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It is probably true quite generally that in the history of human
thinking the most fruitful developments frequently take place at
those points where two different lines of thought meet. Hence, if
they actually meet, that is, if they are at least so much related
to each other that a real interaction can take place, then one may
hope that new and interesting developments may follow. Werner
Heisenberg This volume contains papers presented at the August 1992
NATO Advanced Study Institute on Wavelets and Their Applications.
The conference was held at the beautiful Il Ciocco resort near
Lucca, in the glorious Tuscany region of northern Italy. Once again
we gathered at this idyllic spot to explore and extend the
reciprocity between mathematics and engineering. The dynamic
interaction between world-renowned scientists from the usu ally
disparate communities of pure mathematicians and applied
scientists, which occurred at our 1989 and 1991 ASI's, continued at
this meeting. Wavelet theory and technology is in an important
growth stage at which theoretical and practical results are being
compared with existing methods. There have been spectacular wavelet
successes and sobering comparisons with traditional ideas-but still
there is a wide expanse of scientific problems to explore. Since
these problems lie at the forefront of both pure mathematics and
applied science, our NATO ASI was especially pertinent at this
time."
Probability has been an important part of mathematics for more than
three centuries. Moreover, its importance has grown in recent
decades, since the computing power now widely available has allowed
probabilistic and stochastic techniques to attack problems such as
speech and image processing, geophysical exploration, radar, sonar,
etc. -- all of which are covered here. The book contains three
exceptionally clear expositions on wavelets, frames and their
applications. A further extremely active current research area,
well covered here, is the relation between probability and partial
differential equations, including probabilistic representations of
solutions to elliptic and parabolic PDEs. New approaches, such as
the PDE method for large deviation problems, and stochastic optimal
control and filtering theory, are beginning to yield their secrets.
Another topic dealt with is the application of probabilistic
techniques to mathematical analysis. Finally, there are clear
explanations of normal numbers and dynamic systems, and the
influence of probability on our daily lives.
Wavelets and wavelet packets provide a theory analogous to Fourier
analysis and tools analogous to coherent state methods. Among their
numerous applications, wavelets have been used to data compression
in both image and sound processing. They are intimately related to
splines, and wavelet applications in spline theory are significant.
Wavelets have become a tool in analyzing fractals and iterative
schemes associated with dynamical systems. Signal processing
methods such as quadrature mirror filters go hand in hand with
wavelet techniques in studying a host of communcations problems.
The profound issues of classical turbulence are being studied using
wavelet packets. Both wavelet packet software and wavelet transform
microchips are now available. There are also applications of
wavelet theory in theoretical physics, oil exploration, irregular
sampling, and singular integral operators. Many of the world's
experts in the field of wavelets were principal speakers at the
ASI, and their papers appear in this volume. Furthermore, these
renowned scientists addressed their talks to an audience which
consisted of a broad spectrum of pure and applied mathematicians,
as well as a diverse group of engineers and scientists. Thus, the
reader has the opportunity to both learn or reinforce the
fundamental concepts from the individuals who have created and
developed the blossoming field of wavelets, and to see them discuss
in accessible terms their profound contributions and ideas for
future research.
Probability has been an important part of mathematics for more than
three centuries. Moreover, its importance has grown in recent
decades, since the computing power now widely available has allowed
probabilistic and stochastic techniques to attack problems such as
speech and image processing, geophysical exploration, radar, sonar,
etc. -- all of which are covered here. The book contains three
exceptionally clear expositions on wavelets, frames and their
applications. A further extremely active current research area,
well covered here, is the relation between probability and partial
differential equations, including probabilistic representations of
solutions to elliptic and parabolic PDEs. New approaches, such as
the PDE method for large deviation problems, and stochastic optimal
control and filtering theory, are beginning to yield their secrets.
Another topic dealt with is the application of probabilistic
techniques to mathematical analysis. Finally, there are clear
explanations of normal numbers and dynamic systems, and the
influence of probability on our daily lives.
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