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On discrepancy theorems with applications to approximation theory

Hans-Peter Blatt (1995)

Banach Center Publications

We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turán on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.

On log-subharmonicity of singular values of matrices

Bernard Aupetit (1997)

Studia Mathematica

Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by s 1 , . . . , s n the decreasing sequence of singular values of a matrix, we prove that the functions l o g s 1 ( F ( λ ) ) + . . . + l o g s k ( F ( λ ) ) and l o g + s 1 ( F ( λ ) ) + . . . + l o g + s k ( F ( λ ) ) are subharmonic on Ω for 1 ≤ k ≤ n.

On strong tracts of subharmonic functions of infinite lower order

I. I. Marchenko, A. Szkibiel (2007)

Annales Polonici Mathematici

The notion of a strong asymptotic tract for subharmonic functions is defined. Eremenko's value b(∞,u) for subharmonic functions is introduced and it is used to provide an exact upper estimate of the number of strong tracts of subharmonic functions of infinite lower order. It is also shown that b(∞,u) ≤ π for subharmonic functions of infinite lower order.

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