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\begin{document}
\title[Fractal Measures]{Orthogonal harmonic analysis \\
of fractal measures}

\author{Palle E. T. Jorgensen}
\address{Department of Mathematics, University of Iowa,
Iowa City, IA 52242}
\email{jorgen@math.uiowa.edu}

\author{Steen Pedersen}
\address{Department of Mathematics, Wright State University,
Dayton, OH 45435}
\email{steen@math.wright.edu}

%\date{\the\time, \the\day--\the\month--\the\year}
\subjclass{Primary 28A75, 42B10, 42C05; Secondary 47C05, 46L55}

\issueinfo{4}{06}{}{1998}
\dateposted{May 5, 1998}
\pagespan{35}{42}
\PII{S 1079-6762(98)00044-4}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}

\date{October 13, 1997}

\commby{Yitzhak Katznelson}

\keywords{Spectral pair, tiling, Fourier basis, self-similar measure,
          fractal, affine iteration, spectral resolution, Hilbert space}
\begin{abstract}
    We show that certain iteration systems lead to fractal measures
    admitting an exact orthogonal harmonic analysis.
\end{abstract}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Overview}\label{S:summary}
We study properties of pairs of Borel measures on $\mathbb{R}^{d}$
simultaneously generalizing Fourier series and the Fourier transform.
We show that certain fractal measures fall within the class of
measures admitting generalized Fourier series.

The class of fractal measures considered in this paper
are obtained from an affine iteration
construction leading to self-affine measures $\mu$ with support in
$\mathbb{R}^{d}$. The affine maps are determined by a given expansive
$d\times d$ matrix and a finite set of translation vectors. We show
that the corresponding $L^2$-space $L^2(\mu)$ has an orthonormal
basis of exponentials $e^{i2\pi\,\lambda\cdot x}$, indexed by vectors
$\lambda$ in $\mathbb{R}^{d}$, provided certain geometric conditions
hold for the affine system.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{S:introduction}
\subsection{Segal's question}

Let $\Omega$ be a Lebesgue measurable subset of $d$-dimensional
Euclidean space $\mathbb{R}^{d}$, $d\geq 1$. Let $L^{2}(m_{\Omega})$
be the corresponding Hilbert space of square integrable functions with
the inner product
\begin{equation*}
 \langle f,g \rangle:=\int\overline{f(x)}\,g(x)\,dm_{\Omega}(x)
\end{equation*}
where $m$ denotes Lebesgue measure on $\mathbb{R}^{d}$ and
$m_{\Omega}(\Delta):=m(\Omega\cap\Delta)$ is Lebesgue measure
restricted to the set $\Omega$. Motivated by a question
raised by I. E. Segal,
and a paper \cite{Fugl74} by B. Fuglede the problem of deciding for
which $\Omega$ of finite measure, the space $L^{2}(m_{\Omega})$ admits
an orthogonal basis
$\{e_{\lambda}(x):=\exp(i2\pi\,\lambda\cdot x):\lambda\in\Lambda\}$
of
exponentials, has been studied; see, e.g., \cite{Fugl74}, \cite{Jorg82},
\cite{Pede87}, \cite{JoPe92}, \cite{Pede96}, \cite{LaWa97}. It is
known \cite{Fugl74}, \cite{Pede87} that a connected open set $\Omega$
with finite measure admits an orthogonal basis of exponentials if and
only if there exists commuting (in the sense of commuting spectral
projections) self-adjoint extension operators
$H_{j}$, $1\leq j \leq d$ of the minimal partial derivative
operators $-i\frac{\partial}{\partial x_{j}}$ acting on
$C_{c}^{\infty}(\Omega)$, the space of all smooth functions compactly
supported in $\Omega$.

