``` +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ *Letter from a reader*: ```

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Thanks for the copy of your new book. It looks very interesting and it is beautifully presented. I do not know what the picture on the cover is meant to represent, but it looks like fragments of knees and buttocks rising out of the bath water. What does the dedication mean? Who or what is the imaginary friend? ``` ```

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```*Palle Jorgensen's suggested answer*: ```

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The theme of the book, and its title, refer to the fact that central properties of wavelets depend on the spectrum of a certain operator R, named after David Ruelle, and also called the wavelet-transfer operator.

Spread out over the two sides of the book-cover, you have a family of wavelets, each generating function (scaling function) determined by two parameters. Having the algorithm for the scaling function, an extra step automatically gets us the wavelet itself; see, for example, the flip-book of wavelets on the interval [0,5] and their associated scaling functions [large PDF file].

In the background, in the picture on the book-cover, you see variations of these wavelets, as the parameters move around and cover a period-square. The part of the spectrum of R which determines the shape of each wavelet is a finite set of points. As each point in this part of the spectrum is a function of the two parameters, the eigenvalues may be ordered, and presented in sketch as functions of two variables. The graphs represent surfaces starting with a top eigenvalue (a Perron-Frobenius eigenvalue equal to 1 for all values of the parameters). One of the surfaces, two steps down from the top, is pictured in the round 'Looking Glass frame' on the book cover. Someone else said it looks like a butterfly, but there is a constant eigenvalue 1/2 that gives the appearance of mountains rising from lakes, or Christopher Columbus's hat, or a body in bathwater if you prefer! (And then the top eigenvalue equal to 1 is the ceiling of the bathroom?)

Other eigenvalue surfaces in this series are shown in figures within the book, and one surface with branch cuts, used as a sort of frontispiece (see Excerpts), was referred to by the authors as "Yosemite," after the famous "Half Dome" of Yosemite National Park in California. This closed-form eigenvalue, when compared to the eigenvalue surfaces sorted by absolute value, coincides with the second eigenvalue from the top in part of the parameter range, and with smaller eigenvalues in other parts of the range.

The parameters representing the lower parts of the surfaces are important: for example, they give algorithms that are faster. And wavelets that are more regular tend to be located in the region of the period-square which have a concentration of low-lying spectrum. Hence the flat portions of the "bathwater" surface signify an abundance of "good" wavelets.

As for the dedication, I thought this one is more "democratic" than most standard dedications. As a reader of other books, I often have difficulties relating to various named family members of the authors. Here, for our book, each reader is free to have his/her own answer to the question: "What does the dedication mean? Who or what is the imaginary friend?"--(answer): I have received a number of suggested answers from friends, some referring to math symbols (1, 2), one to the quantum computer which processes registers of qubits (technologically imaginary, as yet), and others referring to long lost childhood memories.