Ruelle operators: functions which are harmonic with respect to a transfer operator

Abstract

Abstract. Let N ∈ N, N ≥ 2, be given. Motivated by wavelet analysis, we consider a class of normal representations of the C * -algebra A N on two unitary generators U , V subject to the relationThe representations are in one-to-one correspondence with solutions h ∈ L 1 (T), In multiresolution wavelet theory, there is a fundamental interplay and interconnection between the following two operators: M and R, where M is the cascade refinement operator and R is the corresponding transfer operator. We also denote the second of these, R, the Ruelle operator, because of its close connection to an operator that David Ruelle used first in his study of phase transitions in quantum statistical mechanics lattice models; see [Rue68], [May80],and [Rue78a]. We first recall the two operators here in a simple wavelet context, but the scope will be 1991 Mathematics Subject Classification. Primary 46L60, 47D25, 42A16, 43A65; Secondary 46L45, 42A65, 41A15.

Radial Multiresolution, Cuntz Algebras Representations and an Application to Fractals

“…The generalized translation reveals the group convolution of radial functions in R 3 . By Lemma 4.10 in [30], given φ ∈ L 2 (H) as an orthonormal scaling function of a radial multiresolution, the associated filter function G satisfies with (ω α ) defined in (18). Hence G is essentially bounded and contained in L 2 ([0, 1]), which allows us to develop it into a cosine series…”

“…is the Bessel function of order α, α ∈ R. Let H be the Hilbert space L 2 R + , ω (α) , where the measure ω (α) on R + is given by (18) …”

“…(For a variety of other examples, the reader is referred to [18]. The setup there applies to dynamical systems of N -to-1 Borel measurable selfmaps: for example, those of complex dynamics and Julia sets.)…”

Positive Definite Maps, Representations and Frames

“…In harmonic analysis unitary representations of groups can be constructed when a function of positive type is present (see [Fol]). Representations are ubiquitous also in the theory of wavelets and frames (see [HL], [Jor98]). We will see how these various results have in fact a common ground -a classical theorem of Kolmogorov (theorem 2.2), also known in the literature as the Kolmogorov decomposition of positive definite kernels.…”

“…Then we apply it to Gabor type unitary systems and we obtain unitary representations and for wavelets we get the cyclic representations introduced in [Jor98].…”