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Toeplitz Sequences

In: Generalized Locally Toeplitz Sequences: Theory and Applications

Author

Listed:
  • Carlo Garoni

    (University of Insubria, Department of Science and High Technology)

  • Stefano Serra-Capizzano

    (University of Insubria, Department of Science and High Technology)

Abstract

This chapter is devoted to Toeplitz matrices. Of course, we do not pretend to cover here all the details of this extensive topic, which has been the subject of several books [23, 24, 28, 29, 70]. On the contrary, we focus on specific properties of the sequences of Toeplitz matrices generated by an $$L^1$$ function. These sequences, together with the sequences of diagonal sampling matrices (to be introduced afterwards) and the zero-distributed sequences (already studied in Sect. 3.4 ), should be regarded as theBuilding blocks of the theoryof GLT sequences ‘building blocks’ of the theory of GLT sequences. Despite its conciseness, this chapter contains all the results we shall need about Toeplitz matrices to fully develop the theory of GLT sequences. In particular, we will give an a.c.s.-based proof of the $$L^1$$ versions of Szegő’s first limit theorem and the Avram–Parter theorem about the singular value and spectral distribution of Toeplitz sequences.

Suggested Citation

  • Carlo Garoni & Stefano Serra-Capizzano, 2017. "Toeplitz Sequences," Springer Books, in: Generalized Locally Toeplitz Sequences: Theory and Applications, chapter 0, pages 95-114, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-53679-8_6
    DOI: 10.1007/978-3-319-53679-8_6
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