Low-pass filters and representations of the Baumslag Solitar group

Abstract

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.

Reducibility of the wavelet representation associated to the Cantor set

“…When one takes the QMF filter m 0 = 1 the situation is very different. As shown in [Dut06], the representation can be realized on a solenoid, and in this case it is irreducible. The result holds even for more general maps r if they are ergodic (see [DLS09]).…”

“…This representation is also reducible and its direct integral decomposition is similar to the one for L 2 (R). See [BDP05,Dut06]. When one takes the QMF filter m 0 = 1 the situation is very different.…”

“…The general theory of the decomposition of wavelet representations into irreducible components was given in [Dut06], but there is a large class of examples where it is not known whether these representations are irreducible or not.…”

“…The measure μ ∞ on the solenoid T ∞ can also be computed; see [Dut06]. It is supported on the embedding of R in the solenoid T ∞ .…”

“…They were used to construct wavelet bases and multiresolutions on fractal measures and Cantor sets [DJ06] or on solenoids [Dut06].…”