When $\Omega=[0,1]^{d}$ is a cube in $\mathbb{R}^{d}$, then
the class of all possible commuting extension operators was the focus
of attension in \cite{JoPe97}; the results are particularly satisfying
for $d\leq 3$. Domains $\Omega$ admitting extension operators
satisfying a weaker form of commutativity were studied by J. Friedrich
\cite{Frie87}, in the case $d=2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fuglede's Conjecture}
Let $\Omega$ be a Lebesgue measurable subset of $\mathbb{R}^{d}$ with
finite measure. If there exists a set $\Lambda$ such that
$\{e_{\lambda}:\lambda\in\Lambda\}$ is an orthogonal basis for
$L^{2}(m_{\Omega})$, then we say that $\Omega$ is a \emph{spectral set},
$\Lambda$ is a \emph{spectrum}, and $(\Omega,\Lambda)$ is a
\emph{spectral pair}. If there exists a set $T$ so that up to sets of
measure zero $\{\Omega+t:t\in T\}$ is a partition of
$\mathbb{R}^{d}$, then we say that $\Omega$ is a \emph{tile} and $T$
is a \emph{tiling set}.
\begin{conjecture}[Spectral Set Conjecture \cite{Fugl74}]
   Let $\Omega$ be a set of finite non-zero measure. Then $\Omega$ is
   a spectral set if and only if $\Omega$ is a tile.
\end{conjecture}
This conjecture is open in both directions even for $d=1$. A set $S$
which is the affine image of a set of the form $\mathbb{Z}^{d}+A$ for
some finite set $A$ so that $(A-A)\cap\mathbb{Z}^{d}=\emptyset$ is
called \emph{periodic} or a \emph{lattice with a base}.
\begin{conjecture}[Periodic Spectral Set Conjecture \cite{Pede97}]
   Let $\Omega$ be a set of fini\-te non-zero measure. Then $\Omega$ is
   a spectral set admitting a periodic spectrum if and only if
   $\Omega$ is a tile admitting a periodic tiling set.
\end{conjecture}
It is known \cite{Fugl74}, \cite{Jorg82}, \cite{Pede87}
that $\Omega$ is a spectral set with spectrum
$\mathbb{Z}^d$ if and only if $\Omega$ is a tile with tiling set
$\mathbb{Z}^d$.
In the papers \cite{Pede96} and \cite{LaWa97} the Periodic
Spectral Set Conjecture is reduced
to certain questions about finite subsets of
the integer lattice $\mathbb{Z}^{d}$. For $d=1$ some progress towards
a resolution of these questions is made in \cite{PeWa97}.
These results support the view that certain specific classes of
spectra correspond to certain corresponding classes of tiling sets.
This is further confirmed
by some results in \cite{JoPe97}, where we show that every periodic
spectrum for the cube $\Omega=[0,1]^{d}$ is also a tiling set for the
cube, and conversely that any periodic tiling set for the cube is a
spectrum for the cube. Very recently the periodicity hypothesis
in the cube result has been removed \cite{IoPe98}, \cite{LRW98}.

The results mentioned in the following will be established in a series
of papers by the co-authors; the first two papers in this series are
\cite{JoPe97} and \cite{JoPe97b}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pairs of measures}\label{S:measures}
\subsection{New pairs from old pairs}
While studying the problems described in Section \ref{S:introduction}
it turned out to be necessary to study spectral pairs in more general
situations. For these reasons we introduced in \cite{JoPe97} the
following more general formulation. Let $\mu$ and $\nu$ be Borel
measures on $\mathbb{R}^d$. We say that $(\mu,\nu)$ is a
\emph{spectral pair} if the map
\begin{equation*}
   Ff(\xi):=\int f(x)\,\overline{e_{\xi}(x)}\,d\mu(x)
\end{equation*}
defined for $f\in L^{1}\cap L^{2}(\mu)$,
extends by continuity to an isometric isomorphism mapping $L^{2}(\mu)$
onto $L^{2}(\nu)$. It was shown in \cite{Pede87} that if $\mu$ is the
restriction of Lebesgue measure to a connected open set of infinite
measure, then the connection to
commuting self-adjoint extensions of the directional
derivatives described in Section \ref{S:introduction} remains valid.
One of the nice features of the more general definition of a spectral
pair is that if $(\mu,\nu)$ is a spectral pair, then so is
$(\nu,\mu)$.

