NORBERT WIENER AND THE DEVELOPMENT OF MATHEMATICAL ENGINEERING

So wrote Norbert Wiener in 1949, in an obituary of G. H. Hardy. Here we shall describe how one particular concrete problem in Wiener’s own work solving the Wiener-Hopf equations encountered in astrophysics - led him, and then a vast host of followers, to chart out several new areas of investigation, and to develop a very significant body of knowledge, which can well go by the name Mathematical Engineering. In the era of the PC, the Internet and the World Wide Web, few of us can be unaware that mathematical engineering has come to play a major role in the world around us. And with this has come an increasing recognition of the seminal role of Norbert Wiener’s ideas and influence in these developments.

[1]  L. Ljung,et al.  Scattering theory and linear least squares estimation , 1976 .

[2]  F. Noether,et al.  Über eine Klasse singulärer Integralgleichungen , 1920 .

[3]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[4]  S. Mitter,et al.  New results on the innovations problem for non-linear filtering , 1981 .

[5]  R. Redheffer On the Relation of Transmission-Line Theory to Scattering and Transfer† , 1962 .

[6]  J. L. Jackson,et al.  LINEAR FILTER OPTIMIZATION WITH GAME THEORY CONSIDERATIONS , 1955 .

[7]  N. Wiener,et al.  Nonlinear Problems in Random Theory , 1964 .

[8]  M. Morf,et al.  Inverses of Toeplitz operators, innovations, and orthogonal polynomials , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[9]  T. Kailath,et al.  Scattering theory and linear least-squares estimation, part III: The estimates , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[10]  Thomas Kailath,et al.  An RKHS approach to detection and estimation problems- III: Generalized innovations representations and a likelihood-ratio formula , 1972, IEEE Trans. Inf. Theory.

[11]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[12]  Thomas Kailath,et al.  From Kalman Filtering to Innovations, Martingales, Scattering and Other Nice Things , 1991 .

[13]  J. L. Hock,et al.  An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space , 1984 .

[14]  Paul Van Dooren,et al.  Numerical Linear Algebra Techniques for Systems and Control , 1994 .

[15]  P. Lancaster,et al.  Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review , 1991 .

[16]  L. Ljung,et al.  Generalized Krein-Levinson Equations for Efficient Calculation of Fredholm Resolvents of Non-Displacement Kernels , 1978 .

[17]  Thomas Kailath,et al.  Likelihood ratios for Gaussian processes , 1970, IEEE Trans. Inf. Theory.

[18]  Stephen P. Boyd,et al.  Linear controller design: limits of performance , 1991 .

[19]  H. W. Bode,et al.  A Simplified Derivation of Linear Least Square Smoothing and Prediction Theory , 1950, Proceedings of the IRE.

[20]  T. Kailath The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures , 1971 .

[21]  J. Doob Time Series and Harmonic Analysis , 1949 .

[22]  T. Kailath Some extensions of the innovations theorem , 1971 .

[23]  J. Doob Stochastic processes , 1953 .

[24]  J. Willems,et al.  The Dissipation Inequality and the Algebraic Riccati Equation , 1991 .

[25]  ON A CLASS OF VARIATIONAL PROBLEMS , 1957 .

[26]  L. Zadeh,et al.  An Extension of Wiener's Theory of Prediction , 1950 .

[27]  Ali H. Sayed,et al.  Displacement Structure: Theory and Applications , 1995, SIAM Rev..

[28]  P. Meyer Sur un probleme de filtration , 1973 .

[29]  T. Kailath The innovations approach to detection and estimation theory , 1970 .

[30]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[31]  R. E. Kalman,et al.  Contributions to the Theory of Optimal Control , 1960 .

[32]  V. E. Bene On kailath's innovations conjecture hold , 1976, The Bell System Technical Journal.

[33]  D.A. Mindell Automation's Finest Hour: Bell Labs and Automatic Control in World War II , 1995, IEEE Control Systems.

[34]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[35]  X and Y Functions for Planetary Atmospheres with Lambert Law Reflecting Surfaces , 1975 .

[36]  T. Kailath,et al.  Linear estimation in Krein spaces. I. Theory , 1996, IEEE Trans. Autom. Control..

[37]  T. Kailath A Note on Least Squares Estimation by the Innovations Method , 1972 .

[38]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications , 1949 .

[39]  L. A. Zadeh,et al.  From Circuit Theory to System Theory , 1962, Proceedings of the IRE.

[40]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[41]  T. Kailath Remarks on the origin of the displacement-rank concept , 1991 .

[42]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[43]  Malcolm H. Davis,et al.  Exact and approximate filtering in signal detection - An example , 1977 .

[44]  H. Kunita,et al.  Stochastic differential equations for the non linear filtering problem , 1972 .

[45]  Albert Wilansky,et al.  Topics in Functional Analysis , 1967 .

[46]  Mark H. A. Davis Linear estimation and stochastic control , 1977 .

[47]  T. Kailath,et al.  Inertia conditions for the minimization of quadratic forms in indefinite metric spaces , 1996 .

[48]  I. V. Girsanov On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures , 1960 .

[49]  N. Levinson The Wiener (Root Mean Square) Error Criterion in Filter Design and Prediction , 1946 .

[50]  L. Ljung,et al.  Scattering theory and linear least squares estimation—Part I: Continuous-time problems , 1976, Proceedings of the IEEE.

[51]  J. Doob The Elementary Gaussian Processes , 1944 .