Harmonic analysis for resistance forms

Kigami, Jun

doi:10.1016/s0022-1236(02)00149-0

Cited by 124 publications

(213 citation statements)

References 20 publications

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“…In [Ki5], a work submitted after our paper was completed, Kigami proves some extensions of results from his book [Ki4] to a wider context which includes all the Dirichlet forms considered in this paper.…”

Section: Effective Resistance Metric Green's Function and Capacity Omentioning

confidence: 71%

Dirichlet forms on the Sierpiński gasket

Meyers¹,

Strichartz²,

Teplyaev³

2004

Pacific J. Math.

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We study not necessarily self-similar Dirichlet forms on the Sierpiński gasket that can be described as limits of compatible resistance networks on the sequence of graphs approximating the gasket. We describe the compatibility conditions in detail, and we also present an alternative description, based on just 3 conductance values and the 3-dimensional space of harmonic functions. In addition, we show how to parameterize all the Dirichlet forms by a set of independent variables.

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Section: Effective Resistance Metric Green's Function and Capacity Omentioning

confidence: 71%

Dirichlet forms on the Sierpiński gasket

Meyers¹,

Strichartz²,

Teplyaev³

2004

Pacific J. Math.

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“…The function ψξ is 0-harmonic, i.e. harmonic, so by Proposition 5.12. of [13] it is Lipschitz with respect to the resistance metric R(·, ·) on K. Therefore…”

Section: Harmonic Functions and Besov-lipschitz Spacesmentioning

confidence: 98%

“…In particular, one can prove that harmonic functions are Lipschitz with respect to the so-called resistance metric on fractals, see [13].…”

Section: Introductionmentioning

confidence: 99%

Harmonic functions representation of Besov-Lipschitz functions on nested fractals

Bodin¹,

Pietruska-Pałuba²

2012

Ann. Acad. Sci. Fenn. Math.

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“…See e.g. [DJ06b], [DJ06c], [Jor06], [JS07b], [JS07a], [JP98c], [JP98a], [JP98b], [Kig01], [Kig03], [LP06], [OS07], [Str06b], [Str06a].…”

Section: (B) Operators and Hilbert Spacementioning

confidence: 99%

Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

Jørgensen

Song

2009

Acta Appl Math

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The purpose of this paper is to show that algorithms in a diverse set of applications may be cast in the context of relations on a finite set of operators in Hilbert space. The Cuntz relations for a finite set of isometries form a prototype of these relations. Such applications as entropy encoding, analysis of correlation matrices (Karhunen-Loève), fractional Brownian motion, and fractals more generally, admit multi-scales. In signal/image processing, this may be implemented with recursive algorithms using subdivisions of frequency-bands; and in fractals with scale similarity. In Karhunen-Loève analysis, we introduce a diagionalization procedure; and we show how the Hilbert space formulation offers a unifying approach; as well as suggesting new results.

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Harmonic analysis for resistance forms

Cited by 124 publications

References 20 publications

Dirichlet forms on the Sierpiński gasket

Dirichlet forms on the Sierpiński gasket

Harmonic functions representation of Besov-Lipschitz functions on nested fractals

Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

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