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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 mean value properties involving a logarithm-type weight

Nikolai G. Kuznecov (2024)

Mathematica Bohemica

Two new assertions characterizing analytically disks in the Euclidean plane ℝ 2 are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

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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