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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.

2.10.2 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.

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.

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.

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.

This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the books examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers. < Features updated information on LMI parameterizations of sum-of-squares trigonometric polynomials; Contains applications in optimization of 1-D and 2-D filter design, orthogonal filterbanks; Includes a new chapter dedicated to applications in super-resolution theory that connects to a currently very active research area.