Recall that the \emph{convolution}
$\mu:=\mu _{1}*\mu _{2}$ of Borel measures $\mu _{j}$ on
$\mathbb{R}^{d}$ is given by
\begin{equation*}
   \int_{\mathbb{R}^{d}}f(x)\,d\mu (x)
   =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}
      f(x_{1}+x_{2})\,d\mu _{1}(x_{1})d\mu _{2}(x_{2}).
\end{equation*}
We will call a convolution $\mu _{1}*\mu _{2}$
\emph{\ non-overlapping}
if the map $f(x_{1},x_{2}):=g(x_{1}+x_{2})$ determines an isometric
isomorphism
$g\rightarrow f$ mapping $L^{2}(\mu _{1}*\mu _{2})$ onto
$L^{2}(\mu_{1}\times \mu _{2})$. The following result allows us to
construct a large class of spectral pairs.

\begin{theorem}[Convolution Theorem]\label{T:convolution}
   Suppose $(\mu _{j},\nu _{j})$ are
   spectral pairs in $\mathbb{R}^{d}$. If $\mu _{1}*\mu _{2}$,
   $\nu _{1}*\nu_{2}$
   are non-overlapping and
   \begin{equation*}%\label{E:compatibility}
      (\mu _{1}\times \nu _{2})
         \{(x_{1},\lambda _{2}):x_{1}\lambda _{2}\not\in\mathbb{Z}\}=0,
   \end{equation*}
   then $(\mu _{1}*\mu _{2},\nu _{1}*\nu _{2})$ is a spectral pair.
\end{theorem}

This result generalizes results from \cite{JoPe92}, \cite{JoPe94}, and
\cite{LaWa97}. All known examples of spectral pairs can be generated
using this result and multiplicative (see Section
\ref{S:applications} for a definition) spectral pairs.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Which measures are possible?}
It turns out that the class of measures that can be part of a spectral
pair is fairly limited; for example, we have

\begin{theorem}[Uncertainty Principle]
   Suppose $(\mu,\nu)$ is a spectral pair. Let $f\in L^2(\mu)$,
   $f\neq 0$, and $A$, $B\subset\mathbb{R}^d$. If
   $\| f-\chi_A f\|_{\mu}\leq\varepsilon$ and
   $\|Ff-\chi_B Ff\|_{\nu}\leq \delta$, then
   $(1-\varepsilon-\delta)^2\leq\mu(A)\nu(B)$.
\end{theorem}

\begin{theorem}[Local Translation Invariance]
   Suppose $(\mu,\nu)$ is a spectral pair and $t\in \mathbb{R}^d$.
   If $\mathscr{O}$ and $\mathscr{O}+t$ are subsets of the support of
   $\mu$, then $\mu(\mathscr{O})=\mu(\mathscr{O}+t)$.
\end{theorem}

M. N. Kolountzakis and J. C. Lagarias in \cite{KoLa96} discuss tilings
of the real line $\mathbb{R}$ by a function. Some given measurable
function $f$
\emph{tiles} the real line with \emph{tile set} $T$ if there exists a
constant $c$ such that
\begin{equation*}
   \sum_{t\in T}f(x+t)=c
\end{equation*}
for almost every $x\in \mathbb{R}$.
It follows from Local
Translation Invariance that for such a function to come from a
spectral pair it must be a multiple of the characteristic function of
some set. In particular, a natural but naive generalization of the
Spectral Set Conjecture to the setting of \cite{KoLa96} is false.

The following result establishes a direct connection to the
spectral pairs discussed in Section \ref{S:introduction}.

\begin{theorem}
   Suppose $(\mu,\nu)$ is a spectral pair. If
   $\mu(\mathbb{R}^d)<\infty$, then $\nu$ is a counting measure with
   uniformly discrete support.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fractal measures}\label{S:fractals}

