TY - JOUR
T1 - The Szego{combining double acute accent} and Avram-Parter theorems for general test functions
AU - Böttcher, Albrecht
AU - Grudsky, Sergei M.
AU - Maksimenko, Egor A.
N1 - Funding Information:
The second author acknowledges financial support by CONACYT grant 046936-F.
PY - 2008/7
Y1 - 2008/7
N2 - The Szego{combining double acute accent} and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. We show that, surprisingly, these theorems are not true for every continuous, nonnegative, and monotonously increasing test function and thus do not hold whenever they make sense. On the other hand, we prove the two theorems in a general form which includes all versions known so far. To cite this article: A. Böttcher et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).
AB - The Szego{combining double acute accent} and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. We show that, surprisingly, these theorems are not true for every continuous, nonnegative, and monotonously increasing test function and thus do not hold whenever they make sense. On the other hand, we prove the two theorems in a general form which includes all versions known so far. To cite this article: A. Böttcher et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).
UR - http://www.scopus.com/inward/record.url?scp=46449119064&partnerID=8YFLogxK
U2 - 10.1016/j.crma.2008.06.002
DO - 10.1016/j.crma.2008.06.002
M3 - Artículo
SN - 1631-073X
VL - 346
SP - 749
EP - 752
JO - Comptes Rendus Mathematique
JF - Comptes Rendus Mathematique
IS - 13-14
ER -