Multidimensional Systems Theory and Applications

List of Acronyms. List of Notations. Preface. Acknowledgements. Introduction. 1: Trends in Multidimensional Systems Theory. 1. Introduction. 2. Multidimensional Systems Stability. 3. Multivariate Realization Theory. 4. n-D Problem of Moments and its Applications in Multidimensional Systems Theory. 5. Role of Irreducible Polynomials in Multidimensional Systems Theory. 6. Hilbert Transform and Spectral Factorization. 7. Conclusions. 8. Updates. 2: Causal and Weakly Causal 2-D Filters with Applications in Stabilization. 1. Scalar 2-D Input / output Systems. 2. Stability. 3. Structural Stability. 4. Multi-Input Multi-Output Systems. 5. Stabilization of Scalar Systems. 6. Characterization of Stabilizers for Scalar Systems. 7. Stabilization of Strictly Causal Transfer Matrices. 8. Characterization of Stabilizers for MIMO Systems. 9. Stabilization of Weakly Causal Systems. 10. Stabilization of MIMO Weakly Causal Systems. 11. Conclusions. 12. Updates. 3: The Equation Ax = b over the Ring C[z, w]. 1. Introduction. 2. Sufficient Condition for Solution. 3. Appendix A. Zero-Dimensional Polynomial Ideals. 4: Grobner Bases: An Algorithmic Method in Polynomial Ideal Theory. 1. Introduction. 2. Grobner Bases. 3. Algorithmic Construction of Grobner Bases. 4. An Improved Version of the Algorithm. 5. Application: Canonical Simplification, Decision of Ideal Congruence and Membership, Computation in Residue Class Rings. 6. Application: Solvability and Exact Solution of Systems of Algebraic Equations. 7. Application: Solution of Linear Homogeneous Equations with Polynomial Coefficients. 8. Grobner Bases for Polynomial Ideals over the Integers. 9. Other Applications. 10. Specializations, Generalizations, Implementations, Complexity. 11. Updates. 5: Multivariate Polynomials, Matrices, and Matrix-Fraction Descriptions. 1. Introduction. 2. Relative Primeness and GCD Extraction from Multivariate Polynomials. 3. Polynomial Matrix Primitive Factorization in the Bivariate Case. 4. Multivariate Polynomial Matrix Factorization. 5. Computations for Coprimeness Using Grobner Bases. 6. Generalization of the Serre Conjecture and its Consequences. 7. Factorization as a Product of Elementary Matrix Factors. 8. Applications in Multidimensional Systems Stabilization. 9. Behavioral Approach. 10. Conclusions. 6: Recent Impacts of Multidimensional Systems Research. 1. Introduction. 2. Inference of Stability of Sets of Multidimensional Systems from Subsets of Low Cardinality. 3. Multiple Deconvolution Operators for Robust Superresolution. 4. Multisensor Array-Based Superresolution. 5. Wavelets for Superresolution. 6. Other Recent Applications. 7. Conclusions. 7: Multivariate Rational Approximants of the Pade Type. 1. Introduction and Motivation. 2. Multivariate Pade-Type Approximants (Scalar Case). 3. Pade Type Matrix Approximants. 4. Conclusions. 5. Updates. 8: Old Problem Solutions and Some New Problems. 1. Progress towards Solution of Open Problems since 1985. 2.New Open Problems. References. Index.