\subsection{Dual iteration systems}
Consider a triplet $(R,B,L)$ such that $R$ is an expan\-sive $d\times d$
matrix with real entries, and $B$ and $L$ are subsets of
$\mathbb{R}^{d}$ so that
\begin{align}
   N &:=\#B=\#L;  \label{B=L} \\
   R^{n}b\cdot l &\in \mathbb{Z},
      \text{ for any }n\in \mathbb{N},\ b\in B,\ l\in L;
     \label{Compatibility} \\
   H_{B,L} &:=N^{-1/2}
      \left( e^{i2\pi b\cdot l}\right) _{b\in B,l\in L}
      \text{is a unitary }N\times N\text{ matrix}.
      \label{Hadamard}
\end{align}
We introduce two dynamical systems,
\begin{align*}
   \sigma _{b}(x) &:=R^{-1}x+b, \\
   \tau _{l}(x) &:=R^{*}x+l,
\end{align*}
and the corresponding ``attractors'',
\begin{equation*}
   X_{\sigma }
   :=\left\{\sum_{k=0}^{\infty}R^{-k}b_{k}:b_{k}\in B\right\}
\end{equation*}
and
\begin{equation}
   \mathcal{L} =X_{\tau }
   :=\left\{ \sum_{k=0}^{n}R^{*k}l_{k}:n\in \mathbb{N},
      l_{k}\in L\right\}.
   \label{eq:Ldef}
\end{equation}
The set $X_{\sigma}$ is the support of the unique probability measure
solving the equation
\begin{equation}
   \mu =N^{-1}\sum_{b\in B}\mu \circ \sigma _{b}^{-1}.
   \label{eq:mu}
\end{equation}
Our goal is to show that under appropriate assumptions the
exponentials
$\{e_{\lambda}:\lambda\in\mathcal{L}\}$
form an orthogonal basis for $L^2(\mu)$. It follows from the
assumptions (\ref{B=L})--(\ref{Hadamard})
on $(R,B,L)$ that the exponentials
$(e_{\lambda})_{\lambda\in\mathcal{L}}$
are orthogonal, so the question is
whether or not they span all of $L^2(\mu)$. If we set
\begin{equation}
   \chi _{B}(t):=N^{-1}\sum_{b\in B}e_{b}(t),
   %\label{eq:chaB}
\end{equation}
then expansiveness of $R$ implies that we have an explicit formula
for the Fourier transform of $\mu$,
\begin{equation}
   \widehat{\mu }(t)
   :=\int \overline{e_{t}(x)}\,d\mu(x)
   =\prod_{k=0}^{\infty }\chi _{B}(R^{*-k}t),
   \label{E:InfiniteProd}
\end{equation}
the convergence being uniform on bounded subsets of $\mathbb{R}^{d}$.
To facilitate the discus\-sion we introduce the function
\begin{equation*}
  Q(t):=\sum_{\lambda \in \mathcal{L}}
     \left| \widehat{\mu }(t-\lambda )\right| ^{2},\quad t\in \mathbb{R}^d,
  %\label{eq:Qdef}
\end{equation*}
and the operator $C$ given by
\begin{equation}
   \left( Cq\right) (t):=\sum_{l\in L}%
      \left| \chi _{B}(t-l)\right| ^{2} q(\rho_{l}(t))%
   \label{eq:Cdef}
\end{equation}
where $\rho_{l}(x):=R^{*-1}(x-l)$. The attractor
\begin{equation}
   X_{\rho}
   :=\left\{\sum_{k=1}^{\infty}-R^{*-k}l_{k}:l_{k}\in L\right\}
   \label{rho-recur-dom}
\end{equation}
corresponding to the
system of $\rho_{l}$'s will be used below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orthogonal bases}
Let $H_{2}(\mathcal{L})$ denote the subspace of $L^{2}(\mu )$ spanned
by the orthonormal set $\{e_{\lambda }:\lambda \in \mathcal{L}\}$. Any
$e_{t}$, $t\in \mathbb{C}^{d}$, is in $L^{2}(\mu )$, so
$H_{2}(\mathcal{L})$ is a subspace of $L^{2}(\mu )$. We will show that
$H_{2}(\mathcal{L})=L^{2}(\mu )$ for certain systems $(R,B,L)$
satisfying (\ref{B=L})--(\ref{Hadamard}).

