Preface; Contents; 1 Positive Polynomials; 1.1 Types of Polynomials; 1.2 Positive Polynomials; 1.3 Toeplitz Positivity Conditions; 1.4 Positivity on an Interval; 1.5 Details and Other Facts; 1.5.1 Chebyshev Polynomials; 1.5.2 Positive Polynomials in mathbbR[t] as Sum-of-Squares; 1.5.3 Proof of Theorem 1.11; 1.5.4 Proof of Theorem 1.13; 1.5.5 Proof of Theorem 1.15; 1.5.6 Proof of Theorem 1.17; 1.5.7 Proof of Theorem 1.18; 1.6 Bibliographical and Historical Notes; References; 2 Gram Matrix Representation; 2.1 Parameterization of Trigonometric Polynomials

2.2 Optimization Using the Trace Parameterization2.3 Toeplitz Quadratic Optimization; 2.4 Duality; 2.5 Kalman
Yakubovich
Popov Lemma; 2.6 Spectral Factorization from a Gram Matrix; 2.6.1 SDP Computation of a Rank-1 Gram Matrix; 2.6.2 Spectral Factorization Using a Riccati Equation; 2.7 Parameterization of Real Polynomials; 2.8 Choosing the Right Basis; 2.8.1 Basis of Trigonometric Polynomials; 2.8.2 Transformation to Real Polynomials; 2.8.3 Gram-Pair Matrix Parameterization; 2.9 Interpolation Representations; 2.10 Mixed Representations; 2.10.1 Complex Polynomials and the DFT

2.10.2 Cosine Polynomials and the DCT2.11 Fast Algorithms; 2.12 Details and Other Facts; 2.12.1 Writing Programs with Positive Trigonometric Polynomials; 2.12.2 Proof of Theorem 2.16; 2.12.3 Proof of Theorem 2.19; 2.12.4 Proof of Theorem 2.21; 2.13 Bibliographical and Historical Notes; References; 3 Multivariate Polynomials; 3.1 Multivariate Polynomials; 3.2 Sum-of-Squares Multivariate Polynomials; 3.3 Sum-of-Squares of Real Polynomials; 3.4 Gram Matrix Parameterization of Multivariate Trigonometric Polynomials; 3.5 Sum-of-Squares Relaxations; 3.5.1 Relaxation Principle; 3.5.2 A Case Study

3.5.3 Optimality Certificate3.6 Gram Matrices from Partial Bases; 3.6.1 Sparse Polynomials and Gram Representation; 3.6.2 Relaxations; 3.7 Gram Matrices of Real Multivariate Polynomials; 3.7.1 Gram Parameterization; 3.7.2 Sum-of-Squares Relaxations; 3.7.3 Sparseness Treatment; 3.8 Pairs of Relaxations; 3.9 The Gram-Pair Parameterization; 3.9.1 Basic Gram-Pair Parameterization; 3.9.2 Parity Discussion; 3.9.3 LMI Form; 3.10 Polynomials with Matrix Coefficients; 3.11 Details and Other Facts; 3.11.1 Transformation Between Trigonometric and Real Nonnegative Polynomials

3.11.2 Pos3Poly Program with Multivariate Polynomials3.11.3 A CVX Program Using the Gram-Pair Parameterization; 3.12 Bibliographical and Historical Notes; References; 4 Polynomials Positive on Domains; 4.1 Real Polynomials Positive on Compact Domains; 4.2 Trigonometric Polynomials Positive on Frequency Domains; 4.2.1 Gram Set Parameterization; 4.2.2 Gram-Pair Set Parameterization; 4.3 Bounded Real Lemma; 4.3.1 Gram Set BRL; 4.3.2 BRL for Polynomials with Matrix Coefficients; 4.3.3 Gram-Pair Set BRL; 4.4 Positivstellensatz for Trigonometric Polynomials; 4.5 Proof of Theorem 4.11