AN OPERATOR-FRACTAL

作者：Jorgensen Palle E T; Kornelson Keri A^*; Shuman Karen L

来源：Numerical Functional Analysis and Optimization, 2012, 33(7-9): 1070-1094.

DOI：10.1080/01630563.2012.682127

• C*-algebras
• Cuntz relations
• Fractals
• Fourier basis
• Multiresolution
• Spectrum
• Unitary operators

Certain Bernoulli convolution measures mu are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the ONB need not be unique; indeed, there are often families of such spectral bases. %26lt;br%26gt;Let lambda = 1/2n for a natural number n and consider the Bernoulli measure with scale factor lambda. It is known that L-2(mu(lambda)) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. %26lt;br%26gt;When L-2(mu(lambda)) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.

• 出版日期2012