Let $Y$ denote the convex hull of the attractor $X_{\rho }$ given by
(\ref{rho-recur-dom}), let
$\left\| q\right\| _{\infty }:=\sup_{y\in Y}\left| q(y)\right| $ and
\begin{equation}
   \left\| q\right\| _{Y,\infty }
   :=\left\| \left| \nabla q\right| _{2}\right\|_{\infty }
   \label{eq:Y-sup-norm}
\end{equation}
where
$\left| z\right| _{2}
:=\left( \sum_{j=1}^{d}\left| z_{j}\right|^{2}\right) ^{1/2}$ is the
usual Hilbert norm on $\mathbb{C}^{d}$. We begin by showing that,
if the
operator norm of $C$ acting on a suitable set of smooth functions is
less than one, then $\mu$ has the basis property.

\begin{theorem}\label{T:Cbound}
   Let $(R,B,L)$ be a system in $\mathbb{R}^{d}$
   satisfying \eqref{B=L}--\eqref{Hadamard}, $0\in L$.
   Let $C$ be the operator given by \eqref{eq:Cdef},
   let $Y$ denote the convex hull of the attractor $X_{\rho }$
   given by \eqref{rho-recur-dom}, and let
   $\left\| q\right\| _{Y,\infty }$ be given by
   \eqref{eq:Y-sup-norm}. Suppose $L$ spans $\mathbb{R}^{d}$; if
   there exists $\gamma <1$ so that
   $\left\| Cq\right\| _{Y,\infty }
      \leq \gamma\left\| q\right\| _{Y,\infty }$
   for all $q$ in a set of $C^{1}$-functions
   containing $\mathbf{1}-Q$,
   then $H_{2}(\mathcal{L})=L^{2}(\mu )$.
\end{theorem}

%\noindent
The following result allows us to compute an operator norm bound for
$C$ in terms of the data $(R,B,L)$ in a fairly straightforward
manner.

\begin{theorem}\label{T:estimate}
   Let $(R,B,L)$ be a system in $\mathbb{R}^{d}$
   satisfying \eqref{B=L}--\eqref{Hadamard}, $0\in L$. Let $C$ be the
   operator given by \eqref{eq:Cdef}, and let $Y$ denote the convex
   hull of the attractor $X_{\rho }$ given by \eqref{rho-recur-dom}.
   If $\left\| q\right\| _{Y,\infty }$ is given by
   \eqref{eq:Y-sup-norm} and
   \begin{equation*}
      \beta :=2\pi \diam(B)
        \max_{\substack{{b,b^{\prime }\in B}\\{l\in L}}}
           \left\|\sin(2\pi(b-b^{\prime})(\cdot-l))\right\|_{\infty},
   \end{equation*}
   then we have
   \begin{equation*}
      \left\| Cq\right\| _{Y,\infty }
      \leq \left[ \left( N-1\right) ^{2}N^{-1}\beta
         \left\| R^{-1}\right\| _{op}\max_{l\in L}\left| l\right|_{2}
      +\left\|R^{-1}\right\| _{hs}\right] \,
         \left\| q\right\|_{Y,\infty },
   \end{equation*}
   for any $C^{1}$-function $q$ such that $q(0)=0$. Here
   $\left\| T\right\| _{op}$ is the operator norm and
   $\left\| T\right\| _{hs}:=\left(\sum\nolimits_{j,k=1}^{d}
      \left| t_{j,k}\right|^{2}\right)^{1/2}$
   is the Hilbert-Schmidt norm of a $d\times d$ matrix $T$.
\end{theorem}

%\noindent
As a consequence of Theorem \ref{T:Cbound} and
Theorem \ref{T:estimate} we have

\begin{corollary}
   Let $(R,B,L)$ satisfy \eqref{B=L}--\eqref{Hadamard} and for
   $r\in \mathbb{N}$ let
   \begin{equation*}
   \mathcal{L}_{r}
   :=\left\{ \sum_{k=0}^{n}
      \left( rR^{*}\right) ^{k}l_{k}:n\in \mathbb{N},
         l_{k}\in L\right\},
   \end{equation*}
   and let $\mu _{r}$ be the probability measure solving
   \begin{equation*}
      \mu _{r}=N^{-1}\sum_{b\in B}\mu _{r}\circ \sigma _{r,b}^{-1}
   \end{equation*}
   where $\sigma _{r,b}(x):=\left( rR\right) ^{-1}x+b$. If $L$ spans
   $\mathbb{R}^{d}$ and $0\in L$, then
   $\{e_{\lambda }:\lambda \in \mathcal{L}_{r}\}$ is an orthonormal
   basis for $L^{2}(\mu _{r})$ provided $r$ is sufficiently large.
\end{corollary}

