite and d_{x₀, ξ₀}(x, ξ) = √((x - x₀)² + (ξ - ξ₀)²) is minimal and complete, culminating in Theorem 3.2.18. We then analyze when the weighted system ℰ(g, ℤ² \ F) is exact for arbitrary g ∈ L²(Q), obtaining sufficient and necessary conditions (Theorems 3.4.4 and Theorem 3.5). Finally, Chapter 4 presents a minor result in frame theory: Theorem 4.3.2 proves that no frame exists whose every infinite subset is also a frame.

Date

2025-07-28

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Text

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Dissertation

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