Wavelets on Fractals

By Dorin Ervin Dutkay, Palle E. T. Jorgensen. 2003

170 citations recorded by the source corpus.

We show that there are Hilbert spaces constructed from the Hausdorff measures Hs on the real line R with 0 < s < 1 which admit multiresolution wavelets. For the case of the middle-third Cantor set C I [0,1], the Hilbert space is a separable subspace of L2(R, (dx)s) where s = log3(2). While we develop the general theory of multiresolutions in fractal Hilbert spaces, the emphasis is on the case of scale 3 which covers the traditional Cantor set C.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25945c6f2734ae22d8a0620a65ea03dba0842e67

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In Applied Mathematics

In Computer Vision and Pattern Recognition

In Mathematical Physics

Citations in the corpus

Construction of Parseval wavelets from redundant filter systems (2004, unknown): “In addition our multiplicity analysis has recently been used in papers by Dutkay and Jorgensen [ DJ1 , DJ2, DJ3, DJ4] in the study of nonlinear dynamical models.”
ON THE SPECTRA OF A CANTOR MEASURE (2008, background): “These fractal measures also have very close connections with the theory of mutiresolution analysis in wavelet analysis (see e.g., [DJ07c, DJ06b ]).”
Analysis of orthogonality and of orbits in affine iterated function systems (2006, methodology): “ole in a certain image processing algorithm. This fact is also based in part on a certain design (due to the co-authors) of a multiresolution analysis for the present aﬃne fractal systems (AIFS); see [DuJo06b]. With our introduction here of scaling in the small and in the large, our present paper ha”
Cuntz–Krieger Algebras and Wavelets on Fractals (2009, methodology): “In particular, this means that one can apply to the Sierpinski set SA all the techniques for constructions of wavelets on fractals from representations of Cuntz algebras developed, for instance, in [3,4,12–14,16–19,26–28], etc.”
MARTINGALES, ENDOMORPHISMS, AND COVARIANT SYSTEMS OF OPERATORS IN HILBERT SPACE (2004, background): “ich satisﬁes (2.11). 110 DUTKAY AND JORGENSEN Examples of measures µ on R/Zwhich satisfy (2.10) but not (2.11) are •µ =δ0, the Dirac mass at x =0; •µ =µC, the Cantor middle-third measure on [0,1](see [DutJo]), i.e., µC is determined by – 1 2 R f x 3 +f x+2 3 dµC(x)= R f(x)dµC(x), – µC([0,1])=1, – µC”
FOURIER SERIES ON FRACTALS: A PARALLEL WITH WAVELET THEORY (2007, background): “FSs) in Rd, i.e., for ﬁnite contractive function systems in Rd, so in dreal dimensions. One of the applications of these IFSs, and their spectral theory, is to image processing, see e.g., [Bar06] and [DJ06a]. The viewpoint in [Bar06] is that a variety of images, e.g., large bit-size satellite photog”
A class of Gaussian processes with fractional spectral measures (2010, unknown): “The spaces L 2 ( dσ ) For later use, we review two results from [5, 6, 7, 8] and [24].”