Some problems concerning the test functions in the Szegö and Avram-Parter theorems

Albrecht Böttcher; Sergei Grudsky; Moshe Schwartz

doi:10.1007/978-3-7643-8893-5_3

Some problems concerning the test functions in the Szegö and Avram-Parter theorems

Albrecht Böttcher
, Sergei Grudsky
, Moshe Schwartz

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

The Szeg¨o and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity. This paper is concerned with some questions that arise when the test functions do not satisfy the known growth restrictions at infinity or when the test function has a logarithmic singularity within the range of the symbol. Several open problems are listed and accompanied by a few new results that illustrate the delicacy of the matter.

Original language	English
Title of host publication	Recent Advances in Operator Theory and Applications
Editors	Tsuyoshi Ando, Raúl E. Curto, Il Bong Jung, Woo Young Lee
Publisher	Birkhäuser Basel
Pages	81-93
Number of pages	13
ISBN (Electronic)	978-3-7643-8893-5
ISBN (Print)	978-3-7643-8892-8
DOIs	https://doi.org/10.1007/978-3-7643-8893-5_3
State	Published - 2008
Externally published	Yes

Keywords

Toeplitz matrix
Toeplitz determinant
test function
singular value
Eigenvalue

Access to Document

10.1007/978-3-7643-8893-5_3

Cite this

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abstract = "The Szeg¨o and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity. This paper is concerned with some questions that arise when the test functions do not satisfy the known growth restrictions at infinity or when the test function has a logarithmic singularity within the range of the symbol. Several open problems are listed and accompanied by a few new results that illustrate the delicacy of the matter.",

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publisher = "Birkh{\"a}user Basel",

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Some problems concerning the test functions in the Szegö and Avram-Parter theorems. / Böttcher, Albrecht; Grudsky, Sergei; Schwartz, Moshe.
Recent Advances in Operator Theory and Applications. ed. / Tsuyoshi Ando; Raúl E. Curto; Il Bong Jung; Woo Young Lee. Birkhäuser Basel, 2008. p. 81-93.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

TY - CHAP

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AU - Böttcher, Albrecht

AU - Grudsky, Sergei

AU - Schwartz, Moshe

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N2 - The Szeg¨o and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity. This paper is concerned with some questions that arise when the test functions do not satisfy the known growth restrictions at infinity or when the test function has a logarithmic singularity within the range of the symbol. Several open problems are listed and accompanied by a few new results that illustrate the delicacy of the matter.

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KW - Toeplitz matrix

KW - Toeplitz determinant

KW - test function

KW - singular value

KW - Eigenvalue

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DO - 10.1007/978-3-7643-8893-5_3

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BT - Recent Advances in Operator Theory and Applications

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PB - Birkhäuser Basel

ER -

Some problems concerning the test functions in the Szegö and Avram-Parter theorems

Abstract

Keywords

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