rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Positive trigonometric polynomials and signal processing applications
  name:
    name part: Dumitrescu, Bogdan
    role:
      role term: creator
    role:
      role term: author.
  type of resource: text
  genre: bibliography
  genre: elektronické knihy
  genre: electronic books
  origin info:
    place:
      place term: sz
    date issued: 2017
    edition: Second edition.
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: remote
    extent: 1 online resource (xvi, 276 pages) : illustrations
  abstract: 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.
  table of contents: 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.
  table of contents: 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.
  table of contents: 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.
  table of contents: 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.
  table of contents: 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.
  note: Bogdan Dumitrescu.
  note: Plný text je dostupný pouze z IP adres počítačů Univerzity Tomáše Bati ve Zlíně nebo vzdáleným přístupem pro zaměstnance a studenty
  note: Includes bibliographical references and index.
  subject:
    topic: Polynomials
  subject:
    topic: Signal processing
    topic: Mathematical models
  subject:
    topic: Toeplitz matrices
  related item:
    title info:
      title: Positive trigonometric polynomials and signal processing applications
    name:
      name part: Dumitrescu, Bogdan.
    origin info:
      publisher: Dordrecht : Springer, 2017
      edition: Second edition.
    identifier: (OCoLC)968763242
  related item:
    title info:
      title: Signals and communication technology
  identifier: 9783319536880
  identifier: 9783319536873
  identifier: https://proxy.k.utb.cz/login
  access condition: Plný text je dostupný pouze z IP adres počítačů Univerzity Tomáše Bati ve Zlíně nebo vzdáleným přístupem pro zaměstnance a studenty
  record info:
    record content source: GW5XE
    record creation date: 170330
    record change date: 20251008112003.0
    record identifier: 99943
    language of cataloging:
      language term: eng