R. S. Strichartz obtained an asymptotic harmonic
analysis for the class of measur\-es considered in this paper; see
\cite{Stri94} for a survey of some of Strichartz's work on
self-similarity in harmonic analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications}\label{S:applications}

\subsection{The Hardy space connection}
One way to construct systems $(R,B,L)$ satisfying
(\ref{B=L})--(\ref{Hadamard}) is to pick $R$, $B$ and $L$ so that
\begin{equation}
   R\in M_{d}(\mathbb{Z}),\quad
   RB\subset \mathbb{Z}^{d},\quad
   L\subset \mathbb{Z}^{d}.
   \label{Compatibility2}
\end{equation}
In fact (\ref{Compatibility2}) implies (\ref{Compatibility}) since
$R^{n}b\cdot l=Rb\cdot R^{*(n-1)}l$ for $n=1,2,3,\ldots\,$. The only
condition that is hard to satisfy is (\ref{Hadamard}). This
condition is notoriously difficult to study; for example, it is not
known which matrices with entries in the unit circle satisfy
(\ref{Hadamard}) for $N=7$; see e.g. \cite{haag1},
\cite{BjSa95} for some progress
in the study of (\ref{Hadamard}). The condition (\ref{Compatibility2})
is closely related to a condition used in the study of certain
multi-dimensional wavelets. Some properties of systems $(R,B,L)$
satisfying (\ref{B=L}), (\ref{Compatibility2}), and (\ref{Hadamard})
were established in \cite{JoPe96a}.

If $(R,B,L)$ satisfies (\ref{B=L}), (\ref{Compatibility2}), and
(\ref{Hadamard}) and $z\in \mathbb{Z}^{d}$, then $(R,B,L_{z})$ also
satisfies (\ref{B=L}), (\ref{Compatibility2}), and (\ref{Hadamard}),
where $L_{z}:=L+z=\{l+z:l\in L\}$. So we may often assume that
$L\subset \mathbb{N}^{d} $. Therefore, if $R$ has non-negative integer entries,
we will often end up with $\{e_{\lambda }:\lambda \in \mathcal{L}\}$
being an orthonormal basis for $L^{2}(\mu )$ and each element in
$\mathcal{L}$ only having non-negative coordinates. This is an
interesting situation because the basis property leads to
\begin{equation*}
   f=\sum_{\lambda \in \mathcal{L}}
      \left\langle e_{\lambda }\mid f\right\rangle_{\mu }e_{\lambda }
\end{equation*}
for $f\in L^{2}(\mu )$, so setting $z_{j}:=e^{i2\pi x_{j}}$ we see
that
\begin{equation*}
   f(x)=\sum_{\lambda \in \mathcal{L}}
      \left\langle e_{\lambda }\mid f\right\rangle _{\mu }z^{\lambda }
\end{equation*}
where $z^{\lambda }:=\prod_{k=1}^{d}z_{k}^{\lambda _{k}}$; it follows
that $f(x)$, $x\in X_{\sigma }$, gives the boundary values of a function
analytic in the polydisc
$\{z\in \mathbb{C}^{d}:\left| z_{j}\right| <1\}$. Hence our
construction shows that many fractal $L^{2}$-spaces are Hardy spaces.
This is in sharp contrast to the Lebesgue spaces; for example, if
$m_{[0,1]}$ is Lebesgue measure restricted to the unit interval
$[0,1]$, then (essentially) the only set $\Lambda $ such that
$\{e_{\lambda }:\lambda \in \Lambda \}$ is an orthonormal basis for
$L^{2}(m_{[0,1]})$ is $\Lambda =\mathbb{Z}$, and the corresponding
analytic subspace spanned by
$\{e_{\lambda }:\lambda \in \mathbb{N}_{0}\}$ is far from being equal
to $L^{2}(m_{[0,1]})$. Note that $m_{[0,1]}=\mu $ if $R=2$, $B=\{0,1/2\}$,
and $L=\{0,1\}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{First order operators}
Let $(U_{t})_{t\in\mathbb{R}}$ be a one-parameter group  of
unitary operators on some $L^{2}$-space $L^{2}(\mu)$. We say that
$U_{t}$ acts \emph{multiplicatively}, if
\begin{equation*}
   U_{t}(fg)=(U_{t}f)(U_{t}g)
   %\label{E:multiplicative}
\end{equation*}
for all $t\in \mathbb{R}$ and all
$f$, $g\in L^{2}\cap L^{\infty}(\mu)$.

