The minimal eigenfunction

Abstract

This brief chapter serves as a bridge between the iterated function systems (IFS) in Chapter 4 and the more detailed wavelet analysis in Chapters 7 and 8. We wish to examine a variation of scale, i.e., examine the effect on the Perron-Frobenius-Ruelle theory induced by a change of scale in a wavelet basis. After stating a general result (Theorem 6.1.1), we take a closer look at a single example: Recall that Haar’s wavelet is dyadic, i.e., it is a wavelet basis for L² (ℝ) which arises from the operations of translation by the integers ℤ, and by scaling with all powers of two, i.e., scaling by 2^j, as j ranges over ℤ. But the process begins with the unit box function, say supported in the interval from x=0 to x=1. The scaling by 3, i.e., f→f(x/3), stretches the support to the interval [0, 3]. It is natural to ask what happens to an ONB dyadic wavelet under scaling by 3, i.e., f→f(x/3). This is related to over-sampling: The simplest instance of this is the following scaling of the low-pass filter m₀(x), i.e., m₀(x)→m₀(3x).

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