J.S. Byrnes – författare

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.

Almost a century ago, harmonic analysis entered a (still continuing) Golden Age, with the emergence of many great masters throughout Europe. They created a wealth of profound analytic methods, to be successfully exploited and further developed by succeeding generations. This flourishing of harmonic analysis is today as lively as ever, as the papers presented here demonstrate. In addition to its own ongoing internal development and its basic role in other areas of mathematics, physics and chemistry, financial analysis, medicine, and biological signal processing, harmonic analysis has made fundamental contributions to essentially all twentieth century technology-based human endeavours, including telephone, radio, television, radar, sonar, satellite communications, medical imaging, the Internet, and multimedia. This ubiquitous nature of the subject is amply illustrated. The book not only promotes the infusion of new mathematical tools into applied harmonic analysis, but also to fuel the development of applied mathematics by providing opportunities for young engineers, mathematicians and other scientists to learn more about problem areas in today's technology that might benefit from new mathematical insights.

Almost a century ago, harmonic analysis entered a (still continuing) Golden Age, with the emergence of many great masters throughout Europe. They created a wealth of profound analytic methods, to be successfully exploited and further developed by succeeding generations. This flourishing of harmonic analysis is today as lively as ever, as the papers presented here demonstrate. In addition to its own ongoing internal development and its basic role in other areas of mathematics, physics and chemistry, financial analysis, medicine, and biological signal processing, harmonic analysis has made fundamental contributions to essentially all twentieth century technology-based human endeavours, including telephone, radio, television, radar, sonar, satellite communications, medical imaging, the Internet, and multimedia. This ubiquitous nature of the subject is amply illustrated. The book not only promotes the infusion of new mathematical tools into applied harmonic analysis, but also to fuel the development of applied mathematics by providing opportunities for young engineers, mathematicians and other scientists to learn more about problem areas in today's technology that might benefit from new mathematical insights.

This volume contains papers presented at the July, 1989 NATO Advanced Study Institute on Fourier Analysis and its Applications. The conference, held at the beautiful II Ciocco resort near Lucca, in the glorious Tuscany region of northern Italy, created a dynamic in­ teraction between world-renowned scientists working in the usually disparate communities of pure and applied Fourier analysts. The papers to be found herein include important new results in x-ray crystallography by Nobel Laureate Herbert Hauptman, the application of the new concept of bispectrum to system identification by renowned probabilist Athanasios Papoulis, fascinating appli­ cations of number theory in Fourier analysis by eminent electrical engineer Manfred R. Schroeder, and exciting concepts regarding polynomials with restricted coefficients by foremost mathematical problem solver Donald J. Newman. The remaining papers further illustrate the inherent power and beauty of classical Fourier analysis, whether the results presented were sought as an end in themselves, or whether these classical methods were employed as a tool in illustrating and solving a particular applied problem. From antenna design to concert hall acoustics to image and speech processing to unimodular polynomi­ als, each conference participant benefited significantly from his or her exposure, in many cases for the first time, to those scientists on the other end of the spectrum from them­ selves. The purpose of this volume is to pass those benefits on to the reader.

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.

This volume contains papers presented at the July, 1989 NATO Advanced Study Institute on Fourier Analysis and its Applications. The conference, held at the beautiful II Ciocco resort near Lucca, in the glorious Tuscany region of northern Italy, created a dynamic in­ teraction between world-renowned scientists working in the usually disparate communities of pure and applied Fourier analysts. The papers to be found herein include important new results in x-ray crystallography by Nobel Laureate Herbert Hauptman, the application of the new concept of bispectrum to system identification by renowned probabilist Athanasios Papoulis, fascinating appli­ cations of number theory in Fourier analysis by eminent electrical engineer Manfred R. Schroeder, and exciting concepts regarding polynomials with restricted coefficients by foremost mathematical problem solver Donald J. Newman. The remaining papers further illustrate the inherent power and beauty of classical Fourier analysis, whether the results presented were sought as an end in themselves, or whether these classical methods were employed as a tool in illustrating and solving a particular applied problem. From antenna design to concert hall acoustics to image and speech processing to unimodular polynomi­ als, each conference participant benefited significantly from his or her exposure, in many cases for the first time, to those scientists on the other end of the spectrum from them­ selves. The purpose of this volume is to pass those benefits on to the reader.

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.