Positive Definite Maps, Representations and Frames

Dutkay, Dorin Ervin

doi:10.1142/s0129055x04002047

Cited by 13 publications

(10 citation statements)

References 11 publications

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“…Everything follows from Theorem 4.3, Proposition 4.5, Theorem 3.18 in [Dut3], and see also Theorem 2.7.1 in [Jor04]. …”

Section: Corollary 46 There Is a One-to-one Linear And Monotone Cormentioning

confidence: 98%

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Low-pass filters and representations of the Baumslag Solitar group

Dutkay

2006

Trans. Amer. Math. Soc.

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Abstract. We analyze representations of the Baumslag Solitar groupthat admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.

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“…Everything follows from Theorem 4.3, Proposition 4.5, Theorem 3.18 in [Dut3], and see also Theorem 2.7.1 in [Jor04]. …”

Section: Corollary 46 There Is a One-to-one Linear And Monotone Cormentioning

confidence: 98%

“…From (ii) to (iii), the correspondence is f → h = E(f ). From (iii) to (i) the correspondence is given by Theorem 3.18 in [Dut3].…”

Section: Corollary 46 There Is a One-to-one Linear And Monotone Cormentioning

confidence: 99%

Low-pass filters and representations of the Baumslag Solitar group

Dutkay

2006

Trans. Amer. Math. Soc.

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“…For this we will first construct a certain positive definite map, following the ideas in [11]. Recall that a map K : X × X → C is said to be positive definite if for all finite sets F ⊂ X and any x i ∈ X, ξ i ∈ C, with i ∈ F one has i, j∈F…”

Section: General Dilations Of Parseval Waveletsmentioning

confidence: 99%

“…Proof Using Kolmogorov's result mentioned in [11,Theorem 2.2], we obtain a Hilbert space H and a map v :…”

Section: )mentioning

confidence: 99%

Orthonormal dilations of Parseval wavelets

et al. 2008

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We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar groupWe give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics (Theorem 3.14). We prove that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics (Theorem 3.24). We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we construct Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.

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“…The idea of using the Kolmogorov extension principle in wavelet constructions was suggested in three papers by Dorin Dutkay [Dut02,Dut04b,Dut04a].…”

Section: References and Remarksmentioning

confidence: 99%