Kathryn A. Hargreaves - Böcker

Probability has been an important part of mathematics for more than 300 years. 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, all of which are covered in this volume. The book contains three expositions on wavelets, frames and their applications. The book also covers the relation between probability and partial differential equations, including probablistic 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 also explored. Another topic dealt with is the application of probablistic techniques to mathematical analysis. Finally, there are 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 experts in the field of wavelets were principal speakers at the ASI, and their papers appear in this volume.Furthermore, these 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 field of wavelets, and to see them discuss in accessible terms their contributions and ideas for future research.

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.