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\begin{talk}[A. Jensen]{M Krishna}{Transforms of Measures}
    
We look a the Wavelet transform of probability measures
on $\RR$ and recover the components of the measures 
in a Lebesgue decomposition from this transform.  
Our motivation for this work is that
in the spectral theory of selfadjoint operators essentially
only the resolvent operator and the unitary group generated
by the operators are widely used and there are lots of
interesting operators where the spectral theoretic questions
are open where it seems hard to use these functions. Therefore
there was a need to find out if there are other functions
of the selfadjoint operators that could be used to study
spectra and such a question translates the question we
address in this work.

In the following $\psi$ is a (complex valued) bounded continous 
even function with $\psi(0) \neq 0$, that satisfies the decay condition
$$
|\psi(x)| + |x \psi^\prime(x)| \leq C (1+|x|)^{-\delta}, ~~ \delta > 1
$$
for some $\delta > 1$.  Given such a $\psi$, we define
$\psi_a(x) = \psi(\frac{x}{a}), ~~ a>0$ and let $\psi_a*\mu$
to be $\int \psi_a(\cdot -y) d\mu(y)$. 


Then the theorems we prove are the following for which we
normalize $\int \psi(y) dy = 1$.

\begin{thm}
Let $\psi$ be as above and let $\mu$ be a probability measure on $\RR$.  Then
\begin{enumerate}
\item $\displaystyle{\lim_{a \rightarrow 0}} ~ \psi_{a} * \mu(x) = \psi(0) \mu(\{x\})$.
\item For every continuous function $f$ of compact support, the
following is valid.
$$
\lim_{a \rightarrow 0} \int \left(\frac{1}{a}\psi_a * \mu\right)(x) f(x) dx = \int f(x) d\mu(x).
$$
\item
Let 
$d^\alpha_\mu(x) = \lim_{\epsilon \rightarrow 0} \frac{\mu((x-\epsilon, x+\epsilon))}{(2\epsilon)^{\alpha}}$ be finite,
for some $0 < \alpha \leq 1$ and $x$, then
\begin{equation*}
\lim_{a \rightarrow 0} a^{-\alpha} \psi_{a}*\mu(x) = c_\alpha d^\alpha_\mu(x),
\end{equation*}
where $c_\alpha = \int_0^\infty \alpha 2^\alpha y^{\alpha-1} \psi(y) ~ dy$.
\end{enumerate}
\end{thm}

The next theorem is the anologue of the theorems of Simon \cite{si1}
for the case of Borel transforms.

\begin{thm}
Let $\mu$ be a probability measure on $\RR$ and $\psi$ be as
above.  Then for any bounded
interval $(c,d)$ the following are valid.
\begin{enumerate}
\item Let
$$
C = \int_\RR |\psi(x)|^2 ~~ dx.
$$
then
$$\lim_{a \rightarrow 0} \frac{1}{a} \int_c^d  |\psi_a * \mu|^2(x) ~~ dx = C \left(\sum_{x \in (c,d)}  \mu(\{x\})^2 +\frac{1}{2}\left[\mu(\{c\})^2 + \mu(\{d\})^2\right]\right).$$

\item Suppose that for some $p > 1$,
$$
\sup_{a > 0} \int_c^d \left|\frac{1}{a} (\psi_a * \mu)(x)\right|^p  ~ dx < \infty,
$$
then $\mu$ is purely absolutely continuous in $(c, d)$.  In addition,
for any compact subset $S$ of $(c, d)$
$$
\frac{1}{a} \psi_a*\mu \rightarrow  ~ \frac{d\mu_{ac}}{dx}, ~~ in  ~~ L^p(S), ~~\text{as} ~~ a \rightarrow 0.
$$
The converse that if $\mu$ is purely absolutely continuous with
the density $\frac{d\mu_{ac}}{dx}$ in $L^p((c,d))$, then the supremum
above is finite, is also valid.

\item For $0 < p < 1$, we have
$$
\lim_{a \rightarrow 0} \int_c^d \left|\frac{1}{a} \psi_a * \mu\right|^p (x) ~ dx = \int_c^d \left|\frac{d\mu_{ac}}{dx}\right|^p ~ dx.
$$
\end{enumerate}
\end{thm}

And finally for the quantities
$$
C_\mu^\alpha(x) = \displaystyle\limsup_{a \rightarrow 0}  \frac{\psi_a*\mu}{a^\alpha}(x). ~~
D_\mu^\alpha(x) = \displaystyle\limsup_{\epsilon \rightarrow 0}  \frac{\mu((x-\epsilon, x+\epsilon))}{(2\epsilon)^\alpha}, ~~
$$
one has the following theorem. 

\begin{thm}
Let $\mu$ be a probability measure and let $\psi$ be as above. 
Then $C_\mu^\alpha(x)$ is finite for
any $x$, whenever $D_\mu^\alpha(x)$ is
finite for the same $x$ and if $\psi$ is positive then they are both
finite or both infinite.
\end{thm}

There are lots of functions $\psi$ satisfying the conditions
we imposed in the above theorems. Therefore, there is hope
that the above theorems will enlarge the useful techniques available 
to the spectral theory of random and deterministic Schr\"odinger
operators. 

\begin{thebibliography}{1}
\bibitem{hol95}
M.~Holschneider.
\newblock {\em Wavelets. An analysis tool.}
\newblock Oxford Mathematical Monographs. Oxford Science Publications
.
The Clarendon Press, Oxford University Press, New York, 1995.

\bibitem{MR97m:47002}
R.~del Rio, S.~Jitomirskaya, Y.~Last, and B.~Simon.
\newblock Operators with singular continuous spectrum. {I}{V}. {H}aus
dorff
  dimensions, rank one perturbations, and localization.
  \newblock {\em J. Anal. Math.}, 69:153--200, 1996.

  \bibitem{MR97c:47008}
  Barry Simon.
  \newblock Spectral analysis of rank one perturbations and application
  s.
  \newblock In {\em Mathematical quantum theory. II. Schr\"odinger oper
  ators
    (Vancouver, BC, 1993)}, pages 109--149. Amer. Math. Soc., Providenc
    e, RI,
      1995.

      \bibitem{si1} \newblock{B. Simon,}
          \newblock{\em Borel Transforms of Measures,}
	      \newblock{Proceedings of the AMS, {\bf 123}, No 12, 3749-3755, 19
	      95.}
	      \bibitem{93c:42036}\newblock{Robert S. Strichartz,}
	          \newblock{\em Wavelet expansions of fractal measures,}
		      \newblock{ J. Geom. Anal, {\bf 1}, no. 3, 269--289, 1991.}
		      \end{thebibliography}

\end{talk}

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