CONTROL AND DYNAMIC SYSTEMS: ADVANCES IN THEORY AND APPLICATION

By F. BEORA

CONTENTS I BASIC OPERATOR THEORY THE WOLD DECOMPOSITION THE UNILATERAL SHIFT THE EIGENVALUES FOR THE BACKWARD SHIFT THE BILATERAL SHIFT ABSTRACT TOEPLITZ OPERATORS ABSTRACT TOEPLITZ OPERATORS VIEWED AS A COMPRESSION NOTES TOEPLITZ AND LAURENT OPERATORS THE FOURIER TRANSFORM +(E) HARDY SPACES MULTIPLICATION OPERATORS LAURENT OPERATORS TOEPLITZ OPERATORS AND MATRICES TOEPLITZ MATRICES AND H?FUNCTIONS INNER AND OUTER FUNCTIONS THE BEURLING-LAX-HALMOS THEOREM THE INVARIANT SUBSPACES FOR THE BACKWARD SHIFT INNER-OUTER FACTORIZATIONS INVERTIBLE OUTER FUNCTIONS THE DETERMINANT OF INNER AND OUTER FUNCTIONS RATIONAL INNER AND OUTER FUNCTIONS BLASCHKE PRODUCTS STATE SPACE REALIZATIONS FOR RATIONAL INNER FUNCTIONS RATIONAL TWO-SIDED INNER FUNCTIONS RATIONAL OUTER FUNCTIONS INNER-OUTER FACTORIZATION AND MCMILLAN DEGREE INNER-OUTER FACTORIZATION AND FINITE SECTIONS THE DOUGLAS-SHAPIRO-SHIELDS FACTORIZATION THE DOUGLAS-SHAPIRO-SHIELDS FACTORIZATION FOR RATIONAL FUNCTIONS A FACTORIZATION FOR RATIONAL L?FUNCTIONS THE NAIMARK REPRESENTATION THE NAIMARK REPRESENTATION THEOREM THE MAXIMAL OUTER SPECTRAL FACTOR THE EIGENSPACE FOR THE UNITARY PART THE INNER-OUTER FACTORIZATION REVISITED POSITIVE REAL FUNCTIONS MINIMAL ISOMETRIC LIFTINGS THE RATIONAL CASE RATIONAL SYMBOLS THE POSITIVE REAL LEMMA FINITE DIMENSIONAL UNITARY PART A CLASSICAL ERGODIC RESULT ANOTHER ERGODIC RESULT II FINITE SECTION TECHNIQUES THE LEVINSON ALGORITHM AND FACTORIZATION THE CASE WHEN TR IS INVERTIBLE THE CLASSICAL SCHUR INVERSION FORMULA THE SCHUR COMPLEMENT AND TOEPLITZ MATRICES SCHUR COMPLEMENT AND MAXIMAL OUTER FACTOR CARATH´EODORY INTERPOLATION A FINITE SECTIONS APPROACH TO FACTORIZATION A ?NITE SECTION APPROACH TO OUTER FACTORIZATION AN INNER-OUTER FACTORIZATION PROCEDURE RATIONAL CONTRACTIVE ANALYTIC FUNCTIONS A CONTRACTIVE REALIZATION PROCEDURE A SPECTRAL FACTORIZATION APPROACH TO FILTERING 7 10 NOTES ISOMETRIC REPRESENTATIONS AND FACTORIZATION ISOMETRIC REPRESENTATIONS AND FINITE SECTIONS A BASIC OPTIMIZATION PROBLEM THE LOWER TRIANGULAR CHOLESKY FACTORIZATION CHOLESKY FACTORIZATION AND MAXIMAL OUTER SPECTRAL FACTORS SOME EXAMPLES OF THE CHOLESKY FACTORIZATION SCALAR-VALUED OUTER FUNCTION MULTI-INPUT MULTI-OUTPUT SQUARE OUTER FUNCTION NON-SQUARE OUTER FUNCTION NON-SQUARE OUTER FUNCTION AND A UNITARY PART AN INNER-OUTER FACTORIZATION ALGORITHM A SCALAR INNER-OUTER FACTORIZATION EXAMPLE A NON-SQUARE INNER-OUTER FACTORIZATION EXAMPLE AN OUTER FUNCTION AND A UNITARY PART SIGNAL PROCESSING A FUNDAMENTAL OPTIMIZATION PROBLEM SINUSOID ESTIMATION SINUSOID ESTIMATION: CAPON-GERONIMUS A NESTED OPTIMIZATION PROBLEM LIMIT THEOREMS THE OUTER FUNCTION KN(Z, ?) III RICCATI METHODS 10 RICCATI EQUATIONS AND FACTORIZATION 10 1 ALGEBRAIC RICCATI EQUATIONS 10 2 THE CASE WHEN TR IS STRICTLY POSITIVE 10 3 THE RICCATI DI?ERENCE EQUATION 10 4 A RICCATI APPROACH TO INNER-OUTER FACTORIZATIONS 10 5 THE OUTER FACTOR FOR ?2I ?G?G 10 6 DARLINGTON SYNTHESIS 10 7 RICCATI EQUATIONS 10 7 1 THE MINIMUM PRINCIPLE 10 7 2 PROOF OF THEOREM 10 7 1 10 8 NOTES 11 KALMAN AND WIENER FILTERING RANDOM VECTORS LEAST SQUARES ESTIMATION OF A RANDOM VECTOR TIME VARYING STATE SPACE SYSTEMS THE KALMAN FILTER THE STEADY STATE KALMAN FILTER WIDE SENSE STATIONARY PROCESSES 11 6 1 A SINUSOID PROCESS THE SPECTRAL DENSITY JOINTLY WIDE SENSE STATIONARY PROCESSES WIENER FILTERING 11 10 STEADY STATE KALMAN AND WIENER FILTERING 11 10 1 STATE SPACE SYSTEMS WITH X(??) = 0 11 10 2 THE WIENER ?LTER FOR {A, B, C, D} 11 11 NOTES IV INTERPOLATION THEORY 12 TANGENTIAL NEVANLINNA-PICK INTERPOLATION 12 1 AN INTRODUCTION TO NEVANLINNA-PICK INTERPOLATION 12 2 THE FIRST MAIN RESULT ON INTERPOLATION 12 3 PROOF OF THEOREM 12 1 1 12 4 PROOF OF THEOREM 12 1 3 12 5 THE CASE WHEN ? IS STRICTLY POSITIVE 12 6 THE CARATH´EODORY INTERPOLATION PROBLEM 12 7 NOTES 13 CONTRACTIVE NEVANLINNA-PICK INTERPOLATION 13 1 ISOMETRIC REALIZATIONS 13 1 1 RATIONAL CONTRACTIVE REALIZATIONS 13 2 TANGENTIAL NEVANLINNA-PICK INTERPOLATION 13 3 ISOMETRIC REALIZATIONS REVISITED 13 4 THE MAXIMUM PRINCIPLE 13 5 NOTES