# The minimal eigenfunction

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## 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 *L*^{2} (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 *m*_{0}(*x*), i.e., *m*_{0}(*x*)→*m*_{0}(3*x*).

## Keywords

Wavelet Basis Haar Wavelet Tight Frame Iterate Function System Frame Wavelet## Preview

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