Toeplitz Sequences

Carlo Garoni () and Stefano Serra-Capizzano ()
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Carlo Garoni: University of Insubria, Department of Science and High Technology
Stefano Serra-Capizzano: University of Insubria, Department of Science and High Technology

Chapter Chapter 6 in Generalized Locally Toeplitz Sequences: Theory and Applications, 2017, pp 95-114 from Springer

Abstract: 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.

Keywords: Fourier Coefficient; Spectral Distribution; Toeplitz Matrix; Toeplitz Matrice; Circulant Matrix (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-53679-8_6

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