Given a probability measure admitting an orthogonal basis
$\{e_{\lambda}:\lambda\in\Lambda\}$ one can define commuting
extension operators $H_{j}$ by setting
\begin{equation*}
   H_{j}e_{\lambda}:=\lambda_{j}e_{\lambda}
   %\label{E:H_{j}def}
\end{equation*}
for $\lambda\in\Lambda$. By results mentioned in Section
\ref{S:introduction} it is reasonable to think of $H_{j}$ as an
``extension'' of $-i\frac{\partial}{\partial x_{j}}$.
Suppose further that $\mu$ is a purely
singular continuous measure generated as
in Section \ref{S:fractals}.
One can adapt methods from \cite{JoPe92} to show that
none of the $H_{j}$'s can satisfy Leibniz's rule, in the
sense that none of the unitary groups $(U_{j,t})_{t\in\mathbb{R}}$,
given by
\begin{equation*}
   U_{j,t}:=exp(i2\pi H_{j} t),
\end{equation*}
acts multiplicatively. A spectral pair is called
\emph{multiplicative} if the unitary groups
$(U_{j,t})_{t\in\mathbb{R}}$ act multiplicatively.
We may construct a Laplace operator $\Delta$ by setting
\begin{equation*}
   -\Delta:=H_{1}^{2}+\cdots+H_{d}^{2}.
\end{equation*}
This way of constructing a Laplace operator \emph{on} a fractal
complements the construc\-tion considered, for example, by J. Kigami and
M. Lapidus \cite{KiLa93}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Examples}

Using Theorem \ref{T:estimate}, Theorem \ref{T:Cbound}, and equation
(\ref{E:InfiniteProd}) one can
prove the following result.

\begin{theorem}\label{T:DimOne}
   Suppose $d=1$, $N=2$, $B=\{0,a\}$, with
   $a\in \mathbb{R}\setminus \{0\}$, $R$ is an integer with
   $\left| R\right| \geq 2$, and $\mu $ is given by \eqref{eq:mu}. If
   $R$ is odd, then $L^{2}(\mu )$ does not have a basis of
   exponentials for any $a\in \mathbb{R}\setminus \{0\}$. If $R$ is
   even and $\left| R\right| \geq 4$, then $L^{2}(\mu )$ has a basis
   of exponentials for any $a\in \mathbb{R}\setminus \{0\}$.
\end{theorem}


Using the Convolution Theorem (Theorem \ref{T:convolution})
and Theorem \ref{T:DimOne} one can verify the following example.

\begin{example}
   Let $\mu_{0}$ be the probability measure solving (\ref{eq:mu})
   when $R=4$ and
   $B=\{0,1/2\}$. Let $L=\{0,1\}$ and let $\mathcal{L}$ be given
   by (\ref{eq:Ldef}). Set $\Omega:=[0,1]+\mathcal{L}$. If
   \begin{equation*}
      \mu(\Delta):=m(\Delta\cap\Omega)
   \end{equation*}
   and
   \begin{equation*}
      \nu(\Delta):=\sum_{k=-\infty}^{\infty}\mu_{0}(\Delta+k),
   \end{equation*}
   then $(\mu,\nu)$ is a spectral pair, and
   $\Omega$ is a tile with tiling set $-2\mathcal{L}$.
\end{example}

This is an example of a spectral set of infinite measure whose
spectrum is not periodic. This takes us full circle ending up in a
situation discussed in Section \ref{S:introduction}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibliographystyle{amsalpha}
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\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

End-File: ERA.tex
%Repeat Begin-File:   and   End-File: tags for each included file.

--
Steen Pedersen,                phone 937-775-2432
Department of Mathematics,     fax 937-775-2081
Wright State University,       e-mail steen@math.wright.edu
Dayton OH 45435, USA           http://mjollnir.math.wright.edu/


      "Liberty lies in the rights of that person whose views
       you find most odious"  - John Stuart Mill




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