Celebratio Mathematica

I. Daubechies and J. C. Lagarias: “Two-scale difference equations, I: Existence and global regularity of solutions,” SIAM J. Math. Anal. 22 : 5 (1991), pp. 1388–1410. MR 1112515 Zbl 0763.42018 article

A two-scale difference equation is a functional equation of the form \[ f(x) = \sum_{n=0}^N c_n f(\alpha x - \beta_n) ,\] where \( \alpha > 1 \) and \( \beta_0 < \beta_1 < \cdots < \mbox{} \) \( \beta_n \), are real constants, and \( c_n \) are complex constants. Solutions of such equations arise in spline theory, in interpolation schemes for constructing curves, in constructing wavelets of compact support, in constructing fractals, and in probability theory. This paper studies the existence and uniqueness of \( L^1 \)-solutions to such equations. In particular, it characterizes \( L^1 \)-solutions having compact support. A time-domain method is introduced for studying the special case of such equations where \( \{\alpha,\beta_0,\dots \), \( \beta_n\} \) are integers, which are called lattice two-scale difference equations. It is shown that if a lattice two-scale difference equation has a compactly supported solution in \( C^m(\mathbb{R}) \), then \[ m < \frac{\beta_n - \beta_0}{\alpha - 1} - 1 .\]

I. Daubechies and J. C. Lagarias: “Sets of matrices all infinite products of which converge,” Linear Algebra Appl. 161 (January 1992), pp. 227–263. A corrigendum/addendum to this article was published in Linear Algebra Appl. 327:1–3 (2007). MR 1142737 Zbl 0746.15015 article

An infinite product \[ \prod_{i=1}^{\infty}\mathrm{M}_i \] of matrices converges (on the right) if \[ \lim_{i\to\infty}\mathrm{M}_1\cdots\mathrm{M}_i \] exists. A set \[ \Sigma = \{\mathrm{A}_i:i\geq 1\} \] of \( n{\times}n \) matrices is called an RCP set (right-convergent product set) if all infinite products with each element drawn from \( \Sigma \) converge. Such sets of matrices arise in constructing self-similar objects like von Koch’s snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set \( \Sigma \) to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in \( \Sigma \) and finite products of these matrices. Necessary and sufficient conditions are given for a finite set \( \Sigma \) to be an RCP set having a limit function \[ \mathrm{M}_{\Sigma}(\mathbf{d}) = \prod_{i=1}^{\infty}\mathrm{A}_{d_i} ,\] where \( \mathbf{d} = (d_1, \dots, d_n,\dots) \), which is a continuous function on the space of all sequences \( \mathrm{d} \) with the sequence topology. Finite RCP sets of column-stochastic matrices are completely characterized. Some results are given on the problem of algorithmically deciding if a given set \( \Sigma \) is an RCP set.

I. Daubechies and J. C. Lagarias: “Two-scale difference equations, II: Local regularity, infinite products of matrices, and fractals,” SIAM J. Math. Anal. 23 : 4 (1992), pp. 1031–1079. MR 1166574 Zbl 0788.42013 article

This paper studies solutions of the functional equation \[ f(x) = \sum_{n=0}^N c_n f(kx-n), \] where \( k\geqq 2 \) is an integer, and \[ \sum_{n=0}^N c_n = k .\] Part I showed that equations of this type have at most one \( L^1 \)-solution up to a multiplicative constant, which necessarily has compact support in \( [0 \), \( N/k-1] \). This paper gives a time-domain representation for such a function \( f(x) \) (if it exists) in terms of infinite products of matrices (that vary as \( x \) varies). Sufficient conditions are given on \( \{c_n\} \) for a continuous nonzero \( L^1 \)-solution to exist. Additional conditions sufficient to guarantee \( f\in C^r \) are also given. The infinite matrix product representations is used to bound from below the degree of regularity of such an \( L^1 \)-solution and to estimate the Hölder exponent of continuity of the highest-order well-defined derivative of \( f(x) \). Such solutions \( f(x) \) are often smoother at some points than others. For certain \( f(x) \) a hierarchy of fractal sets in \( \mathbb{R} \) corresponding to different Hölder exponents of continuity for \( f(x) \) is described.

I. Daubechies and J. C. Lagarias: “On the thermodynamic formalism for multifractal functions,” pp. 1033–1070 in Special issue dedicated to Elliott H. Lieb, published as Rev. Math. Phys. 6 : 5A. Issue edited by M. Aizenman and H. Araki. World Scientific (Singapore), 1994. Also published in The state of matter (1994). MR 1301365 Zbl 0843.58091 incollection

The thermodynamic formalism for “multifractal” functions \( \phi(x) \) is a heuristic principle that states that the singularity spectrum \( f(\alpha) \) (defined as the Hausdorff dimension of the set \( S_{\alpha} \) of points where \( \phi \) has Hölder exponent \( \alpha \)) and the moment scaling exponent \( \tau(q) \) (giving the power law behavior of \[ \int|\phi(x+t)-\phi(x)|^q\,dx \] for small \( |t| \)) should be related by the Legendre transform, \[ \tau(q) = 1 + \inf_{\alpha\geq 0}[q\alpha-f(\alpha)] .\] The range of validity of this heuristic principle is unknown. Here this principle is rigorously verified for a family of “toy examples” that are solutions of refinement equations. These example functions exhibit oscillations on all scales, and correspond to multifractal signed measures rather than multifractal measures; moreover, their singularity spectra \( f(\alpha) \) are not concave.

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I. Daubechies and J. C. Lagarias: “On the thermodynamic formalism for multifractal functions,” pp. 213–264 in The state of matter: A volume dedicated to E. H. Lieb (Copenhagen, July 1992). Edited by M. Aizenman and H. Araki. Advanced Series in Mathematical Physics 20. World Scientific (Singapore), 1994. Also published in Rev. Math. Phys. 6:5A (1994). MR 1462145 incollection

Jeffrey Clark Lagarias	Related
Michael Aizenman	Related
Elliott Hershel Lieb	Related
Huzihiro Araki	Related

I. Daubechies and J. C. Lagarias: “Corrigendum/addendum to: ‘Sets of matrices all infinite products of which converge’,” Linear Algebra Appl. 327 : 1–3 (April 2001), pp. 69–83. Corrigendum/addendum to article published in Linear Algebra Appl. 161 (1992). MR 1823340 Zbl 0978.15024 article

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